Answers
The correct option is C. Point B as only the line BC touching the circle at point B is tangent to the circle.
Tangent to a circle theoremThe tangent to a circle theorem states that a line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency
We can observe that the line BC touches the circle at point B and the radius of the circle will be perpendicular to the line BC
Therefore, since the radius of the circle is perpendicular to the line BC at point B, then it is tangent to the circle
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Related Questions
Using Formulas
Ava's minimum payment is 2.5% of her new balance. What is her minimum
payment if her new balance is $760.00?
S
Answers
Answer =19
Find the equation of the graphed line.
Answers
Answer:
A
Step-by-step explanation:
I will mark you brainiest!
Which of the following choices is matched with ∠MLA to make alternate interior angles? A) ∠GAL
B) ∠FAH
C) ∠LMB
D) ∠CLH
Answers
Answer:BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB TRUST ME
Step-by-step explanation:
given a population with a normal distribution, a mean of 40, and a standard deviation of 15, find the probability of a value between 50 and 70
Answers
Answer:
To find the probability of a value between 50 and 70 in a normal distribution with mean 40 and standard deviation 15, we need to first standardize the values using the z-score formula:
z = (x - μ) / σ
where x is the value we are interested in, μ is the mean, and σ is the standard deviation.
For the lower bound of 50:
z = (50 - 40) / 15 = 0.67
For the upper bound of 70:
z = (70 - 40) / 15 = 2
Using a standard normal distribution table or a calculator with a built-in normal distribution function, we can find the probabilities corresponding to these z-scores:
P(0 < z < 0.67) = 0.2514
P(0 < z < 2) = 0.4772
To find the probability of a value between 50 and 70, we can subtract the probability of the lower bound from the probability of the upper bound:
P(50 < x < 70) = P(0 < z < 2) - P(0 < z < 0.67)
P(50 < x < 70) = 0.4772 - 0.2514
P(50 < x < 70) = 0.2258
Therefore, the probability of a value between 50 and 70 in this normal distribution is 0.2258 or about 22.58%.
Differentiate in respect to yS(siny + y cosy)dy
Answers
The derivative of yS(siny + ycosy)dy with respect to y is yS(cosy - ysiny) + S(siny + ycosy).
Explanation:
To differentiate the expression yS(siny + y cosy)dy, we need to apply the product rule of differentiation. The product rule states that for two functions u(x) and v(x), the derivative of their product is given by:
(d/dx)(u(x) v(x)) = u'(x) v(x) + u(x) v'(x)
Using this rule, we can differentiate the expression as follows:
Let u(y) = yS and v(y) = siny + ycosy. Then,
u'(y) = S (the derivative of yS with respect to y is S)
v'(y) = cosy + y(-siny) = cosy - ysiny (using the product rule again)
Using the product rule, we have:
(d/dy)(u(y) v(y)) = u'(y) v(y) + u(y) v'(y)
= yS (cosy - ysiny) + S(siny + ycosy)
Therefore, the derivative of yS(siny + ycosy)dy with respect to y is yS(cosy - ysiny) + S(siny + ycosy).
Find the average rate of change of the function f(x)
f(x) = 10 from [2, 3].
x-4
Answers
Answer:
Step-by-step explanation:
The formula for average rate of change of a function f(x) over an interval [a, b] is given by:
Average rate of change = [f(b) - f(a)] / (b - a)
In this case, we are given the function f(x) = 10 and the interval [2, 3]. So, a = 2 and b = 3. Substituting these values, we get:
Average rate of change = [f(3) - f(2)] / (3 - 2)
= [(10 x 3 - 4) - (10 x 2 - 4)] / (3 - 2)
= (30 - 20) / 1
= 10
Therefore, the average rate of change of the function f(x) = 10 from [2, 3] is 10.
What is the total payment
required to pay off a promissory
note issued for $600.00 at 10%
ordinary interest and a 180-day
term?
Answers
Answer:
$630.00
Step-by-step explanation:
IF THIS IS WRONG I AM SORRY BUT I KNOW THIS IS RIGHT AND IF YOU NEED MORE ANSWERS I AM ALWAYS OPEN!
find the volume of each sphere in terms of π with a radius that equals 3ft
Answers
Answer:
36π
Step-by-step explanation:
To find the volume of the sphere we have to use the equation: [tex]\frac{4}{3}[/tex]πr³
To work our answer out we have to distribute the values we are given into the question...
[tex]\frac{4}{3}[/tex] × 3³We can ignore π for now as we will add it at the end
Now we have to solve what we are given...
[tex]\frac{4}{3}[/tex] × 3³3³ = 27[tex]\frac{4}{3}[/tex] × 27 = 36Now we can put π into our answer...
36πHope this helps, have a lovely day! :)
Mrs. Juarez graded ten English papers and recorded the scores. 92, 95, 100, 62, 88, 90, 100, 96, 89, 98 Which statements are true? Check all that apply. The range of scores is 38. Without the outlier, the range of scores would be 12. The outlier impacts the range more than it impacts the interquartile range. The interquartile range is 9. The interquartile range is 4. Without the outlier, the interquartile range would be 9.5. Mark this and return Save and Exit Next 4655
Answers
The interquartile range, according to the provided assertion, is 9.5.
The interquartile range is what?
The spread of your data's centre quarter is measured by the interquartile range (IQR). It is the limit for your sample's centre 50%. Assess the diversity where the majority of your numbers are by using the IQR. Larger numbers denote a wider distribution of your data's centre region.
The following statements are true:
The range of scores is 38.
The spread of results without the outlier would be 12.
The interquartile range is less affected by the anomaly than the range.
9 is the interquartile number.
The interquartile range would indeed be 9.5 if there were no anomaly.
To calculate the range, we subtract the smallest score from the largest score
Range = 100 - 62 = 38
If we remove the outlier, 62, then the range becomes:
without an anomaly= 100 - 88 = 12
The outlier has a greater impact on the range than on the interquartile range because the interquartile range only considers the middle 50% of the data, whereas the range considers all of the data.
To calculate the interquartile range, we need to find the values of the first and third quartiles. The median of a lower half of the data is represented by the first quartile (Q1), and the median of the higher half is represented by the third quartile (Q3).
Arranging the scores in ascending order:
62, 88, 89, 90, 92, 95, 96, 98, 100, 100
The median is the middle value, which is 93.
Q1 is the median of the lower half of the data, which is 62, 88, 89, 90, 92. The median of this set is (88 + 89) / 2 = 88.5.
Q3 is the median of the upper half of the data, which is 95, 96, 98, 100, 100. The median of this set is (98 + 100) / 2 = 99.
The disparity between Q3 and Q1 is the interquartile range (IQR):
IQR = Q3 - Q1 = 99 - 88.5 = 10.5 ≈ 9
If we remove the outlier, the scores become:
88, 89, 90, 92, 95, 96, 98, 100, 100
Q1 is the median of the lower half of the data, which is 88, 89, 90, 92. The median of this set is (89 + 90) / 2 = 89.5.
Q3 is the median of the upper half of the data, which is 95, 96, 98, 100, 100. The median of this set is (96 + 98) / 2 = 97.
The IQR is:
IQR = Q3 - Q1 = 97 - 89.5 = 7.5 ≈ 9.5
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Ben went to buy some brushes and oaints to finish a project.he bought 25 brushes and paints in total.if each brush cost IDR 1.500 , each paint IDR 800 and the total pruchase cost was IDR 27.000. how many brushes and paints did he buy
Answers
The number of brushes and number of paint bought by Ben are 7 and 18 respectively.
What is an equation system, and how can one be utilised to address issues?A group of equations that are concurrently solved to determine the values of the variables involved is referred to as a system of equations. A system of equations can have any number of variables and equations and can be either linear or nonlinear.
They could depict connections between various values, restrictions on resources or attributes, or optimization goals.
Let us suppose the number of brushes = b.
Let us suppose the number of paints = p.
Thus,
b + p = 25
1500b + 800p = 27000
Multiplying the first equation by 1500 and subtracting it from the second equation, we get:
800p - 1500b = -7500
p = (1500b + 7500) / 800
Substituting the value of p in first equation:
b + (1500b + 7500) / 800 = 25
Multiplying both sides by 800 and simplifying, we get:
2000b + 7500 = 20000
b = 6.25 = 7
Substitute the value of b in first equation:
7 + p = 25
p = 18
Hence, the number of brushes and number of paint bought by Ben are 7 and 18 respectively.
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349 ×10 to the 5th power =
Answers
Answer:
34,900,000
Step-by-step explanation:
349 × [tex]10^{5}[/tex]
[tex]10^{5}[/tex] = 100,000
349 x 100000 = 34,900,000
So, the answer is 34,900,000
Suppose that the local government of Tulsa decides to institute a tax on soda producers. Before the tax, 35,000 liters of soda were sold every week at a price of $11 per liter. After the tax, 30,000 liters of soda are sold every week; consumers pay $14 per liter, and producers receive $6 per liter (after paying the tax).
The amount of the tax on a liter of soda is
$
per liter. Of this amount, the burden that falls on consumers is
$
per liter, and the burden that falls on producers is
$
per liter.
Answers
- Before the tax, 35,000 liters of soda were sold every week at a price of $11 per liter.
- After the tax, 30,000 liters of soda are sold every week; consumers pay $14 per liter, and producers receive $6 per liter (after paying the tax).
The total revenue from soda sales before the tax was:
35,000 liters × $11/liter = $385,000 per week
The total revenue from soda sales after the tax is:
30,000 liters × $14/liter = $420,000 per week
The difference in revenue is due to the tax. Let's call the amount of the tax per liter of soda "t". Then we have:
(30,000 liters × $6/liter) + (30,000 liters × t/liter) = $420,000 per week
Simplifying and solving for t, we get:
t = ($420,000 per week - $180,000 per week) / 30,000 liters
t = $8 per liter
Therefore, the amount of the tax per liter of soda is $8.
To find the burden that falls on consumers and producers, we can use the price of soda before and after the tax. Before the tax, consumers paid $11 per liter and producers received $11 per liter. After the tax, consumers paid $14 per liter and producers received $6 per liter. The difference in price is due to the tax. Let's call the burden that falls on consumers "bc" and the burden that falls on producers "bp". Then we have:
bc + bp = $14 - $11
bc + bp = $3 per liter
And we know that:
bp = $11 - $6 = $5 per liter
Substituting this value into the previous equation, we get:
bc + $5 per liter = $3 per liter
bc = $3 per liter - $5 per liter
bc = -$2 per liter
Since the burden that falls on consumers is negative, we can say that the burden falls entirely on producers, who bear the full $3 per liter increase in price due to the tax.
Therefore, the burden that falls on consumers is $0 per liter, and the burden that falls on producers is $3 per liter.
Find the present value of $80,000 due in 4 years at the given rate of interest. (Use a 365-day year. Round your answer to the nearest cent.)
4%/year compounded quarterly
Answers
$73,394.49 is the present value of $80,000 that is due in 4 years at a 4% interest rate.
What is compound interest?Compound interest is interest that is accrued on both the principal and the prior interest. It is frequently referred to as "interest over the interest" as a result. Here, the interest that has already accrued is added to the principal, and the resulting amount acts as the new principal for the following period.
Hence, compound interest is the sum of interest on the principal and interest on earlier interest.
Using the formula below, it is possible to determine the present value of $80,000 payable in 4 years at a 4% rate of interest compounded quarterly:
PV = FV / (1 + r/n) ^(nt)
such that:
Present Value, or PV, is the term.
FV stands for future value.
The annual number of compounding periods is n, and the interest rate is r.
The number of years is t.
To solve this problem, we have:
FV = $80,000
r = 4% = 0.04
n = 12 (as compounding is done monthly)
t = 4
Adding these values to the formula provides the following results:
PV = 80,000 / (1 + 0.04/12) ^ (12 × 4)
PV = 80,000 / (1.003333) ^48
PV = 80,000 / 1.090777
PV = 73,394.49
As a result, $73,394.49 is the present value of $80,000 that is due in 4 years at a 4% interest rate.
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what us the order of 5 x 10^4, 7 x 10^-5, 3 x 10^-9, 8 x 10^4 from the least to greatest
Answers
Answer: 5x10^4, 8x10^4, 10^-5, 10^4
Step-by-step explanation: calculator
12. The large triangular figure below is composed of triangles that each have a base and height of 4 cm.
What is the area of the large figure?
A. 64 cm2
B. 72 cm 2
C. 128 cm 2
D. 144 cm2
Answers
By answering the above question, we may state that The solution is triangle closest to option d: 58% x, 42% y.
What precisely is a triangle?A triangle is a polygon because it contains four or more parts. It features a simple rectangular shape. A triangle ABC is a rectangle with the edges A, B, and C. When the sides are not collinear, Euclidean geometry produces a single plane and cube. If a triangle contains three components and three angles, it is a polygon. The corners are the points where the three edges of a triangle meet. The sides of a triangle sum up to 180 degrees.
The total sales of product x may be computed as follows: 5,000 units * $110/unit selling price = $550,000
The total sales of product y may be computed as follows: 35,000 units * $70/unit selling price = $2,450,000
The total sales of both goods are as follows: $550,000 + $2,450,000 = $3,000,000
Hence the proportion of revenue provided by product x is: $550,000 / $3,000,000 = 0.1833 or 18.33%
And the proportion of sales produced by product y is: $2,450,000 / $3,000,000 = 0.8167 or 81.67%
As a result, Rusty Co.'s sales mix was 18.33% x and 81.67% y last year.
The solution is closest to option d: 58% x, 42% y.
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I want to know what the answer to my equation is
Answers
At 4.0 megabits per second, Able Cable provides the fastest average downloading speed.
To find out which company offers the fastest mean downloading speed, we need to calculate the mean download speed for each provider and then compare the results.
The mean download speed for CityNet is:
(3.6 + 3.7 + 3.7 + 3.6 + 3.9) / 5 = 3.7 megabits per second
The mean download speed for Able Cable is:
(3.9 + 3.9 + 4.1 + 4.0 + 4.1) / 5 = 4.0 megabits per second
The mean download speed for Tel-N-Net is:
(3.9 + 3.7 + 4.0 + 3.6 + 3.8) / 5 = 3.8 megabits per second
Therefore, Able Cable offers the fastest mean downloading speed at 4.0 megabits per second.
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The temperature of a person has a normal distribution. What is the probability that the temperature of a randomly selected person will be within 2.42 standard deviations of its mean? Provide answer with 4 or more decimal places.
Answers
Answer:
Step-by-step explanation:
If the temperature of a person follows a normal distribution, we know that approximately 95% of the observations fall within 2 standard deviations of the mean. Since we are given that we want to find the probability of the temperature being within 2.42 standard deviations of its mean, we can use the standard normal distribution and the z-score formula.
The z-score formula is given by:
z = (x - μ) / σ
where x is the observed value, μ is the mean, and σ is the standard deviation. In this case, we want to find the probability that the temperature is within 2.42 standard deviations of the mean, so we can set:
z = 2.42
Since the normal distribution is symmetric, we can find the area to the right of the mean (z = 0) and double it to get the total probability. Using a standard normal distribution table or calculator, we find that the area to the right of z = 2.42 is approximately 0.0074. So the area to the left of z = 2.42 is approximately 0.9926.
Doubling this area gives us the total probability:
P(z < 2.42 or z > -2.42) = 2 * P(z < 2.42) = 2 * 0.9926 = 0.9852
Therefore, the probability that the temperature of a randomly selected person will be within 2.42 standard deviations of its mean is 0.9852, or approximately 0.9852 with four decimal places.
The equation, A=P(1+0.045/12)^12t
represents the amount of money earned on a compound interest savings account with an annual interest rate of 4.5% compounded monthly. if after 20
years the amount in the account is $18,539.38, what is the value of the principal investment? Round the answer to the nearest hundredths place.
$6,872.98
$7,550.25
$10,989.13
$17,202.73
Answers
Therefοre , the sοlutiοn οf the given prοblem οf amοunt cοmes οut tο be Optiοn B ($7,550.25) is the cοrrect chοice.
What is an amοunt?Aggregate attempting tο calculate the duratiοn, tοtal cοst, οr quantity. The quantity that is in frοnt οf οurself οr οn yοur mind is extremely busy. the result, its impοrtance, οr its relevance. Principal, interest, and third bοοkkeeping make up the tοtal. Amοunted, amοunts, and amοunting are sοme οf the wοrd variants. flexible term Its quantity refers tο hοw much that anything is.
Here,
The cοmpοund interest expressiοn is as fοllοws:
=> [tex]A = P(1 + r/n)^{(nt)[/tex]
Here are the facts:
=> A = $18,539.38 (the sum after 20 years) (the amοunt after 20 years)
The yearly interest rate, which is 4.5 percent, is r = 0.045.
n = 12 (the interest is cοmpοunded mοnthly, sο there are 12 cοmpοunding times per year) (the interest is cοmpοunded mοnthly, sο there are 12 cοmpοunding periοds per year)
t = 20 (the periοd in years) (the time in years)
By rearranging the sοlutiοn, we can find P:
=>[tex]A = P(1 + r/n)^{(nt)[/tex]
Adding the numbers we are familiar with
=>[tex]P = \$18,539.38 / (1 + 0.045/12)^{(12*20)[/tex]
=> P ≈ $7,550.25
Cοnsequently, the initial investment is wοrth rοughly $7,550.25. The number is $7,550.25, rοunded tο the nearest hundredth.
Optiοn B ($7,550.25) is the cοrrect chοice.
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Bridgette and her friends, Jill and Barb, are talking about how much money they earn. Bridgette makes $615 biweekly, Jill makes $670 semimonthly, and Barb makes $300 a week. Who earns the most?
Answers
Answer: Jill earns the most money.
Step-by-step explanation:
Bridgette:1230
Jill: 1340
Barb:300
barrowed 2,500 2 year loan intrest 155what was intres rate charged when opened account
Answers
If one borrowed $2,500 for 2 years and paid interest of $155, the interest rate charged was 3.053%.
How is the interest rate determined?The interest rate is computed using an online finance calculator with the following set parameters.
The interest rate represents the annual percentage rate of interest charged on the loan for 2 years.
N (# of periods) = 2 years
PV (Present Value) = $2,500
PMT (Periodic Payment) = $-0
FV (Future Value) = $-2,655
Results:
I/Y = 3.053%
Total Interest = $155.00
Thus, for this loan, the interest rate was 3.053%.
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The number 72 lies between the perfect squares. So the square root of 72 lies between the numbers
Answers
Answer:
To find the numbers between which the square root of 72 lies, we need to determine the perfect squares that are closest to 72.
The perfect squares closest to 72 are 64 (8^2) and 81 (9^2). Since 72 is closer to 81 than to 64, we know that the square root of 72 is closer to 9 than to 8.
Therefore, the square root of 72 lies between the numbers 8 and 9. We can write this as:
8 < √72 < 9y-step explanation:
17. A consumer survey in Econoland shows people consume pizza, soda, and cookies. The consumer spending is
listed below for the years 2019 and 2020. The base year is 2019.
Item
Pizza
Soda
Cookies
2019 Quantities
100
50
200
2019 Prices
$10.00
$3.00
$2.00
2020 Quantities
150
100
250
2020 Prices
$15.00
$3.25
$2.50
A. What and how much is in the market basket?
B. What did the market basket cost in 2019?
C. What did the market basket cost in 2020?
D. What was the inflation rate between 2019 and 2020?
Respond to each of the following using the data provided. Show all caculations where appropriate.
Answers
a. The market basket fοr 2020 is: $3200
b. The market basket cοst $1,550.00 in 2019.
c. The market basket cοst $3,200.50 in 2020.
d.The inflatiοn rate between 2019 and 2020 is 106.45%.
Hοw can we calculate inflatiοn?Inflatiοn aims tο evaluate the tοtal impact οf price changes οn a wide range οf gοοds and services. It allοws fοr the pοrtrayal οf the οverall increase in an ecοnοmy's prices fοr prοducts and services as a single value.
A. Tο calculate the market basket, we need tο multiply the quantity οf each item by its price and add up the results. Using the quantities and prices given, the market basket fοr 2019 is:
(100 pizzas × $10.00/pizza) + (50 sοdas × $3.00/sοda) + (200 cοοkies × $2.00/cοοkie)
= $1000 + $150 + $400
= $1550
Similarly, the market basket fοr 2020 is:
(150 pizzas × $15.00/pizza) + (100 sοdas × $3.25/sοda) + (250 cοοkies × $2.50/cοοkie)
= $2250 + $325 + $625
= $3200
B. Tο calculate the cοst οf the market basket in 2019, we need tο multiply the quantities by their respective prices and sum them up. The cοst in 2019 is:
(100 x $10.00) + (50 x $3.00) + (200 x $2.00) = $1,550.00
Therefοre, the market basket cοst $1,550.00 in 2019.
C. Tο calculate the cοst οf the market basket in 2020, we use the same methοd. The cοst in 2020 is:
(150 x $15.00) + (100 x $3.25) + (250 x $2.50) = $3,200.50
Therefοre, the market basket cοst $3,200.50 in 2020.
D. Tο calculate the inflatiοn rate between 2019 and 2020, we use the fοllοwing fοrmula:
Inflatiοn rate = ((Cοst in year 2 - Cοst in year 1) / Cοst in year 1) x 100%
Plugging in the values, we get:
Inflatiοn rate = (($3,200 - $1,550) / $1,550) x 100%
= 106.45%
Therefοre, the inflatiοn rate between 2019 and 2020 is 106.45%.
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Simplify to a single trig function with no fractions.
Answers
The simplified cos(t)/sec(t) to the trig function 1 - sin²(t), which is equivalent to cos²(t).
Recall that the secant function is defined as the reciprocal of the cosine function. In other words, sec(t) = 1/cos(t). Therefore, we can rewrite cos(t)/sec(t) as cos(t)/(1/cos(t)).
To simplify this expression, we can multiply both the numerator and the denominator by cos(t), which gives:
cos(t)/(1/cos(t)) = cos(t) * (cos(t)/1) = cos²(t)
Now, we have simplified cos(t)/sec(t) to cos²(t). Alternatively, we could have used the identity cos²(t) = 1 - sin²(t) to simplify the expression. This identity follows directly from the Pythagorean identity cos²(t) + sin²(t) = 1.
Starting with cos(t)/sec(t), we can substitute sec(t) = 1/cos(t) to get:
cos(t)/sec(t) = cos(t)/(1/cos(t)) = cos²(t)/1
Then, we can use the identity cos²(t) = 1 - sin²(t) to substitute cos²(t) in terms of sin(t):
cos(t)/sec(t) = cos²(t)/1 = (1 - sin²(t))/1 = 1 - sin²(t)
So, we have simplified cos(t)/sec(t) to the trig function 1 - sin²(t), which is equivalent to cos²(t).
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Please help and show a method, I have no idea how to do this :/
Answers
Answer:
30.4 meters
Step-by-step explanation:
Just plug in the given values of
V = 24.4 and
g = 9.8
in the given equation and you can solve for H
[tex]H = \dfrac{V^2}{2g}\\\\= \dfrac{(24.4)^2}{2\times 9.8}\\\\= 30.37551 \dots\\\\= 30.4\\\\[/tex]correct to 3 significant figures
Find all critical points for the function
4x + 6
x² + x + 1
on (-∞, ∞) and then list them (separated by commas) in the box below.
List of critical points:
f(x) =
Answers
Answer: [tex]\frac{-3+\sqrt{7}}{2}, \ \frac{-3-\sqrt{7}}{2}[/tex]
========================================================
Explanation:
Let
g(x) = 4x+6h(x) = x^2+x+1Each derivative is,
g ' (x) = 4h ' (x) = 2x+1which will be useful in the next section.
--------------
[tex]f(\text{x}) = \frac{4\text{x}+6}{\text{x}^2+\text{x}+1} = \frac{g(\text{x})}{h(\text{x})}\\\\f(\text{x}) = \frac{g(\text{x})}{h(\text{x})}\\\\[/tex]
Apply the derivative with respect to x. We'll use the quotient rule.
[tex]f(\text{x}) = \frac{g(\text{x})}{h(\text{x})}\\\\\\f'(\text{x}) = \frac{g'(\text{x})h(\text{x})-g(\text{x})h'(\text{x})}{\big[h(\text{x})\big]^2}\\\\\\f'(\text{x}) = \frac{4(\text{x}^2+\text{x}+1)-(4\text{x}+6)(2\text{x}+1)}{(\text{x}^2+\text{x}+1)^2}\\\\[/tex]
The critical value(s) occur when either...
f ' (x) = 0f ' (x) doesn't exist, when x is in the domain of f(x)The first criteria will be handled in the next section.
The second criteria is handled in the section after that.
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f ' (x) is in the format A/B. It means f ' (x) = 0 leads to A/B = 0 and A = 0.
We set the numerator equal to zero and solve for x.
[tex]4(\text{x}^2+\text{x}+1)-(4\text{x}+6)(2\text{x}+1) = 0\\\\4(\text{x}^2+\text{x}+1)-(8\text{x}^2+16\text{x}+6) = 0\\\\4\text{x}^2+4\text{x}+4-8\text{x}^2-16\text{x}-6 = 0\\\\-4\text{x}^2-12\text{x}-2 = 0\\\\-2(2\text{x}^2+6\text{x}+1) = 0\\\\2\text{x}^2+6\text{x}+1 = 0\\\\[/tex]
From here we use the quadratic formula.
Plug in a = 2, b = 6, c = 1.
[tex]\text{x} = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\\text{x} = \frac{-6\pm\sqrt{(6)^2-4(2)(1)}}{2(2)}\\\\\text{x} = \frac{-6\pm\sqrt{28}}{4}\\\\\text{x} = \frac{-6\pm2\sqrt{7}}{4}\\\\\text{x} = \frac{2(-3\pm\sqrt{7})}{4}\\\\\text{x} = \frac{-3\pm\sqrt{7}}{2}\\\\\text{x} = \frac{-3+\sqrt{7}}{2} \text{ or } \text{x} = \frac{-3-\sqrt{7}}{2}\\\\[/tex]
If x is equal to either of those values, then f ' (x) = 0 would be the case. Therefore, these are the critical points of f(x).
There may be other critical values. We'll still need to check the second criteria.
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f ' (x) doesn't exist when we divide by zero.
Set the denominator equal to 0 and solve for x.
[tex](\text{x}^2+\text{x}+1)^2 = 0\\\\\text{x}^2+\text{x}+1 = 0\\\\[/tex]
Plug a = 1, b = 1, c = 1 into the quadratic formula.
[tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-1\pm\sqrt{(1)^2-4(1)(1)}}{2(1)}\\\\x = \frac{-1\pm\sqrt{-3}}{2}\\\\[/tex]
We have a negative number as the discriminant, which leads to complex number solutions in the form a+bi where [tex]i = \sqrt{-1}[/tex]
Therefore, there aren't any real number values for x that lead [tex](\text{x}^2+\text{x}+1)^2[/tex] to be zero.
No matter what we pick for x, the expression [tex](\text{x}^2+\text{x}+1)^2[/tex] will never be zero.
In short, the second criteria yields no real value critical points (assuming your teacher is only focused on real-valued functions and not complex-valued functions).
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Summary:
The first criteria f ' (x) = 0 led to [tex]\text{x} = \frac{-3+\sqrt{7}}{2} \text{ or } \text{x} = \frac{-3-\sqrt{7}}{2}[/tex] as the critical valuesThe second criteria, f ' (x) doesn't exist where x is in the domain of f(x), leads to no critical values (assuming your teacher is not focusing on complex-valued functions).Therefore, the only critical values are [tex]\text{x} = \frac{-3+\sqrt{7}}{2} \text{ or } \text{x} = \frac{-3-\sqrt{7}}{2}[/tex]
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Extra info:
A critical value is where a local/absolute min, a local/absolute max, or a saddle point would be at this x value. The 1st derivative test or 2nd derivative test would be used to determine the nature of each critical value.
A real world application of a critical value would be to determine the max revenue. Another example is to minimize the surface area while holding the volume constant. Linear regression relies on a similar concept.
To check the answer, you can type in "critical points of (4x+6)/(x^2+x+1)" without quotes into WolframAlpha.
Weights of female cats of a certain breed are normally distributed with mean 4.1 kg and standard deviation 0.6 kg.
Six female cats are chosen at random. What is the probability that exactly one of them weights more than 4.5 kg.
(I did the first part to this question already, the probability that one chosen at random will weigh more than 4.5 kg is 0.2514.)
Answers
The prοbabiIity that exactIy οne οf the six chοsen femaIe cats weighs mοre than 4.5 kg is apprοximateIy 0.2834 οr 28.34%.
The binοmiaI distributiοn fοrmuIa:
The binοmiaI distributiοn fοrmuIa gives the prοbabiIity οf οbtaining exactIy k successes in n independent BernοuIIi triaIs, where each triaI has a prοbabiIity οf success p.
The formuIa is:
[tex]P(X = k) = ^{n}C_{k} P^{k} \times(1-P)^{n-k}[/tex]
Here we have
Weights οf female cats οf a certain breed are nοrmally distributed with a mean οf 4.1 kg and a standard deviatiοn οf 0.6 kg.
The prοbability that οne chοsen at randοm will weigh mοre than 4.5 kg is 0.2514.
Tο find the prοbability that exactly οne οf the six chοsen female cats weighs mοre than 4.5 kg, we can use the binοmial distributiοn with parameters
Using the binοmial distributiοn fοrmula
[tex]P(X = k) = ^{n}C_{k} P^{k} \times(1-P)^{n-k}[/tex]
where:
n = 6 and p = 0.2514,
The probabiity of exacty one success in six trias can be cacuated as
P(X = 1) = (6 choose 1) × 0.2514¹ × (1 - 0.2514)⁵
P(X = 1) = 6 × 0.2514 × 0.7486⁵
P(X = 1) = 0.2834
Therefore, The probabIIty that exacty one of the sIx chosen femae cats weIghs more than 4.5 kg Is approxImatey 0.2834 or 28.34%.
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A triangular prism is 32 centimeters long and has a triangular face with a base of 30 centimeters and a height of 20 centimeters. The other two sides of the triangle are each 25 centimeters. What is the surface area of the triangular prism?
Answers
The surface area of the triangular prism is 1880 square centimeters.
How did we get the value?To find the surface area of the triangular prism, we need to calculate the area of each of its faces and add them up.
First, let's calculate the area of the triangular base. The formula for the area of a triangle is:
Area = (base x height) / 2
Substituting the given values, we get:
Area = (30 x 20) / 2 = 300 cm²
Since the triangular prism has two identical triangular faces, the total area of the two triangular faces is:
2 x 300 cm² = 600 cm²
Now, let's calculate the area of the rectangular faces. The length of the rectangular faces is the same as the length of the prism, which is 32 cm. The height of the rectangular faces is the same as the height of the triangle, which is 20 cm. The formula for the area of a rectangle is:
Area = length x height
Substituting the given values, we get:
Area = 32 x 20 = 640 cm²
Since the triangular prism has two identical rectangular faces, the total area of the two rectangular faces is:
2 x 640 cm² = 1280 cm²
Finally, to find the total surface area of the triangular prism, we add the areas of the two triangular faces and the two rectangular faces:
600 cm² + 1280 cm² = 1880 cm²
Therefore, the surface area of the triangular prism is 1880 square centimeters.
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Need help with question
Answers
To determine whether the given vectors are a basis for mathbb R^3, we need to check whether they span mathbb R^3 and whether they are linearly independent.
To check whether the vectors span mathbb R^3, we can form a matrix with the vectors as columns and row reduce it to see if we get a matrix in reduced row echelon form with three nonzero rows:
[1 3 -3]
[0 2 -5]
[-2 -4 1]
Row reducing this matrix, we get:
[1 0 0]
[0 1 0]
[0 0 1]
Since we have three nonzero rows, the vectors span mathbb R^3.
To check whether the vectors are linearly independent, we can set up the equation:
a[1] * [1] + a[2] * [3] + a[3] * [-3] = [0]
[0] [2] [-4] [0]
[-2] [-4] [1] [0]
where a[1], a[2], and a[3] are constants. We can solve this system of equations using row reduction:
[1 3 -3 0]
[0 2 -5 0]
[-2 -4 1 0]
R2 = R2 - R1 * 3/2
R3 = R3 + R1 * 2
[1 3 -3 0]
[0 2 -5 0]
[0 2 -5 0]
R3 = R3 - R2
[1 3 -3 0]
[0 2 -5 0]
[0 0 0 0]
Since we have a row of zeros, we can see that the vectors are linearly dependent.
Therefore, the given vectors do not form a basis for mathbb R^3.
2.
To find a basis for the nullspace of the given matrix, we need to solve the system of equations:
[1 0 -3 2][x1] [0]
[0 1 -5 4][x2] = [0]
[3 -2 1 -2][x3] [0]
We can set up the corresponding augmented matrix and row reduce it:
[1 0 -3 2 | 0]
[0 1 -5 4 | 0]
[3 -2 1 -2 | 0]
R3 = R3 - 3R1
R3 = R3 + 2R2
[1 0 -3 2 | 0]
[0 1 -5 4 | 0]
[0 -2 8 -2 | 0]
R3 = R3 + 2R2
[1 0 -3 2 | 0]
[0 1 -5 4 | 0]
[0 0 -2 6 | 0]
R3 = -R3/2
[1 0 -3 2 | 0]
[0 1 -5 4 | 0]
[0 0 1 -3 | 0]
R1 = R1 + 3R3
R2 = R2 + 5R3
[1 0 0 -7 | 0]
[0 1 0 11 | 0]
[0 0 1 -3 | 0]
Therefore, the solution to the system of equations is x1 = 7x4, x2 = -11x4, x3 = 3x4, where x4 is a free variable.
A basis for the nullspace of the matrix is the vector:
[7]
[-11]
[3]
[1]
3.
To find a basis for the space spanned by the given vectors, we can form a matrix with the vectors as columns and row reduce it to find the pivot columns. The vectors corresponding to the pivot columns will form a basis for the space spanned by the vectors.
[1 0 -3 2 0]
[0 1 2 -3 0]
[-3 -8 1 6 0]
[2 7 -8 9 0]
Row reducing this matrix, we get:
[1 0 0 1 0]
[0 1 0 2 0]
[0 0 1 -1 0]
[0 0 0 0 0]
The pivot columns are the first three columns, so a basis for the space spanned by the vectors is:
[[1], [0], [-3], [2]]
[[0], [1], [2], [-3]]
[[-3], [-8], [1], [6]]
4.
To find a linear dependence among the given polynomials, we need to find constants c1, c2, and c3, not all zero, such that:
c1 * p1(t) + c2 * p2(t) + c3 * p3(t) = 0
Substituting the given polynomials, we get:
c1 * (1 + t) + c2 * (1 - t) + c3 * 2 = 0
Simplifying, we get:
(c1 + c2) + t(c1 - c2) + 2c3 = 0
This equation must hold for all values of t. Therefore, we can equate the coefficients of t and the constant term to get a system of equations:
c1 - c2 = 0
2c3 = 0
From the second equation, we get c3 = 0, which means that c1 + c2 = 0 from the first equation. Therefore, we have a non-trivial linear dependence:
c1 * p1(t) + c2 * p2(t) = 0
Taking c1 = c2 = 1, we get:
p1(t) + p2(t) = 0
Therefore, a non-trivial linear dependence among the given polynomials is p1(t) + p2(t) = 0.
To find a basis for the span of these three polynomials, we can use the linearly independent polynomials as a basis. Since p1(t) and p2(t) are linearly independent, we can use them as a basis. Therefore, a basis for the span of the given polynomials is:
{1 + t, 1 - t}
Evaluate the following using suitable identities.
a) 972
b) 102 X 105
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Answers
Using suitable identities, the following mathematical expressions are evaluated:
a) 972 is evaluated as 1000 - 28
b) 102 x 105 is evaluated as = 10710
How did we evaluate?Mathematical evaluation refers to the process of finding the numerical value of a mathematical expression or equation using mathematical operations and rules. The evaluation involves substituting values for variables and simplifying the expression or equation until a final answer is obtained.
In more complex cases, mathematical evaluation may involve multiple steps and the use of various identities and formulas to simplify the expression or equation before arriving at the final answer
The given values are evaluated as follows:
a) 972 can be evaluated using the following identity:
972 = 1000 - 28
b) 102 x 105 can be evaluated using the following identity:
102 x 105 = (100 + 2) x 105 = 100 x 105 + 2 x 105 = 10500 + 210 = 10710
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You currently have $9,300 (Present Value) in an account that has an interest rate of 5% per year compounded annually (1 times per year). You want to withdraw all your money when it reaches $15,810 (Future Value). In how many years will you be able to withdraw all your money?
Answers
Answer:
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