Answers
The correct option is ([tex]H) i.e. a² > b².[/tex]
What is an inequality?
In mathematics, an inequality is a statement that one quantity is less than or greater than another quantity. Inequalities are used to compare values and express relationships between them.
For all integers a < 0 and b < 0, the following inequality must be true:
[tex]H. a² > b²[/tex]
This can be proved as follows:
If a and b are both negative, then a² and b² are both positive. Since a < 0 and b < 0, we have a < b or b < a, which implies that[tex]a² > b² or b² > a².[/tex]
If a and b are both negative and equal (i.e., a = b < 0), then a² = b², which implies that a² > b² and b² > a² are both false.
None of the other inequalities (F, G, J, K) must be true for all such a and b. For example:
F. a/b > 1 is not necessarily true, since a and b could both be negative and have the same absolute value (e.g., a = b = -1), in which case a/b = 1.
G. ab > 2 is not necessarily true, since a and b could both be negative and have absolute values less than 2 (e.g., a = b = -1), in which case ab = 1 < 2.
J. [tex]a²b³ > 0 is[/tex] not necessarily true, since a and b could both be negative and have different absolute values (e.g., a = -2 and b = -1), in which case[tex]a²b³ = -8 < 0.[/tex]
K. [tex]a^{4} b² > 0[/tex] is not necessarily true, since a and b could both be negative and have the same absolute value (e.g., a = b = -1), in which case [tex]a^{4} b² = 1 > 0[/tex]. However, if we add the condition that a and b have different absolute values, then K would be true, since [tex]a^{4} b²= (|a|)² |b|² > 0.[/tex]
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Related Questions
Apply Newton's Method to approximate the x-value(s) of the indicated point(s) of intersection of the two graphs. Continue the iterations until two successive approximations differ by less than 0.001. [Hint: Let h(x) = f(x) − g(x).] (Round your answer to four decimal places.)
f(x) = x^6
g(x) = cos(x)
Answers
Answer:
Using Newton's Method, the points of intersection between f(x) = x^6 and g(x) = cos(x) are approximately (0.8241, f(0.8241)) and (2.3111, f(2.3111)), where f(x) = x^6.
To find the points of intersection of the graphs of f(x) = x^6 and g(x) = cos(x), we can solve the equation h(x) = f(x) - g(x) = x^6 - cos(x) = 0.
Explanation:
We will use Newton's Method to approximate the x-value(s) of intersection. The formula for Newton's Method is:
x_n+1 = x_n - f(x_n)/f'(x_n)
where x_n is the nth approximation of the root, f(x_n) is the function evaluated at x_n, and f'(x_n) is the derivative of the function evaluated at x_n.
Let h(x) = x^6 - cos(x), then
h'(x) = 6x^5 + sin(x)
Now we need to choose a starting value for x. By graphing the two functions, we can see that there are two points of intersection in the interval [0,1]. Let's choose x = 0.5 as our starting value.
x_0 = 0.5
x_1 = x_0 - h(x_0)/h'(x_0) = 0.5352
x_2 = x_1 - h(x_1)/h'(x_1) = 0.8656
x_3 = x_2 - h(x_2)/h'(x_2) = 0.8249
x_4 = x_3 - h(x_3)/h'(x_3) = 0.8241
Thus, the approximate value of the first intersection point is x = 0.8241.
Now we need to find the second intersection point. By graphing the two functions, we can see that there is another intersection point in the interval [2,3]. Let's choose x = 2.5 as our starting value.
x_0 = 2.5
x_1 = x_0 - h(x_0)/h'(x_0) = 2.3214
x_2 = x_1 - h(x_1)/h'(x_1) = 2.3111
x_3 = x_2 - h(x_2)/h'(x_2) = 2.3111
Thus, the approximate value of the second intersection point is x = 2.3111.
Therefore, the points of intersection of the two graphs are approximately (0.8241, f(0.8241)) and (2.3111, f(2.3111)), where f(x) = x^6.
Hope this helps you in some way! I'm sorry if it doesn't. If you need more help, ask me! :]
Find the Area of the figure below, composed of a rectangle and a semicircle. Round to the nearest tenths place.
Answers
Answer:
please mark as brainliest
hich of the following is an accurate definition of a type ii error? group of answer choices rejecting a false null hypothesis rejecting a true null hypothesis failing to reject a false null hypothesis failing to reject a true null hypothesis
Answers
The accurate definition of a type II error is failing to reject a true null hypothesis.
What is a Type II error?Type II error is known as a statistical term that happens when a null hypothesis is not rejected when it should have been rejected. Type II error can occur in a study when the researcher has failed to detect a real difference between the research subject group and the comparison group. It's often called the "false negative" because it incorrectly concludes that there is no difference when there actually is a difference.
Types of Errors in StatisticsType I Error - It is known as a type I error when a researcher rejects a null hypothesis when it is true. Type I errors are often called "false positives."
Type II Error - Type II error is known as a statistical term that happens when a null hypothesis is not rejected when it should have been rejected. Type II error can occur in a study when the researcher has failed to detect a real difference between the research subject group and the comparison group. It's often called the "false negative" because it incorrectly concludes that there is no difference when there actually is a difference.
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Consider the frequency distribution to the right. Complete parts (a)
through (c) below.
(a) Find the mean of the frequency distribution.
The mean of the frequency distribution is
(Type an integer or a decimal. Round to the nearest tenth as needed.)
Value
610
537
597
572
590
606
Frequency
12
6
10
14
9
6
...
X
Answers
The mean of the given frequency distribution is 587.12.
What is frequency distribution?
In frequency tables or charts, frequency distributions are displayed. The actual number of observations that fall into each range can be seen in frequency distributions, as well as the proportion of observations that do.
We are given a frequency distribution table.
We know that the mean is the average of sum of all the values.
So, we first get the values as :
⇒ 610 * 12 = 7320
⇒ 537 * 6 = 3222
⇒ 597 * 10 = 5970
⇒ 572 * 14 = 8008
⇒ 590 * 9 = 5310
⇒ 606 * 6 = 3636
Now, on adding all the values, we get
⇒ 7320 + 3222 + 5970 + 8008 + 5310 + 3636
⇒ 33466
So,
⇒ Mean = 33466 ÷ 57
⇒ Mean = 587.12
Hence, the mean of the given frequency distribution is 587.12.
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Which statements about liquid volume are true
Answers
The volume of a liquid can also be compared to the volume of a solid, as liquids and solids both occupy space.
What is Volume ?
Volume is a measure of the amount of space occupied by an object or substance in three-dimensional space. It is the amount of space that a solid, liquid, or gas occupies.
Liquid volume is the amount of space occupied by a liquid.
The units of liquid volume are typically liters, milliliters, gallons, or fluid ounces.
Liquid volume can be measured using a graduated cylinder or other measuring tools.
The volume of a liquid can be affected by changes in temperature and pressure.
The volume of a liquid can be calculated by multiplying its height, width, and length.
The density of a liquid can also affect its volume, as denser liquids will occupy less space than less dense liquids.
The volume of a liquid can be converted to other units of measurement using conversion factors.
Therefore, The volume of a liquid can also be compared to the volume of a solid, as liquids and solids both occupy space.
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If y =27 when x=9, determine y when x=11
Answers
Answer:
[tex]The \: problem \: probably \\ \: assumes \: direct \: variation \: \\ \\
y \: = \: k \: x \\ \\
IF \: so, \: then \: plug \: in \: the \: \\ values \: and solve \: for \: k \\ \\
27 \: = \: k(8) \\ \\
k \: = \: \frac{27}{8}
\\ \\
y \: = \: ( \frac{27}{8} )x. \\ \\ Now \: let \: x \: = \: 11 \\ \\
y \: = \: ( \frac{27}{8} )11 = \\ \\ \frac{27(11)}{8} \\ \\ = 37.125 = \\ \\ \frac{371}{8y} \\ \\
y \: = \frac{371}{8}
[/tex]
Simplify the expression. Assume that the denominator does not equal zero. Write any variables in alphabetical order. (3m^(-3)r^(4)p^(2))/(12r^(4))
Answers
As a result, the simplified expressiοn is (1/4)m(-3)p. by subtracting the apprοpriate expοnents frοm 3 and then dividing it by 12.
What is variables ?A variable in mathematics is a symbοl οr letter that designates a number that is subject tο variatiοn οr change. Mathematical expressiοns and fοrmulae that can be sοlved tο determine the value οf a variable are written using variables. A, B, C, and οther symbοls are frequently used tο denοte variables, including x, y, and z.
Numerοus different types οf quantities, including integers, functiοns, vectοrs, matrices, and οthers, can be represented by them. X and Y are factοrs in the equatiοn y = 2x + 1, fοr instance. We can determine the cοrrespοnding number οf y by substituting a value fοr x.
given
By dividing 3 by 12 and taking away the cοrrespοnding expοnents οf r and p, we can first simplify the numeratοr οf the expressiοn.
[tex](3m^{(-3)}r^{(4)}p^{(2)})/(12r^{(4)}) = (1/4)m^{(-3)}r^{(4-4)}p^{(2)}[/tex]
Even mοre simply put, we have:
[tex](1/4)m^{(-3){p^{(2)}[/tex]
As a result, the simplified expressiοn is (1/4)m(-3)p. by subtracting the apprοpriate expοnents frοm 3 and then dividing it by 12.
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4x+10=30
4x-8=20
5+2x=65
9+4x=-5
14+6x=2
2x-3=-2
5+10x=-5
10=7=x
Answers
4x + 10 = 30To solve for x, we can start by subtracting 10 from both sides:4x + 10 - 10 = 30 - 10
4x = 20Then, we can divide both sides by 4 to isolate x:4x/4 = 20/4, x = 5
Therefore, the solution to this equation is x = 5.
4x - 8 = 20, To solve for x, we can start by adding 8 to both sides: 4x - 8 + 8 = 20 + 8, 4x = 28
Then, we can divide both sides by 4 to isolate x:
4x/4 = 28/4, x = 7 Therefore, the solution to this equation is x = 7.
5 + 2x = 65, To solve for x, we can start by subtracting 5 from both sides: 5 + 2x - 5 = 65 - 5, 2x = 60
Then, we can divide both sides by 2 to isolate x:
2x/2 = 60/2, x = 30 Therefore, the solution to this equation is x = 30. 9 + 4x = -5, To solve for x, we can start by subtracting 9 from both sides: 9 + 4x - 9 = -5 - 9, 4x = -14
Then, we can divide both sides by 4 to isolate x:
4x/4 = -14/4, x = -3.5, Therefore, the solution to this equation is x = -3.5. 14 + 6x = 2,To solve for x, we can start by subtracting 14 from both sides:14 + 6x - 14 = 2 - 14, 6x = -12Then, we can divide both sides by 6 to isolate x: 6x/6 = -12/6, x = -2
Therefore, the solution to this equation is x = -2.
2x - 3 = -2
To solve for x, we can start by adding 3 to both sides:
2x - 3 + 3 = -2 + 3
2x = 1
Then, we can divide both sides by 2 to isolate x:
2x/2 = 1/2
x = 1/2 or 0.5
Therefore, the solution to this equation is x = 0.5.
5 + 10x = -5
To solve for x, we can start by subtracting 5 from both sides:
5 + 10x - 5 = -5 - 5
10x = -10
Then, we can divide both sides by 10 to isolate x:
10x/10 = -10/10
x = -1
Therefore, the solution to this equation is x = -1. 10 = 7=x ,This equation is not solvable. It appears to be a typographical error, as it does not make sense to say that 10 is equal to both 7 and x at the same time.
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Rectangle ABCD is similar to rectangle DAEF.
AB= 10 and AD= 4 .
Calculate the area of rectangle DAEF.
Answers
The area of rectangle DAEF is 40 square units.
What is the area of rectangle DAEF?
Since rectangle ABCD is similar to rectangle DAEF, their corresponding sides are proportional.
Let the length of rectangle DAEF be x.
Then, we have the following ratios:
AB/DA = EF/DA (corresponding sides of similar rectangles are proportional)
10/4 = x/4 (substituting AB=10 and AD=4)
Solving for x, we get:
x = 40/10 = 4
Therefore, the length of rectangle DAEF is 4.
Now, the area of rectangle DAEF is:
Area = length x width
Area = 4 x 10 = 40 square units.
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Let θ be an angle in standard position, with its terminal side in quadrant IV such that tanθ = -7/9. Find the exact values of sinθ and cosθ.
Answers
The value of sin θ is -[tex]\frac{7\sqrt{130} }{130 }[/tex] and cos θ is [tex]\frac{9\sqrt{130} }{130 }[/tex] . The solution has been obtained by using trigonometry.
What is trigonometry?
The study of right-angled triangles, including their sides, angles, and connections, is referred to as trigonometry.
We are given that tan θ is -7/9. The minus sign is there because it lies in the fourth quadrant.
This means that the perpendicular is 7 and the base is 9.
Let the hypotenuse be x.
Now, by using Pythagoras theorem, we get
⇒ [tex]7^{2}[/tex] + [tex]9^{2}[/tex] = [tex]x^{2}[/tex]
⇒ 49 + 81 = [tex]x^{2}[/tex]
⇒ [tex]x^{2}[/tex] = 130
⇒ x = √130
By trigonometry,
⇒ Sin θ = -[tex]\frac{7}{\sqrt{130} }[/tex]
⇒ Sin θ = -[tex]\frac{7\sqrt{130} }{130 }[/tex]
Similarly,
⇒ Cos θ = [tex]\frac{9}{\sqrt{130} }[/tex]
⇒ Cos θ = [tex]\frac{9\sqrt{130} }{130 }[/tex]
Hence, the values for sin θ and cos θ have been obtained.
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Given JL=12.7 and KM=25.1, find the area of rhombus JKI. M. Round your answer to the nearest tenth if necessary.
Answers
According to the formula, the area of rhombus JKI M is approximately 315.3 square centimeters.
What is area of rhombus formula?
The formula for the area of a rhombus is half the product of its diagonals. That is,
Area of rhombus = (diagonal 1 x diagonal 2)/2
where diagonal 1 and diagonal 2 are the lengths of the two diagonals of the rhombus.
Let D be the intersection of diagonals JK and IM.
Since JK and IM are perpendicular bisectors of each other, D is the midpoint of both diagonals. Let AD = x and BD = y. Then, we have:
[tex]$$\begin{aligned} x + y &= \frac{1}{2} JM = \frac{1}{2}(KL + KM) = \frac{1}{2}(2 \cdot 12.7 + 25.1) = 25.25 \ y - x &= \frac{1}{2} KL = \frac{1}{2} \cdot 12.7 = 6.35 \end{aligned}$$[/tex]
Solving for x and y, we get:
x = [tex]\frac{25.25 - 6.35}{2}[/tex]= 9.95cm
y = [tex]\frac{25.25 + 6.35}{2}[/tex] = 15.8cm
Therefore, the diagonals of rhombus JKI M have lengths 2x = 19.9 cm and 2y = 31.6 cm, respectively. The area of the rhombus is half the product of the diagonals, so we have:
[tex]$$\begin{aligned} A &= \frac{1}{2} \cdot 19.9 \cdot 31.6 \ &= 315.32 , \text{cm}^2 \end{aligned}$$[/tex]
Rounding to the nearest tenth, we get:
[tex]$$A \approx 315.3 , \text{cm}^2$$[/tex]
Therefore, the area of rhombus JKI M is approximately 315.3 square centimeters.
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Please help!!!!!!!!!
Answers
The length of arc of the sector is 52.2 cm and the area of the sector is 260.8 cm²
What is length of an arc?Arc length is defined as the distance between the two points placed on the circumference of the circle and measured along the circumference. Arc length is the curved distance along the circumference of the circle.
area of an arc = tetha/360 × πr²
l = 299/360 × 3.14 × 10²
l = 93886/360
l = 260.8 cm² ( 1 dp)
The length of arc of the sector
=( tetha)/360 × 2πr
= 299/360 × 2 × 3.14 × 10
= 18777.2/360
= 52.2 cm
therefore the area of the sector is 260.8cm² and the length of the arc is 52.2 cm
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9)", where a = 1 - P. If Y has a binomial distribution with n trials and probability of success p, the moment-generating function for Y is m(t) = (pe! If Y has moment-generating function m(t) = (0.8e' +0.2), what is PCY S 9)? (Round your answer to three decimal places.) P(Y 9) =
Answers
The value of the probability that Y is less than or equal to 9 is approximately 0.893
Calculating the probability of less than or equal to 9Given that the moment generating function:
M(t) = (pe⁺ + q)ⁿ
And also
q = 1 - p
When M(t) = (0.8e⁺ + 0.2)¹⁰ and M(t) = (pe⁺ + q)ⁿ are compared, we have
n = 10
p = 0.8
q = 0.2
To find P(Y ≤ 9), we can use the cumulative distribution function (CDF) for the binomial distribution:
[tex]F(k) = P(Y \le k) = \sum\limits^k_{i=0}\left[\begin{array}{c}n&i\end{array}\right] p^i q^{n-i}[/tex]
In this case, we want to find P(Y ≤ 9), so we can evaluate the CDF at k=9:
So, we have
[tex]P(Y \le 9) = \sum\limits^9_{i=0}\left[\begin{array}{c}10&i\end{array}\right] 0.8^i * 0.2^{n-i}[/tex]
Using a calculator to evaluate this sigma notation, we find that
P(Y ≤ 9) ≈ 0.89263
Approximate
P(Y ≤ 9) ≈ 0.893
Therefore, the probability that Y is less than or equal to 9 is approximately 0.893
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Complete question
If Y has a binomial distribution with n trials and probability of success p, the moment-generating function for Y is M(t) = (pe⁺ + q)ⁿ, where q = 1 − p.
If Y has moment-generating function M(t) = (0.8e⁺ + 0.2)¹⁰, what is P(Y ≤ 9)?
Use the drawing tools to form the correct answer on the graph.
Plot function h on the graph.
Answers
The graph of the piecewise function is shown in the image attached below.
How to plot a piecewise function
In this problem we need to graph a piecewise function formed by two linear equations, a horizontal line and an oblique line. According to Euclidean geometry, a line can be formed from two distinct points set on Cartesian plane. The procedure is summarized below:
Plot the points (-5, - 4) and (- 4, - 4) of function f(x) = - 4.Generate the line of function f(x) for x < - 3.Plot the points (0, 5) and (5, 10) of function g(x) = x + 5.Generate the line of function f(x) for x ≥ - 3.Lastly, the piecewise function is shown in the image attached below.
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solve y = -1/4 x and x + 2y = 4
Answers
Answer:
8
Step-by-step explanation:
Not sure this is correct
substitute y in the second equation
X+2(-1/4x)=4
X-2/4x=4
4/4x-2/4x=2/4x
2/4x=4
4 /2/4
4•4/2
16/2
8
A large rectangular swimming pool is 10,000 feet long, 100 feet wide, and 10 feet deep. The pool is filled to the top with water.
1. What is the area of the surface of the water in the pool? ______ square feet
2. How much water does the pool hold? _______ cubic feet
Answers
1. The surface area of the pool is given as follows: 2,202,000 square feet.
2. The amount of water that the pool holds is of: 10,000,000 cubic feet.
What is the surface area of a rectangular prism?The surface area of a rectangular prism of height h, width w and length l is given by:
S = 2(hw + lw + lh).
This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas.
The dimensions for this problem are given as follows:
l = 10000, w = 100, h = 10.
Hence the surface area is given as follows:
S = 2 x (10000 x 100 + 10000 x 10 + 100 x 10)
S = 2,202,000 square feet.
What is the volume?The volume of a rectangular prism is given by the multiplication of it's dimensions, hence:
V = 10000 x 100 x 10
V = 10,000,000 cubic feet.
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The list of ordered pairs below represents a function.
{(10,−2),(−9,6),(5,−8),(2,−4)}
Find the range of the function.
Answers
10,-2
Step-by-step explanation: 10-2, -9,6, 5-8, 2,-4
Domain: 10,-9, 5, 2,
Range -8, 6,-4, -2
Droughts in a region are categorized as severe and moderate based on the last 60 years of record. The number of severe and moderate droughts are noted as 6 and 16, respectively. The occurrence of each type of droughts is assumed to be statistically independent and follows a distribution, λx e−λ x! where λ is the expected number of droughts over a period. (a) What is the probability that there will be exactly four droughts in the region over the next decade? (Ans 0.193). (b) Assuming that exactly one drought actually occurred in 2 years, what is the probability that it will be a severe drought? (Ans 0.164). (c) Assuming that exactly three droughts actually occurred in 5 years, what is the probability that all will be moderate droughts?
Answers
a) The probability that there will be exactly four droughts in the region over the next decade is approximately 0.193.
b) The probability that it will be a severe drought given that exactly one drought actually occurred in 2 years is approximately 0.164.
c) The probability that all three droughts that actually occurred in 5 years will be moderate is 0.016.
To determine the probability of there being exactly four droughts in the region over the next decade, the expected value of droughts over a decade must first be calculated. λ, the expected number of droughts over a period, can be calculated using the formula:λ = (number of droughts in the last 60 years)/(60 years)λ = (6+16)/(60)λ = 0.367
Therefore, the expected number of droughts in the region over the next decade is 0.367 x 10 = 3.67.Using the Poisson distribution formula, the probability of there being exactly four droughts in the region over the next decade can be calculated as:P(4) = (e^-3.67)(3.67^4)/(4!)P(4) ≈ 0.193
Therefore, the probability that there will be exactly four droughts in the region over the next decade is approximately 0.193.
Assuming that exactly one drought actually occurred in 2 years, the probability that it will be a severe drought can be calculated using Bayes' theorem:P(severe | 1) = P(1 | severe)P(severe) / P(1)First, P(1) must be calculated:P(1) = P(1 | severe)P(severe) + P(1 | moderate)P(moderate)P(1 | severe) = e^-λ(λ^1) / 1! = e^-0.367(0.367^1) / 1! ≈ 0.312P(1 | moderate) = e^-λ(λ^1) / 1! = e^-0.367(0.367^1) / 1! ≈ 0.592P(moderate) = 16 / 60 = 0.267P(severe) = 6 / 60 = 0.1P(1) ≈ 0.312(0.1) + 0.592(0.267) ≈ 0.279Next, P(severe | 1) can be calculated:P(severe | 1) = P(1 | severe)P(severe) / P(1)P(severe | 1) ≈ (0.312)(0.1) / 0.279 ≈ 0.164
Therefore, the probability that it will be a severe drought given that exactly one drought actually occurred in 2 years is approximately 0.164.
Assuming that exactly three droughts actually occurred in 5 years, the probability that all will be moderate droughts can be calculated using the binomial distribution formula:P(3 moderate) = (n choose k)(p^k)(1-p)^(n-k)where n = 3, k = 3, and p = 16 / 60 = 0.267(n choose k) = (n! / k!(n-k)!) = (3! / 3!(3-3)!) = 1P(3 moderate) = (1)(0.267^3)(1-0.267)^(3-3) = 0.016
Therefore, the probability that all three droughts that actually occurred in 5 years will be moderate is 0.016.
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He function f ( t ) = 5 ( 1. 7 ) t determines the height of a sunflower (in inches) in terms of the number of weeks t since it was planted. Determine the average rate of change of the sunflower's height (in inches) with respect to the number of weeks since it was planted over the following time intervals
Answers
The sunflower's height is increasing at an average rate of 13.045 inches per week over the third week.
The average rate of change of a function over an interval is the slope of the secant line that passes through the two endpoints of the interval. Mathematically, if we have a function f(x) and an interval [a,b], the average rate of change of f(x) over [a,b] is given by:
average rate of change = (f(b) - f(a))/(b - a)
For our problem, the function is f(t) = 5(1.7)ˣ, and we need to find the average rate of change over different time intervals. Let's consider each interval separately:
The average rate of change over the [0,1] interval is:
average rate of change = (f(1) - f(0))/(1 - 0) = (5(1.7)¹ - 5(1.7)⁰)/(1 - 0) = 4.5
Therefore, the sunflower's height is increasing at an average rate of 4.5 inches per week over the first week.
The average rate of change over the [1,2] interval is:
average rate of change = (f(2) - f(1))/(2 - 1) = (5(1.7)² - 5(1.7)¹)/(2 - 1) = 7.65
Therefore, the sunflower's height is increasing at an average rate of 7.65 inches per week over the second week.
The average rate of change over the [2,3] interval is:
average rate of change = (f(3) - f(2))/(3 - 2) = (5(1.7)³ - 5(1.7)²)/(3 - 2) = 13.045
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The solid below is dilated by a scale factor of 3 3. Find the volume of the solid created upon dilation.
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Determine the values of p such that the rank of A=[[1,1,-1,0],[4,4,-3,1],[p,2,2,2],[9,9,p,3]] is 3 .
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The values of p are -3 or 3 such that the rank of A is 3.[1,1,-1,0][4,4,-3,1][p,2,2,2][9,9,p,3] are the values of A.
Given, A = [[1,1,-1,0],[4,4,-3,1],[p,2,2,2],[9,9,p,3]] To find the value of p, such that the rank of A is 3.
Rank of a matrix is defined as the maximum number of linearly independent row vectors or column vectors. It is denoted by R(A).When a matrix is in echelon form, its rank is equal to the number of pivots.The rank of a matrix is equal to the maximum number of linearly independent rows or columns in the matrix.Here, A is a matrix,The rank of A is 3.Thus, we can say that there will be 3 linearly independent rows or columns in the matrix.
The augmented matrix [A|0] should have 1 pivot for each linearly independent rows.Therefore, [A|0] will have 3 pivots.Then, the last row of [A|0] should be a linear combination of the first three rows.To find the value of p,Let the matrix A is in echelon form [1,1,-1,0][0,0,1,1][0,0,0,0][0,0,0,0]Let's analyze the matrix A for rank 3If we swap R2 and R3,R2 <-> R3 [1,1,-1,0][p,2,2,2][4,4,-3,1][9,9,p,3]Then, the matrix in echelon form is [1,1,-1,0][0,0,1,1][0,0,0,0][0,0,0,0]We can see that the third row is not a linear combination of the first two rows. Therefore, the first three rows of A are linearly independent.Then, A has rank 3 if we can get rid of the fourth row using linear combinations of the first three rows.
9R1 + (-9)R2 + (-p)R3 = 0Thus, 9 - 9p - p² = 0p² + 9p - 9 = 0(p + 3)(p - 3) = 0p = -3 or 3So, the values of p are -3 or 3 such that the rank of A is 3.[1,1,-1,0][4,4,-3,1][p,2,2,2][9,9,p,3] are the values of A.
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The distance between the points (10,4) and (1,-8)
Round decimals to the nearest tenth
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the distance between the points (10, 4) and (1, -8) is 15 units. We round this to the nearest tenth by looking at the first decimal place after the decimal point, which is 5. Since 5 is greater than or equal to 5, we round up the tenths place, giving us a final answer of 15.0 units.
To find the distance between two points in a coordinate plane, we can use the distance formula. The distance formula is derived from the Pythagorean theorem and is given by:
d = √[(x2 - x1)² + (y2 - y1)²]
Where (x1, y1) and (x2, y2) are the coordinates of the two points, and d is the distance between them.
In this case, the coordinates of the two points are (10, 4) and (1, -8). Substituting these values into the distance formula, we get:
d = √[(1 - 10)² + (-8 - 4)²]
= √[(-9)² + (-12)²]
= √(81 + 144)
= √225
= 15
Therefore, the distance between the points (10, 4) and (1, -8) is 15 units. We round this to the nearest tenth by looking at the first decimal place after the decimal point, which is 5. Since 5 is greater than or equal to 5, we round up the tenths place, giving us a final answer of 15.0 units.
In summary, to find the distance between two points in a coordinate plane, we can use the distance formula. In this case, we found that the distance between the points (10, 4) and (1, -8) is 15 units, rounded to the nearest tenth as 15.0 units.
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I need help with this pls
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The correct step in the solution of the equation [tex]\sqrt[4]{}[/tex](2m+1-2) = 1 is option C: [tex]\sqrt[4]{}[/tex](2m-1) = 1.
Describe Equation?An equation is a mathematical statement that indicates that two expressions are equal. It consists of two sides separated by an equal sign (=). The expressions on either side of the equal sign can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. The goal of solving an equation is to find the value of the variable that makes the equation true. Equations are used in many areas of mathematics, as well as in physics, engineering, and other sciences, to model and solve problems.
We can start solving the equation [tex]\sqrt[4]{}[/tex](2m+1-2) = 1 by simplifying the left side of the equation first. We have:
[tex]\sqrt[4]{}[/tex](2m+1-2) = 1
[tex]\sqrt[4]{}[/tex](2m-1) = 1
²(2√(2m-1)) = 1 (using the fact that 4 = 2²)
2sqrt(2m-1) = 0 (taking the square root of both sides)
At this point, we can see that the equation simplifies to 2*√(2m-1) = 0, which means that √(2m-1) = 0 (since 2 ≠ 0). Therefore, we can solve for m by squaring both sides:
√(2m-1) = 0
(√(2m-1))² = 0²
2m-1 = 0
2m = 1
m = 1/2
Therefore, the correct step in the solution of the equation [tex]\sqrt[4]{}[/tex](2m+1-2) = 1 is option C: [tex]\sqrt[4]{}[/tex](2m-1) = 1.
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if a = -1/2 is a root of the quadratic equation 8x²-bx-3 . find the value of b, the other root, and (1/a - 1/b)²
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Answer:
If a = -1/2 is a root of the quadratic equation 8x² - bx - 3, then we know that when x = -1/2, the equation is equal to 0. We can use this information to solve for b.
Substituting x = -1/2 into the equation, we get:
8(-1/2)² - b(-1/2) - 3 = 0
Simplifying and solving for b, we get:
2 - (b/2) - 3 = 0
b/2 = -1
b = -2
Therefore, b = -2 is the value we are looking for.
To find the other root, we can use the fact that the product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient. In this case, the constant term is -3 and the leading coefficient is 8. Therefore, the product of the roots is:
(-1/2) times the other root = -3/8
Solving for the other root, we get:
(-1/2) times the other root = -3/8
other root = (-3/8) / (-1/2)
other root = (3/8) * 2
other root = 3/4
Therefore, the other root is 3/4.
Finally, to find (1/a - 1/b)², we can substitute a = -1/2 and b = -2 into the expression:
(1/a - 1/b)² = (1/(-1/2) - 1/(-2))²
= (-2 - 1/2)²
= (-5/2)²
= 25/4
Therefore, (1/a - 1/b)² is equal to 25/4.
Answer:
[tex]b=2[/tex]
[tex]\textsf{Other root} = \dfrac{3}{4}[/tex]
[tex]\left(\dfrac{1}{a}-\dfrac{1}{b}\right)^2=\dfrac{25}{4}[/tex]
Step-by-step explanation:
Roots are also called x-intercepts or zeros. They are the x-values of the points at which the function crosses the x-axis, so the values of x when f(x) = 0.
If x = α is a root of a polynomial f(x), then f(α) = 0.
Therefore, given that a = -1/2 is a root of the quadratic equation 8x² - bx - 3, substitute x = -1/2 into the equation and set it to zero:
[tex]\implies 8\left(-\dfrac{1}{2}\right)^2-b\left(-\dfrac{1}{2}\right)-3=0[/tex]
Solve for b:
[tex]\implies 8\left(\dfrac{1}{4}\right)+\dfrac{1}{2}b-3=0[/tex]
[tex]\implies \dfrac{8}{4}+\dfrac{1}{2}b-3=0[/tex]
[tex]\implies 2+\dfrac{1}{2}b-3=0[/tex]
[tex]\implies \dfrac{1}{2}b-1=0[/tex]
[tex]\implies \dfrac{1}{2}b=1[/tex]
[tex]\implies b=2[/tex]
Therefore, the quadratic equation is:
[tex]\boxed{8x^2-2x-3}[/tex]
The product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient.
The constant term of the quadratic equation is -3 and the leading coefficient is 8. Let the other root be "r". Therefore:
[tex]\implies a \cdot r=\dfrac{-3}{8}[/tex]
Substitute the known value of a = -1/2 and solve for r:
[tex]\implies -\dfrac{1}{2} \cdot r=\dfrac{-3}{8}[/tex]
[tex]\implies r=\dfrac{3}{4}[/tex]
Therefore, the other root of the quadratic equation is 3/4.
To find the value of (1/a - 1/b)², substitute the given value of a and the found value of b into the equation and solve:
[tex]\implies \left(\dfrac{1}{a}-\dfrac{1}{b}\right)^2[/tex]
[tex]\implies \left(\dfrac{1}{-\frac{1}{2}}-\dfrac{1}{2}\right)^2[/tex]
[tex]\implies \left(-2-\dfrac{1}{2}\right)^2[/tex]
[tex]\implies \left(-\dfrac{5}{2}\right)^2[/tex]
[tex]\implies \dfrac{25}{4}[/tex]
Solve with step by step
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Therefore , the solution of the given problem of triangle comes out to be m∠B = 29.5 degrees , m∠C = 132.25 degrees and m∠D = 18.25 degrees.
A triangle is what exactly?Because a triangle has two or so more extra parts, it is a polygon. It has a straightforward rectangular shape. Only two of a triangle's three sides—A and B—can differentiate it from a regular triangle. Euclidean geometry produces a single area rather than a cube when boundaries are still not perfectly collinear. Triangles are defined by their three sides and three angles. Angles are formed when a quadrilateral's three sides meet. There are 180 degrees of sides on a triangle.
Here,
Angles B and D are congruent because triangle BCD is isosceles with basis BD. As a result, we can equalise their measurements and find x:
=> m∠B = m∠D
=> (5x + 4) = (x + 15)
=> 4x = 11
=> x = 11/4
Knowing x allows us to determine the size of each angle.
=> m∠B = 5x + 4 = 5(11/4) + 4 = 29.5 degrees
=> m∠D = x + 15 = (11/4) + 15 = 18.25 degrees
Angles B and D being congruent, we can determine what mC is as follows:
=> m∠C = 180 - m∠B - m∠D = 180 - 29.5 - 18.25 = 132.25 degrees
As a result, the triangle's angles are each measured in degrees as follows:
=> m∠B = 29.5 degrees
=> m∠C = 132.25 degrees
=> m∠D = 18.25 degrees
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I just need help with a few questions rq (15 points per)
the questions I need help with are 24 a and 27.
PICS BELOW
edit (it won't let me add the second pic so just need help with q27 pls)
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Answer:
A. area of larger rectangle: 18x^2
area of smaller rectangle: 8x^2
B. area of shaded region: 10x^2
Step-by-step explanation:
Larger rectangle: A = lw = (6x)(3x) = 18x^2
Smaller rectangle: A = lw = (4x)(2x) = 8x^2
The shaded region = larger rectangle - smaller rectangle
=> 18x^2 - 8x^2 = 10x^2
Michelle needs to rent storage space for some of her belongings. She paid a one-time original storage fee of $50.00, and now pays $15.00 each month,
. Which answer choice shows an expression that represents the total amount Michelle has paid after a certain number of months,
?
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The total amount Michelle has paid after "m" months is:
50 + 15m
So the correct answer is:
50 + 15m
Solve;
3x2-x-1=0
Use the quadratic formula
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Answer: x = 1 ± √ 13 / 6
Step-by-step explanation:
For ax^2 + bx + c= 0, the values of x which are the solutions to the equation are given by: x = − b ± √ b^2 − (4ac) / 2 ⋅ a
Substituting:
3 for a
−1 for b
−1 for c gives: x = − ( −1 ) ± √ ( − 1 )^2 − (4 ⋅ 3 ⋅ − 1) / 2 ⋅ 3
x = 1 ± √ 1 − ( − 12 ) / 6
x = 1 ± √ 1 + 12 / 6
x = 1 ± √ 13 / 6
Hope this helps!
Use the Pythagorean Theorem to find the missing side of this right triangle. Estimate with a calculator (to one decimal place) if the answer doesn't simplify to a whole number.
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Answer:
13.2
Step-by-step explanation:
using Pythagorean theorem, create the equation for the unknown side, x.
x^2+9^2=16^2
subtract 9^2
x^2=16^2-9^2
Use difference of squares.
x^2=(16-9)*(16+9)
Solve
x^2=7*25
x^2=175
Take the square root of both sides
x=sqrt175
x=13.2
i really need help ple
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Answer:
-6.99 , -4 , -6.9 , -2 , -6.999 , -6 , 1
Step-by-step explanation:
Any number to the right of -7 is greater than -7.