The approximate probability that between 12 and 20 patients, inclusive will experience weight gain as a side effect is 0.4147.
Normal approximation can be used to approximate the binomial distribution when the sample size is large enough (n >= 30) and the probability of success (p) and failure (q=1-p) are not too small or too large. In this case, we have a sample size of 500, which is sufficiently large, and the probability of success (p=0.03) and failure (q=0.97) are not too small or too large.
To approximate the probability that 15 or fewer users will experience weight gain as a side effect, we can use the normal approximation to the binomial distribution with mean (μ) = np = 500 x 0.03 = 15 and standard deviation (σ) = sqrt(npq) = sqrt(500 x 0.03 x 0.97) = 3.814. Then, we can use the normal cumulative distribution function (normalcdf) to calculate the probability that X ≤ 15, where X is the number of users who experience weight gain.
normalcdf(–1E99,15.5,15, 3.814) = 0.5522
Therefore, the approximate probability that 15 or fewer users will experience weight gain as a side effect is 0.5522.
To approximate the probability that 24 or more users experience weight gain as a side effect, we can use the normal approximation to the binomial distribution with the same mean and standard deviation as before. Then, we can use the normal complementary cumulative distribution function (normalccdf) to calculate the probability that X ≥ 24.
normalccdf(23.5,15,3.814) = 0.0097
Therefore, the approximate probability that 24 or more users experience weight gain as a side effect is 0.0097.
To approximate the probability that between 12 and 20 patients, inclusive will experience weight gain as a side effect, we can use the normal approximation to the binomial distribution with the same mean and standard deviation as before. Then, we can use the normal cumulative distribution function (normalcdf) to calculate the probability that 12 ≤ X ≤ 20.
normalcdf(11.5,20.5,15,3.814) = 0.6081 - 0.1934 = 0.4147
Therefore, the approximate probability that between 12 and 20 patients, inclusive will experience weight gain as a side effect is 0.4147.
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#4- Find the volume of the right prism. Round your answer to two decimal places, if necessary.
Thank you
I’m a bit confused. I know the formula is V=Bh
The base is the 2 rectangles on the side right? I just can’t find the height.
To find the volume of the right prism, we used the Pythagorean theorem to determine the height of the triangular base is 1.197 inches. We then used the formula V = Bh to calculate the volume, which was approximately 2.70 cubic inches.
To find the height of the prism, we need to use the information provided about the triangular base. Since the triangular base is equilateral with a dimension of 1.74 inches, the height of the triangle (and therefore, the height of the prism) can be found by using the Pythagorean theorem.
If we draw a line from the center of the base to the midpoint of one of the sides, we create a right triangle with hypotenuse 1.74 in (which is also the height of the triangle) and one leg equal to half the length of one of the sides of the triangle (since the base of the prism is a square with dimension 1.5 in).
Using the Pythagorean theorem, we can solve for the height of the triangle (and prism)
(1.74/2)² + (1.5/2)² = h²
0.8725 + 0.5625 = h²
h² = 1.435
h ≈ 1.197 inches
Now, we can use the formula V = Bh to find the volume of the prism
V = (1.5 x 1.5) x 1.197 ≈ 2.70 cubic inches
Therefore, the volume of the right prism is approximately 2.70 cubic inches.
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Dr. Searcy was entering grades for the last summative test into his gradebook. Here are the scores.
90, 88, 95, 98, 85, 82, 92, 75, 82, 65, 97, 85
What is the range? And explain it.
Answer:
The range of a set of data is the difference between the highest and lowest values. In this case, the highest value is 98 and the lowest value is 65, so the range is 98-65 = 33. This means that the scores on the test ranged from 65 to 98, a difference of 33 points.
The range is a measure of the spread of the data. In this case, the range is relatively large, which means that the scores were spread out over a wide range of values. This suggests that the test was challenging and that there was a wide range of student abilities.
The density function of the random variable X is:
-
p(x) =
=
0,
x <1;
1
(x - 1), 1
12
1
3< x < 6;
6
5 1
x, 6< x <10;
12 24
0,
x >10
1
X
a)Make a drawing showing the value of the function depending on the detection area.
b)Write down the corresponding calculation formula and find the average value. (Convert conversions and calculations in detail.)
The expected value of X is 8.
a) Here is a sketch of the density function p(x) with respect to the detection area:
|
|
|
|
|
|
|
|
|
_____________|_____________
1 1.5 3 6 10
b) The formula for the expected value (or mean) of a continuous random variable X with density function p(x) is:
E(X) = ∫xp(x)dx
To find the expected value of X for the given density function, we need to split the integral into several parts based on the different intervals where p(x) takes different forms:
E(X) = ∫_(-∞)^1 xp(x)dx + ∫_1^2 xp(x)dx + ∫_2^3 xp(x)dx + ∫_3^6 xp(x)dx + ∫_6^10 xp(x)dx + ∫_10^∞ xp(x)dx
Note that the first and last integrals are both zero, since p(x) = 0 for x < 1 and x > 10. The other integrals can be evaluated as follows:
∫_1^2 xp(x)dx = ∫_1^2 (x-1)dx = [x^2/2 - x]_1^2 = 1/2
∫_2^3 xp(x)dx = ∫_2^3 (x-1)dx = [x^2/2 - x]_2^3 = 3/2
∫_3^6 xp(x)dx = ∫_3^6 (1/3)dx = 1
∫_6^10 xp(x)dx = ∫_6^10 (x/12)dx = [x^2/24]_6^10 = 5/2
Therefore, we have
E(X) = 0 + 1/2 + 3/2 + 1 + 5/2 + 0 = 8
So the expected value of X is 8.
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I don't understand this problem
Answer:
1. a
2. a
3. b
Step-by-step explanation:
Q1. The equation is 3x+6=30. b c d are all good answers so it has to be a. If you want more details on why a is wrong, if you expand it becomes 3x+18=30 which is wrong
Q2. This one is 4(x+6) = 40 because you have x+6 4 times and it tells you the total is 40. So the one that matches is a.
Q3.
You have 6 plus 4 x which makes a total of 40.
So 6 + 4x = 40. The equation that matches is b.
Hmu if you need more explanation
Pls Reply before tommorrow
1. A bathtub is being filled at a rate of 2.5 gallons per minute. The bathtub will
hold 20 gallons of water.
a. How long will it take to fill the bathtub?
b. Is the relationship described linear, inverse, exponential, or neither? Write
an equation relating the variables.
2. Suppose a single bacterium lands on one of your teeth and starts reproducing
by a factor of 4 every hour.
a. After how many hours will there be at least 1,000,000 bacteria in the new
colony?
b. Is the relationship described linear, inverse, exponential, or neither? Write
an equation relating the variables.
1.
a.
It will take 8 minutes to fill the bathtub.
b.
The relationship described is linear.
The equation is 20 = 2.5t + 0.
2.
a.
It will take approximately 4.807 hours to have at least 1,000,000 bacteria in the new colony.
b.
The relationship described is exponential,
The equation is Number of bacteria = initial number of bacteria x (reproduction factor)^(time/hour)
We have,
1.
a.
To fill the bathtub, we need 20 gallons of water.
The rate at which the water is being filled is 2.5 gallons per minute.
Using the formula:
time = amount/rate
we get:
time = 20/2.5 = 8 minutes
b.
The relationship described is linear.
The equation relating the variables can be written as:
amount of water = rate x time + initial amount
where the rate is 2.5 gallons per minute, the initial amount is 0 gallons, and the amount of water is 20 gallons.
So, the equation is:
20 = 2.5t + 0
where t is the time in minutes.
2.
a.
The relationship described is exponential.
The equation relating the variables can be written as:
number of bacteria = initial number of bacteria x (reproduction factor)^(time/hour)
where the initial number of bacteria is 1, the reproduction factor is 4, and we need to find the time it takes to reach 1,000,000 bacteria.
So, we have:
1,000,000 = 1 x 4^(time/hour)
Taking the logarithm of both sides, we get:
log(1,000,000) = log(4^(time/hour))
6 = (time/hour) x log(4)
time/hour = 6/log(4)
time = (6/log(4)) x hour
time ≈ 4.807 hours
b.
The relationship described is exponential, and the equation relating the variables is:
Number of bacteria = initial number of bacteria x (reproduction factor)^(time/hour)
where the initial number of bacteria is 1, the reproduction factor is 4, and t is the time in hours.
Thus,
1.
a.
It will take 8 minutes to fill the bathtub.
b.
The relationship described is linear.
The equation is 20 = 2.5t + 0.
2.
a.
It will take approximately 4.807 hours to have at least 1,000,000 bacteria in the new colony.
b.
The relationship described is exponential,
The equation is Number of bacteria = initial number of bacteria x (reproduction factor)^(time/hour)
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I need help fast please
The probability that the person chosen belonged to Group Y is 69/164.
As, Out of 200 persons in the sample, those having at least one dream are 200− those who had no dream are
= 200−36
=164
Now, out of 164 people belonged to group Y
= 100−21
=79
So, the probability that the person chosen belonged to Group Y become
= 69/164
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need extreme help with my math
Given that a ski set is being sold at 10% discount at $325, we need to find its original price,
Let the original price be x,
Therefore,
90% of x = 325
0.9x = 325
x = 361.11
Hence the original price of the ski set is $361.11.
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Can you please help me with these three problems? I’m really confused about this unit.
The angles are 11°, 42° and 35°.
Given are circles, we need to find the missing angles,
1) ∠1 = 1/2 [119° - (360° - (119°+174°)]
= 1/2 [119° - 97°]
∠1 = 11°
2) ∠1 = 1/2[360°-138°-138°]
∠1 = 1/2 x 84
∠1 = 42°
3) ∠1 = 1/2[111°-360°-(111°+104°+104°)]
∠1 = 1/2 x 70
∠1 = 35°
Hence the angles are 11°, 42° and 35°.
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Need help asap I don’t understand at all please nd thanks
The two points that a line of best fit would go through would be B. (3, 5) and ( 5, 6 ).
Why would a line of best fit go through these ?The line of best fit is determined by the data points that have the least sum of squared distances from the line. These chosen points provide a clear representation of the general trend of the data and effectively facilitate precise future predictions concerning upcoming data points.
The line of best fit would therefore go through (3, 5) and ( 5, 6 ) because it would lead to four points being above the line, and four points being below which would be a good line of best fit.
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A fenced backyard has a length
of 20 feet, and width of 25 feet,
and a diagonal of 30 feet. Does
the backyard have a 90 degree
angle in its corner?
Answer:no it doesn’t it makes a trapezoid which doesn’t have 90 degree angles or right angles
Step-by-step explanation:
The alternating series test can be used to show convergence of which of the following alternating series?I. 4−19+1−181+14−1729+116−...,+an+...,where an={82nif n is odd−13nif n is evenII. 1−12+13−14+15−16+17−18+...+an+...,where an(−1)n+1nIII. 23−35+47−59+611−713+815−...+an+...,where an=(−1)n+1n+12n+1(A) I only(B) II only(C) III only(D) I and II only(E) I, II, and III
The alternating series test can be used to show convergence of the alternating series I, II, and III given in the options and the correct answer to this question is Option A. I only.
The alternating series test is a method used to determine the convergence or divergence of alternating series. According to the alternating series test, an alternating series converges if the absolute value of its terms decreases monotonically to zero. In other words, if the absolute value of the terms in an alternating series eventually becomes smaller and smaller until it is less than or equal to a certain positive number, then the series converges.
In series, I, the absolute value of the terms decreases monotonically to zero since the terms eventually become smaller and smaller. Therefore, series I converge by the alternating series test.In series II, the absolute value of the terms does not decrease monotonically to zero, since the terms eventually increase in magnitude. Therefore, the alternating series test cannot be used to show the convergence or divergence of series II.In series III, the absolute value of the terms decreases monotonically to zero since the terms eventually become smaller and smaller. Therefore, series III converges by the alternating series test.In conclusion, the alternating series test can be used to show the convergence of series I and III, but not for series II. Therefore, the answer is (A) I only.
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on a roulette table, you can bet on a single number (a single square, corresponding to a single slot on the wheel). there are 38 numbers, so the odds are 37 to 1 against you. if you risk $1 on a single square and win, you get it back plus $35 in winnings. (a) if you bet on a single number for 25 rounds, what is the expected value of your net gain? (b) the standard deviation of your net gain? (c) estimate the chance you come out ahead.
(a) If you bet on a single number for 25 rounds, the expected value of your net gain is -$1.32.
(b) The standard deviation of your net gain is 28.0126.
(c) The estimate the chance you come out ahead is 48.17%.
(a) The probability of winning a single bet is 1/38 and the expected net gain for each bet is -1 + (35/1)1/38 = -0.0526. Therefore, the expected value of your net gain after 25 rounds is 25(-0.0526) = -$1.32.
(b) The variance of the net gain for each bet is [(-1 - (-0.0526))^2*(37/38) + (35 - (-0.0526))^2*(1/38)] = 31.3728. So, the variance of the net gain for 25 rounds is 25*31.3728 = 784.32, and the standard deviation is the square root of the variance, which is 28.0126.
(c) The chance of coming out ahead can be estimated using the normal distribution with mean -1.32 and standard deviation 28.0126. We want to find the probability that the net gain is greater than zero, which is equivalent to finding the probability that a standard normal random variable Z is greater than (0 - (-1.32))/28.0126 = 0.0471.
Using a standard normal table or calculator, we find that this probability is approximately 0.4817 or 48.17%. So, there is about a 48.17% chance of coming out ahead after 25 rounds of betting on a single number.
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What is the value of the expression −3 1/3÷(−2.4) ?
Answer:
First, we need to convert the mixed number −3 1/3 to a fraction. −3 1/3 = −(3 + 1/3) = −(10/3).
Now, we can divide the fraction by the decimal. −(10/3) ÷ (-2.4) = −(10/3) ÷ (-24/10) = −(10/3) x (10/-24) = 10/-7.2 = -1.388888889.
Therefore, the value of the expression is −1.388888889.
Step-by-step explanation:
1. Convert the mixed number to a fraction.
```
-3 1/3 = -(3 + 1/3) = -(10/3)
```
2. Multiply the numerator and denominator of the fraction by -1.
```
-(10/3) = (-1)(10/3) = -10/3
```
3. Divide the numerator and denominator of the fraction by -24.
```
-10/3 = (-10/3) ÷ (-24/10) = 10/-7.2 = -1.388888889
```
Therefore, the value of the expression is −1.388888889.
Here is a visual representation of the steps:
```
-3 1/3 ÷ (-2.4)
= -(10/3) ÷ (-24/10)
= -(10/3) x (10/-24)
= 10/-7.2
= -1.388888889
```
You purchase a box of 50 scarves wholesale for $7. 00 per scarf. If you then resell each scarf at an 18% markup,
1.26 more than the scarf originally so 8.26 for each scarf and 430 total made from the markup and the total profit is 63 dollars.
14 1 point If two parents are homozygous for a genetically inherited recessive trait, what is the probability that they will have a child who does not have this trait in his or her phenotype?
The child will always have the recessive trait in their phenotype.
If both parents are homozygous for a recessive trait, it means they both carry two copies of the recessive allele. Let's assume that the dominant allele is represented by 'A' and the recessive allele by 'a'. Since both parents are homozygous for the recessive trait, their genotype must be 'aa'.
When these parents have children, they will each contribute one 'a' allele, resulting in all of their children inheriting the recessive allele. The probability that their child will have the trait is therefore 100%. The probability of not inheriting the trait is 0%.
Therefore, the answer to the question is 0%. The child will always have the recessive trait in their phenotype.
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Circumference of 1/8th of a circle. There is 1/8 of a circle. The radius of the circle is 30 centimeters. The radius of the circle is 30 centimeters. Find the distance of the figure. Give steps
The circumference of 1/8th of a circle with a radius of 30 centimeters is 11.78 centimeters.
The formula for the circumference of a circle is C = 2πr, where C is the circumference, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.
To find the circumference of 1/8th of a circle, we need to divide the circumference of the full circle by 8. So, the formula becomes:
C = (1/8) * 2πr
Substituting the given value of the radius, we get:
C = (1/8) * 2π(30)
Simplifying, we get:
C = (1/4) * π(30)
C = (1/4) * 30π
C = 7.5π
Approximating π as 3.14, we get:
C = 7.5 * 3.14
C = 23.55/2
C = 11.78
Therefore, the circumference of 1/8th of a circle with a radius of 30 centimeters is 11.78 centimeters.
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Complete Question:
Circumference of 1/8th of a circle. There is 1/8 of a circle. The radius of the circle is 30 centimeters. The radius of the circle is 30 centimeters. Find the distance of the figure. Give steps.
calculate the slope of the lines that pass through 5,-8 and -3,-4
Slope of the lines that pass through points (5,-8) and (-3,-4) is -1/2
The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is
m=y₂-y₁/x₂-x₁
We have to find slope of the lines that pass through (5,-8) and (-3,-4)
Slope = -4-(-8)/-3-5
=-4+8/-8
=-4/8
=-1/2
Hence, slope of the lines that pass through (5,-8) and (-3,-4) is -1/2
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You flip a coin.
What is P(heads)?
The calculated value of the probability P(head) is 0.5 i.e. one half
How to determine P(heads).From the question, we have the following parameters that can be used in our computation:
Sections = 2
Sections = head and tail
Using the above as a guide, we have the following:
Head = 1
When the head section is flipped, we have
P(head) = head/section
The required probability is
P(head) = 1/2
Evaluate
P(head) = 0.5
Hence, the value is 0.5
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For the IVP: (t-4) cos ty" – In(t-1)y'+√7+5y=e-', y(2) = 1, y'(2) = 1 determine the largest interval in which the solution is certain to exist
a. (-5,4)
b. (π/2,4)
c. (1,[infinity])
d. (1,π/2)
We can conclude that the largest interval in which the solution is certain to exist is (1,π/2).
To determine the largest interval in which the solution is certain to exist, we need to check the coefficients and initial values for any discontinuities or singularities.
Notice that the coefficient of the second derivative term, (t-4)cos(ty''), becomes zero at t=4, which can cause a singularity in the solution. Moreover, the coefficient of the first derivative term, In(t-1), becomes negative for t<1, which can cause instability issues in the solution.
Since the initial value problem is given for t=2, the interval of certain existence must contain t=2. Therefore, we can eliminate option a (-5,4) and option b (π/2,4) since neither of them contain t=2.
For option c (1,[infinity]), the coefficient of the first derivative term becomes negative for t<1, which violates the condition for the existence of a solution. Therefore, option c can also be eliminated.
The only remaining option is d (1,π/2). This interval contains t=2 and does not cause any discontinuity or instability issues in the coefficients. Therefore, we can conclude that the largest interval in which the solution is certain to exist is (1,π/2).
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Directions: Answer the following questions. Use the text entry box or file uploads to submit your answers.
1. How many hours and minutes elapsed from 8:00 a.m. to 2:30 p.m.?
2. How many hours and minutes elapsed from 7:40 p.m. to 1:10 a.m.?
3. How many hours and minutes elapsed from 12:00 noon to 4:59 p.m.?
4. How many hours and minutes elapsed from 1:23 a.m. to 7:35 a.m.?
5. How many hours and minutes elapsed from 11:28 p.m. to 5:30 a.m.?
The hours and minutes elapsed from 8:00 a.m. to 2:30 p.m is 6 hours and 30 minutes.
How to explain the TimeThe hours and minutes elapsed from 7:40 p.m. to 1:10 a.m. is 5 hours and 30 minutes.
The hours and minutes elapsed from 12:00 noon to 4:59 p.m is 4 hours and 59 minutes.
The hours and minutes that elapsed from 1:23 a.m. to 7:35 a.m is 6 hours and 12 minutes.
The hours and minutes that belapsed from 11:28 p.m. to 5:30 a.m is 6 hours and 2 minutes.
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The base of this right triangular prism is a right triangle with legs that are 7 in. and 8 in. The height of the prism is 5 in.
What is the volume of this right triangular prism?
plsss help
Step-by-step explanation:
Area of base ( 1/2 * L1 * L2 ) * height = volume
1/2 ( 7)(8) * 5 = 140 in^3
If 5 liters of a solution are 20% acid, how much of the solution is acid?
0. 2 liters
1 liter
2 liters
Answer:
0.2
Step-by-step explanation:
1. Find any extrema or saddle points of f(x,y) = x^3 + 12xy - 3y^2 - 27x + 34 2. A company plans to manufacture closed rectangular boxes that have a volume of 16 ft? Without using Lagrange multipliers, find the dimensions that will minimize the cost if the material for the top and bottom costs twice as much as the material for the sides
The dimensions that minimize the cost subject to the volume constraint are [tex]L = 4 ft, W = 2 ft,[/tex] and [tex]H = 2 ft[/tex] using surface area.
Assuming that the cost of material is proportional to the surface area, we can write the cost function as:
[tex]C = k(2LW + 2LH + WH)[/tex]
where k is a constant of proportionality that depends on the cost of the material. We are given that the cost of the material for the top and bottom is twice the cost of the material for the sides, so we can take k = 3 for simplicity (since the cost of the material for the sides is then 1).
Using the volume constraint as before, we can eliminate one of the variables:
[tex]H = 16/LW[/tex]
When this is used as a cost function substitution,
[tex]C = 3(2LW + 2LH + WH) = 6LW + 96/L + 48/W[/tex]
To find the critical points of C, we need to find where the partial derivatives are zero:
[tex]dC/dL = 6W - 96/L^2 = 0[/tex]
[tex]dC/dW = 6L - 48/W^2 = 0[/tex]
When we simultaneously solve these equations, we obtain:
L = 4 ft
W = 2 ft
H = 2 ft
Therefore, the dimensions that minimize the cost subject to the volume constraint surface area are L = 4 ft, W = 2 ft, and H = 2 ft.
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can someone help me with this?? it’s properties of quadratic relations
The table should be completed with the correct key features as follows;
Axis of symmetry (1st graph): x = 1.
Vertex (1st graph): (1, -9).
Minimum (1st graph): -9.
y-intercept (1st graph): (0, -8).
Axis of symmetry (2nd graph): x = 2.
Vertex (2nd graph): (2, 16).
Maximum (2nd graph): 16.
y-intercept (2nd graph): (0, 12).
What is the graph of a quadratic function?In Mathematics and Geometry, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. Based on the first graph of a quadratic function, we can logically deduce that the graph is an upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero (0).
Based on the second graph of a quadratic function, we can logically deduce that the graph is a downward parabola because the coefficient of x² is negative and the value of "a" is less than zero (0).
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Si la ciudad de Dallas tiene un impuesto sobre las ventas del 9,75 % en todas las compras en línea, ¿cuál es el costo total cuando compras un artículo en línea que cuesta $200,00?
The total cost of the online purchase of $200.00 in Dallas, including the 9.75% sales tax is approximately $219.50.
To calculate the total cost, we first need to find the amount of sales tax. We do this by multiplying the cost of the item by the sales tax rate:
$200.00 x 0.0975 = $19.50
Then, we add the sales tax amount to the cost of the item to get the total cost:
$200.00 + $19.50 = $219.50
Therefore, the total cost of the online purchase of $200.00 in Dallas, including the 9.75% sales tax, is $219.50.
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Complete Question:
If the City of Dallas has a 9.75% sales tax on all online purchases, what is the total cost when you buy an item online that costs $200.00?
Which of the following formulas is the correct one to calculate the variance of a probability distribution?μ = nπσ2 = Σ[(x - μ)2 P(X)]number of trials and P(success)
The correct formula to calculate the variance of a probability distribution is σ2 = Σ[(x - μ)2 P(X)].
The formula is σ2 = Σ[(x - μ)2 P(X)],
where σ2 represents the variance, Σ represents the sum of, x represents the possible outcomes, μ represents the mean or expected value of the distribution, and P(X) represents the probability of each outcome.
The number of trials and the probability of success are not directly involved in this formula, but they may be used to calculate the probabilities of each outcome.
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4. ([2]) Find the radius of convergence R of the series 2n=1 (22)" n2
The radius of convergence comes out as 1/4. To get the radius of convergence, we can use the ratio test.
Step:1. Let's call the nth term in the series a_n, where a_n = 2^(2n)/n^2.
Step:2. Using the ratio test, we take the limit as n approaches infinity of |a_(n+1)/a_n|:
|a_(n+1)/a_n| = (2^(2(n+1))/(n+1)^2) * (n^2/2^(2n))
Step:3. Simplifying this expression, we can cancel out the 2^n terms and get: |a_(n+1)/a_n| = 4((n^2)/(n+1)^2)
Step:4. Taking the limit as n approaches infinity, we get:
lim n→∞ |a_(n+1)/a_n| = 4
Since this limit is less than 1, the series converges.
Step:5. Now we just need to find the radius of convergence, which is given by:
R = 1/lim sup n→∞ |a_n|^(1/n)
Step:6. Taking the limit superior of |a_n|^(1/n), we get:
lim sup n→∞ |a_n|^(1/n) = lim sup n→∞ (2^(2n)/n^(2n/n))^(1/n)
= lim sup n→∞ 2^2 = 4
So the radius of convergence is:
R = 1/lim sup n→∞ |a_n|^(1/n) = 1/4
Therefore, the radius of convergence is 1/4.
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Exercises : Find a solution for the following an (1 а. a = 1 an = n 2 anni +1 (2) a = 1, 9, = 2, 11 Van an-z 4 a n- n 2 2 (3) Hard Problem *te a = 6, 0,= 17, a +5na, +6nen-ida n-1 M-2
For problem 1, the solution is an = n.
For problem 2, the solution is an = 3n - 1.
For problem 3 (the hard problem), we can solve for the values of a, b, and c in the quadratic equation: [tex]an^2 + bn + c = 0[/tex], where a = 5, b = 6n - 1, and c = -2.
Using the quadratic formula, we get:
[tex]n= \frac{-b±\sqrt{b^{2}-4ac } }{2a}[/tex]
Substituting the values of a, b, and c, we get:
[tex]n= \frac{-(6n-1)±\sqrt{(6n-1)^{2}-4(5)(-2) } }{2(5)}[/tex]
Simplifying, we get:
[tex]n = \frac{(-6n+1 ± \sqrt{36n^{2}-48n+49 } ) }{10}[/tex]
Therefore, the solution for problem 3 is:
[tex]an= 5n^{2} + \frac{-6n+1 + \sqrt{36n^{2}-48n+49 } }{10}[/tex]
or
[tex]an= 5n^{2} + \frac{-6n+1 - \sqrt{36n^{2}-48n+49 } }{10}[/tex]
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10. What is the radius of a sphere with a volume of 4186 in³ to the nearest tenth of an inch?
Answer:10
Step-by-step explanation:
10
Solve the separable differential equation dy / dx = − 0. 6 y , and find the particular solution satisfying the initial condition y (0) = − 9. Y(x) =___
The differential equation solution is y(x) = [tex]-9e^_{(-0.6x)[/tex] for the initial condition y(0) = -9.
The given differential condition is dy/dx = -0.6y. To address this condition, we can isolate the factors by partitioning the two sides by y and duplicating the two sides by dx:
dy/y = -0.6dx
Then, we can incorporate the two sides. On the left side, we get ln|y|, and on the right side, we get -0.6x+C, where C is an inconsistent steady of joining:
ln|y| = -0.6x+C
To find the specific arrangement that fulfills the underlying condition y(0) = -9, we can substitute x = 0 and y = -9 into the situation:
ln|-9| = -0.6(0)+C
Rearranging, we get:
C = ln(9)
In this manner, the specific arrangement that fulfills the underlying condition is:
y(x) = [tex]-9e^_{(-0.6x)[/tex]
This is the last response.
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The complete question is:
Solve the separable differential equation dy/dx = 0.9y, and find the particular solution satisfying the initial condition y(0) = -9. y(x) = e^((0.9/2)x^2-ln(9)).