The prism is completely filled with 135 cubes that have edge length of 13ft. What is the volume of the prism?Enter your answer in the box

The probability of choosing a vowel from the word PROBABILITY, expressed as a decimal, is approximately 0.364.

To find the probability of choosing a vowel from the word PROBABILITY, you'll need to follow these steps:

1. Identify the total number of letters in the word: There are 11 letters in the word PROBABILITY.
2. Identify the number of vowels in the word: There are 4 vowels (O, A, I, and I).
3. Calculate the probability by dividing the number of vowels by the total number of letters: Probability = (number of vowels) / (total number of letters) = 4/11.

A decimal indication that the probability of selecting a vowel from the word PROBABILITY is roughly 0.364.

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The probability of being assigned all hardback books is approximately 0.33 or 33%.

This can be done by combination formula C(n,r) = n!/(r!(n-r)!)

The total number of ways to choose three books from a list of 14 books is given by the combination formula,

C(14,3) =14!/(3!(14-3)!) =  (141312) / (321) = 364.

To find the probability of selecting all hardback books, we need to determine the number of ways to select 3 books from the 10 hardback books. This is given by the combination formula,

C(10,3) = 10!/(3!(10-3)!) = 1098 / (321) = 120.

Therefore, the probability of selecting all hardback books is:

P(all hardback) = C(10,3) / C(14,3) = 120/364 = 0.3297

So, the probability of being assigned all hardback books is approximately 0.33 or 33%. This means that out of all possible combinations of 3 books, about 33% will consist of only hardback books.

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La probabilidad de sacar un producto defectuoso al azar de las tres cajas es aproximadamente 0.0917, o un 9.17%.

La probabilidad de sacar un producto defectuoso al azar de las tres cajas se puede calcular utilizando la fórmula de la probabilidad.

Primero, calculemos la probabilidad de sacar un producto defectuoso de cada caja:

1. En la primera caja, hay 3 productos defectuosos entre 20 productos en total. La probabilidad es 3/20.

2. En la segunda caja, hay 2 productos defectuosos entre 16 productos en total. La probabilidad es 2/16.

3. En la tercera caja, no hay productos defectuosos entre 10 productos en total. La probabilidad es 0/10.

Para encontrar la probabilidad total, sumamos las probabilidades de cada caja y luego dividimos por el número total de cajas:

(3/20 + 2/16 + 0/10) / 3 ≈ (0.15 + 0.125 + 0) / 3 ≈ 0.0917

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Answer:

image dosent load

Step-by-step explanation:

Answer:

a = 80°, b = 100°, c = 80°, d = 100°

Step-by-step explanation:

You know a + b + c + d = 360°.

Plugging b + c + d = 280° into the formula above:

a + 280° = 360°.

a = 80°.

You know a + d = 180° because of angles on a straight line. Solve for d:

80° + d = 180°.

d = 100°.

For the same reason, you know a + b = 180°. b must be equal to d, which is 100°.

Lastly, c + d = 180° because they make up the second line. Solve for c:

c + 100° = 180°.

c = 80°.

a = 80°, b = 100°, c = 80°, d = 100°.

The required answer is Jocelyn travels approximately 0.83 miles in 5 minutes.

Jocelyn's car tires are spinning at a rate of 120 revolutions per minute. If her car's tires are 28 inches in diameter, we can calculate the distance traveled in one revolution by finding the circumference of the tire:

Circumference = π x diameter
Circumference = 3.14 x 28 inches
Circumference ≈ 87.92 inches

So in one revolution, the car travels approximately 87.92 inches. To find out how many miles Jocelyn travels in 5 minutes, we need to multiply the number of revolutions in 5 minutes (which is 120 revolutions per minute x 5 minutes = 600 revolutions) by the distance traveled in one revolution (87.92 inches).

Distance traveled in 5 minutes = 600 revolutions x 87.92 inches/revolution

Distance traveled in 5 minutes = 52,752 inches

To convert inches to miles, we can use the conversion factor given: 1 mile = 63,360 inches.

Distance traveled in 5 minutes = 52,752 inches ÷ 63,360 inches/mile

Distance traveled in 5 minutes ≈ 0.83 miles

Therefore, Jocelyn travels approximately 0.83 miles in 5 minutes with her car tires spinning at a rate of 120 revolutions per minute. Rounded to the nearest hundredth, the answer is 0.83 miles.
To find out how many miles Jocelyn travels in 5 minutes, follow these steps:

1. Calculate the circumference of one tire: Circumference = Diameter × π.
Circumference = 28 inches × π ≈ 87.96 inches.

2. Determine the distance traveled in one revolution: One revolution covers the circumference of the tire, which is 87.96 inches.

3. Calculate the distance traveled in one minute: 120 revolutions per minute × 87.96 inches per revolution ≈ 10,555.2 inches per minute.

4. Determine the distance traveled in 5 minutes: 10,555.2 inches per minute × 5 minutes = 52,776 inches.

5. Convert the distance from inches to miles: 52,776 inches ÷ 63,360 inches per mile ≈ 0.83 miles.

So, Jocelyn travels approximately 0.83 miles in 5 minutes.

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Therefore, the correct sequence of transformations is:

Dilation by a scale factor of 2, centered at the origin.

Translation of 7 units to the left.

This is equivalent to option D.

To move AABC onto ADEF, we need to perform a sequence of transformations that will map each point of AABC onto the corresponding point of ADEF. Let's first label the vertices of AABC and ADEF as shown:

We can see that A and D have the same coordinates, so we only need to map B, C, and A' to E, F, and D', respectively.

Option A suggests a dilation by a scale factor of 2, centered at the origin, followed by a reflection over the y-axis. This transformation will map B to E, but it will also map C to C' which is not the same as F. Moreover, A' will be mapped to a point that is reflected over the y-axis and does not coincide with D. Therefore, option A is not the correct answer.

Option B suggests a translation of 7 units to the right, followed by a dilation by a scale factor of 2 centered at the origin. This transformation will not map B to E, but it will map A' to D', and C to a point that is 7 units to the right of F. Therefore, option B is not the correct answer either.

Option C suggests a dilation by a scale factor of 2, centered at the origin, followed by a translation of 7 units to the right. This transformation will map A' to D', and B to a point that is 7 units to the right of E. However, it will also map C to a point that is 7 units to the right of F, which is not the same as F. Therefore, option C is not the correct answer.

Option D suggests a dilation by a scale factor of 2, centered at the origin, followed by a translation of 7 units to the left. This transformation will map B to E, and A' to D', but it will also map C to a point that is 7 units to the left of F. However, if we reflect the resulting figure over the y-axis, we obtain the desired result.

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50% is the percentage the snake is longer if the new measurement is 111 inches and the previous one was 74 inches.

The growth percent refers to the percent of the growth in the length of the snake. It is calculated by the growth divided by the original length multiplied by 100.

Growth rate = [tex]\frac{L-l}{l}*[/tex] 100

where L is the new length

l is the original length

L = 111 inches

l = 74 inches

Growth = 111 - 74

= 37 inches

Growth percent = [tex]\frac{37}{74}[/tex] * 100

= 50%

The growth rate of Abigail's pet snake comes out to be 50%

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The weight of the heaviest 20% of fruits can be estimated using the z-score corresponding to the 80th percentile, which is 0.84.

The weight can be calculated by adding the z-score multiplied by the standard deviation to the mean weight. Therefore, the weight of the heaviest 20% of fruits is approximately 405 grams.

To calculate this, we first need to find the z-score for the 80th percentile, which is 0.84. Then, we can use the formula z = (x - mu) / sigma, where x is the weight we want to find, mu is the mean weight, and sigma is the standard deviation.

Rearranging this formula to solve for x, we get x = z * sigma + mu. Plugging in the values we have, we get x = 0.84 * 20 + 374 = 405.2 grams. Rounding this to the nearest gram, we get the answer of 405 grams.

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The percent of error in the measurement is 8%. This means that the measured value of 5.52 miles is 8% less than the actual distance of 6 miles.

To find the percent of error, we need to calculate the difference between the measured value and the actual value, divide that by the actual value, and then multiply by 100 to convert to a percentage.

Actual distance around the neighborhood = 6 miles

Measured distance around the neighborhood = 5.52 miles

Difference = Actual distance - Measured distance

Difference = 6 miles - 5.52 miles

Difference = 0.48 miles

Percent of error = (|Difference| / Actual distance) x 100%

Percent of error = (|0.48| / 6) x 100%

Percent of error = 0.08 x 100%

Percent of error = 8%

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Answer:

United States medals to Soviet Union medals:

Fraction: 1211/1010

Decimal: 1.198

Percent: 119.8%

Soviet Union medals to Great Britain medals:

Fraction: 1010/867

Decimal: 1.165

Percent: 116.5%

United States medals to Great Britain medals:

Fraction: 1211/867

Decimal: 1.397

Percent: 139.7%

The number of countries that have won between 2,250 and 2,499 total medals to the number of countries that have won between 0 and 249 total medals:

Fraction: 2/1

Decimal: 2

Percent: 200%

Only one country participating in the Summer Olympics has never won a medal. Write a comparison of the number of countries that have never won a medal to the number of participating countries:

Fraction: 1/205

Decimal: 0.00488

Percent: 0.488%

Step-by-step explanation =

United States medals to Soviet Union medals:

The fraction represents the ratio of medals won by the United States to those won by the Soviet Union. To find it, you can divide the number of medals won by the United States (1,211) by the number of medals won by the Soviet Union (1,010): 1211/1010.

To convert this fraction to a decimal, divide the numerator (1211) by the denominator (1010): 1.198.

To convert the decimal to a percent, multiply it by 100: 119.8%.

Soviet Union medals to Great Britain medals:

The fraction represents the ratio of medals won by the Soviet Union to those won by Great Britain. To find it, you can divide the number of medals won by the Soviet Union (1,010) by the number of medals won by Great Britain (867): 1010/867.

To convert this fraction to a decimal, divide the numerator (1010) by the denominator (867): 1.165.

To convert the decimal to a percent, multiply it by 100: 116.5%.

United States medals to Great Britain medals:

The fraction represents the ratio of medals won by the United States to those won by Great Britain. To find it, you can divide the number of medals won by the United States (1,211) by the number of medals won by Great Britain (867): 1211/867.

To convert this fraction to a decimal, divide the numerator (1211) by the denominator (867): 1.397.

To convert the decimal to a percent, multiply it by 100: 139.7%.

The number of countries that have won between 2,250 and 2,499 total medals to the number of countries that have won between 0 and 249 total medals:

The fractions represent the ratios of the number of countries that have won between two ranges of total medals. To find these fractions, you need to count the number of countries that fall into each range, and then divide one by the other. According to the information provided, there are 2 countries that have won between 2,250 and 2,499 total medals, and 1 country that has won between 0 and 249 total medals. So the fraction is 2/1.

To convert this fraction to a decimal, divide the numerator (2) by the denominator (1): 2.

To convert the decimal to a percent, multiply it by 100: 200%.

Only one country participating in the Summer Olympics has never won a medal. Write a comparison of the number of countries that have never won a medal to the number of participating countries:

The fraction represents the ratio of the number of countries that have never won a medal to the total number of participating countries. According to the information provided, only one country has never won a medal, and there are 205 participating countries. So the fraction is 1/205.

To convert this fraction to a decimal, divide the numerator (1) by the denominator (205): 0.00488.

To convert the decimal to a percent, multiply it by 100: 0.488%.

The chances of success in cycling across the United States is higher on the central route because more than half of the people who started on this route finished it (52%), while on the other routes the chances of successful completion is lower (route north (0%) south (0.02%).

According to the above, to find the probability of success in crossing the United States from coast to coast, we must divide the number of successful attempts by the total number of attempts as shown below:

North Route

0 ÷ 2 = 0

Center Route

10 ÷ 19 = 0.52

South Route

1 ÷ 39 = 0.025

According to the above, the highest probability of success is found in the central route with 0.52, that is to say that more than half of those who tried this route finished.

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The first statement about the surface area in the question are false, the second statement about surface area is false and the third statement about surface area is true respectively.

1. False. The surface area of a cone is the sum of the areas of the circular base and the curved lateral surface.

2. False. The surface area of a sphere is given by 4πr^2, while the surface area of a cube with side length s=r is 6r^2. Since 4πr^2 is less than 6r^2 for any value of r, the surface area of a sphere is actually less than the surface area of such a cube.

3. True. The surface area of a composite figure is the sum of the surface areas of each individual figure that make up the composite figure.

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1. The number of students choosing walking as their preferred method of transportation is 9. Therefore, the correct option is C.

2. The total students participated in the survey are 20. Therefore, the correct option is B.

3. The percentage of the total students choosing skateboarding is 10%. Therefore, the correct option is A.

4. The percentage of the boys choosing walking is 30%. Therefore, the correct option is A.

5. The percentage of the students who chose biking were girls is 37.5%. Therefore, the correct option is B.

1. The number of students who chose walking as their preferred method of transportation based on the information provided is 9. The total number of students who chose walking is given as 9 (3 boys + 6 girls). Hence the correct answer is option C.

2. The total students who participated in the survey based on the information provided are 20. The total number of students is given as 20 (10 boys + 10 girls). Hence the correct answer is option B.

3. The percentage of the total students who chose skateboarding based on the information provided is 10%. 3 students chose skateboarding out of 20 total students, so the percentage is (3/20) * 100% = 10%. Hence the correct answer is option A.

4. The percentage of the boys who chose walking based on the information provided is 30%. 3 boys chose walking out of 10 total boys, so the percentage is (3/10) * 100% = 30%. Hence the correct answer is option A.

5. The percentage of the students who chose biking were girls based on the information provided is 37.5%. 3 girls chose biking out of 8 total students who chose biking, so the percentage is (3/8) * 100% = 37.5%. Hence the correct answer is option B.

Note: The question is incomplete. The complete question probably is: Given the following data:

Activity Boys Girls Total

Walk 3 6 9

Bike 5 3 8

Skateboard 2 1 3

Total 10 10 20

1. How many students chose walking as their preferred method of transportation? A. 6 B. 3 C. 9 D. none of these. 2. How many total students participated in the survey? A. 10 B. 20 C. 15 D. none of these 3. What percentage of the total students chose skateboarding? A. 10% B. 20% C. 30% D. none of these. 4. What percentage of the boys chose walking? A. 30% B. 45% C. 25% D. none of these. 5. What percentage of the students who chose biking were girls? A. 30% B. 37. 5% C. 45% D. none of these.

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The FICA tax will be $3595.5 on an income of $47,000.

Given that the principal amount = $47,000

Given that the FICA is taxed at the percentage of 7.65%

To findout the FICA tax we have to findout the 7.65% of money from the principal money $47,000.

The formula for finding the Y% of money from Z amount is = [tex]\frac{y}{100}[/tex] * Z

From the above formula, we can find the FICA tax.

FICA tax = [tex]\frac{7.65}{100}[/tex] * 47000 = 0.0765 * 47000 = 3595.5.

From the above solution, we can conclude that the FICA tax on an income of $47,000 is $3595.5

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The total mass of the rod is 81622.5 kg. To find the total mass of the rod, you need to integrate the linear density function with respect to the length of the rod.

To find the total mass of the rod, we need to integrate the linear density function over the entire length of the rod.
Let's start by finding the linear density function at the end of the rod, which is a = 9:
p(9) = 3 + 2017 = 2020 kg/m
Now we can integrate the linear density function from a = 0 to a = 9 to find the total mass:
m = ∫₀⁹ p(a) da
m = ∫₀⁹ (3 + 2017a) da
m = [3a + 1008.5a²] from 0 to 9
m = (3(9) + 1008.5(9)²) - (3(0) + 1008.5(0)²)
m = 81622.5 kg
Therefore, the total mass of the rod is 81622.5 kg.

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The factorization of the given algebraic expression will give us:

4πr²(1 + ¹/₃r)

In order for us to factorize an algebraic expression, the highest common factors of the terms of the given algebraic expression would first be found and then we will group the terms accordingly. In simple terms, the reverse process of expansion of an algebraic expression is referred to as  its factorization.

We are given the algebraic expression as:

4πr² + ⁴/₃πr³

Now, we want to factor 4πr² from both terms and this will give us:

4πr²(1 + ¹/₃r)

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To find the estimated probability that it will take at least five patients to find one patient on whom the medicine would not be effective, we can use the geometric distribution. The probability of not curing the disease is 1 - 0.70 = 0.30.

Let X be the number of patients needed to find one patient on whom the medicine would not be effective. Then X follows a geometric distribution with p = 0.30. The probability that it will take at least five patients to find one patient on whom the medicine would not be effective is:

P(X ≥ 5) = (1 - p)^(5-1) = 0.7^4 ≈ 0.2401

Therefore, the estimated probability that it will take at least five patients to find one patient on whom the medicine would not be effective is approximately 0.2401.

To find the estimated probability that the medicine will be effective on exactly three out of four randomly selected patients, we can use the binomial distribution. Let X be the number of patients out of four who are cured by the medicine. Then X follows a binomial distribution with n = 4 and p = 0.70. The probability that the medicine will be effective on exactly three out of four randomly selected patients is:

P(X = 3) = (4 choose 3) * (0.70)^3 * (1 - 0.70)^(4-3) = 4 * 0.343 * 0.3^1 ≈ 0.4116

Therefore, the estimated probability that the medicine will be effective on exactly three out of four randomly selected patients is approximately 0.4116.

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The test statistic for the hypothesis about a consumer activist decides to test the authenticity of the claim is t = 1.38.

In a hypothesis test, a test statistic—a random variable—is computed from sample data. To decide whether to reject the null hypothesis, you can utilise test statistics. Your results are compared to what would be anticipated under the null hypothesis by the test statistic. The p-value is computed using the test statistic.

A test statistic gauges how closely a sample of data agrees with the null hypothesis. Its observed value fluctuates arbitrarily from one random sample to another. When choosing whether to reject the null hypothesis, a test statistic includes information about the data that is important to consider. The null distribution is the sample distribution of the test statistic for the null hypothesis.

Sample size, n = 20

Sample mean, x = 14.8 pounds

Sample standard deviation, s = 2.6

The null hypothesis is,

[tex]H_o[/tex]: μ ≤ 14

The alternative hypothesis is,

[tex]H_a[/tex] : μ > 14

t-test statistic is defined as:

[tex]t = \frac{x - \mu}{\frac{s}{\sqrt{n} } }[/tex]

[tex]= \frac{14.8 - 14}{\frac{2.6}{\sqrt{20} } }[/tex]

= [tex]\frac{0.8}{0.581}[/tex]

= 1.377

t = 1.38.

Therefore, the test statistic for the hypothesis is 1.38.

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Complete question"

An advertisement for a popular weight-loss clinic suggests that participants in its new diet program lose, on average, more than 14 pounds. A consumer activist decides to test the authenticity of the claim. She follows the progress of 20 women who recently joined the weight-reduction program. She calculates the mean weight loss of these participants as 14.8 pounds with a standard deviation of 2.6 pounds. The test statistic for this hypothesis would be Multiple Choice -1.38 1.38 1.70 -1.70 O O

Identify the best deal for each item

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The smallest angle measures 15 degrees and the other non-right angle measures 75 degrees.

To find the measure of both angles in a right triangle with the given conditions, we will use the information provided:

Let x be the measure of the smallest angle. The problem states that the measure of one of the small angles is 30 less than 7 times the smallest angle, which can be written as:

Other non-right angle = 7x - 30

Since this is a right triangle, the sum of the two small angles must be 90 degrees (because the other angle is 90 degrees, and the sum of angles in a triangle is 180 degrees). So, we can set up the following equation:

x + (7x - 30) = 90

Now, solve for x:

8x - 30 = 90
8x = 120
x = 15

So, the smallest angle is 15 degrees. Now, we can find the measure of the other non-right angle:

Other non-right angle = 7x - 30 = 7(15) - 30 = 105 - 30 = 75 degrees

In summary, the smallest angle measures 15 degrees and the other non-right angle measures 75 degrees.

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Riley and her family move from the Midwest all the way to San Francisco. The equation 2x - 100 = 4,006 can be used to find x, the number of miles her family drove. Riley's family drove 2,053 miles from the Midwest to San Francisco.

Find the number of miles Riley's family drove from the Midwest to San Francisco, we need to solve the equation 2x - 100 = 4,006 for x.
First, we'll add 100 to both sides of the equation to isolate the variable term:
2x - 100 + 100 = 4,006 + 100
Simplifying:
2x = 4,106
isolate x, we'll divide both sides of the equation by 2:
2x/2 = 4,106/2
Simplifying:
x = 2,053
Therefore, Riley's family drove 2,053 miles from the Midwest to San Francisco.

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If the surface area of a can with a particular volume is minimized when the height of the can is 22 cm, we would expect the radius of the can to be the same as the height, given that a cylinder has the smallest surface area when its height and radius are equal.

The surface area of a can with height h and radius r can be given by the formula:

A = 2πr² + 2πrh

The volume of the can is given by:

V = πr²h

If we differentiate the surface area with respect to r and equate it to zero to find the critical point, we get:

dA/dr = 4πr + 2πh(dr/dr) = 0

Simplifying this expression, we get:

2r + h = 0

Since we know that the height of the can is 22 cm, we can substitute h = 22 in the equation to get:

2r + 22 = 0

Solving for r, we get:

r = -11

Since the radius of the can cannot be negative, we discard this solution. Therefore, the radius of the can should be equal to its height, which is 22 cm.

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The domain of the function f(x) = 2x² + 5x - 12 is all real numbers, or (-∞, ∞).

This is because there are no restrictions on the input values of x that would make the function undefined. In other words, we can input any real number into the function and get a valid output.

To determine the domain of a function, we need to consider any restrictions on the independent variable that would make the function undefined.

Common examples of such restrictions include division by zero, taking the square root of a negative number, or taking the logarithm of a non-positive number. However, in this case, there are no such restrictions, and therefore the domain is all real numbers.

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The value of the scale factor for the similar figures is 1/4

From the question, we have the following parameters that can be used in our computation:

The similar figures

The corresponsing sides of the similar figures are

Original = 8

New = 2

Using the above as a guide, we have the following:

Scale factor = New /Original

substitute the known values in the above equation, so, we have the following representation

Scale factor = 2/8

Evaluate

Scale factor = 1/4

Hence, the scale factor for the similar figures is 1/4

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a) It will take approximately 16.85 years for the rabbit population to double.

b) It will take approximately 37.28 years for the rabbit population to reach 1000.

The formula for calculating the population of rabbits in x years, starting with 100 rabbits and a natural growth rate of 4% per year, is given by:

f(x) = 100(1.04)ˣ

(a) To find out how long it will take for the rabbit population to double, we need to solve the following equation:

100(1.04)ˣ = 200

Dividing both sides by 100, we get:

(1.04)ˣ = 2

Taking the logarithm of both sides with base 1.04, we get:

x = log₁.₀₄ 2

Using a calculator, we get:

x ≈ 16.85

(b) To find out how long it will take for the rabbit population to reach 1000, we need to solve the following equation:

100(1.04)ˣ = 1000

Dividing both sides by 100, we get:

(1.04)ˣ = 10

Taking the logarithm of both sides with base 1.04, we get:

x = log₁.₀₄ 10

Using a calculator, we get:

x ≈ 37.28

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Answer:

8 people

Step-by-step explanation:

Tea- 13 people

Coffee- 12 people

Neither- 3 people

20 - 3 = 17

12 + 13 = 25

25 - 17 = 8

The zeros of the quadratic equation are as follows

1. y = -x²+ 6x - 5:

zeros: (1, 0) and (5, 0)

2. y = x² + 2x + 1:

zeros: (-1, 0).

3. y = -x²+ 8x - 17:

zeros: (0, 0) and (0, 0)

4. y = x² - 4:

zeros: (1, 0) and (5, 0)

Zero in a quadratic equation are x values ​​that make the equation equal to zero. In other words, they are the x-intercepts or roots of a quadratic function.

Using graphical method, a zero is the point of intersection of the curve with the x -axis and this is shown in the graph attached.

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The second derivative yⁿ (y'') is 48x² - 6, and the third derivative yᵐ (y''') is 96x.

To calculate the second and third derivatives of the function y = 4x^4 - 3x² + 7x:

1. First, calculate the first derivative, y':
y' = dy/dx = (d/dx)(4x^4 - 3x² + 7x)
Using the power rule for derivatives, we get:
y' = 16x³ - 6x + 7

2. Now, calculate the second derivative, y'' (also denoted as yⁿ when n=2):
y'' = d²y/dx² = (d/dx)(16x³ - 6x + 7)
Applying the power rule again:
y'' = 48x² - 6

3. Finally, calculate the third derivative, y''' (also denoted as yᵐ when m=3):
y''' = d³y/dx³ = (d/dx)(48x² - 6)
Using the power rule one more time:
y''' = 96x

So, the second derivative yⁿ (y'') is 48x² - 6, and the third derivative yᵐ (y''') is 96x.

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Object B was launched with the greatest vertical speed and Object A was launched from the greatest height of 50 meters.

a. The vertical speed of an object launched can be calculated using the derivative of the distance function with respect to time. Taking the derivative of the distance function of Object A with respect to time, we get:

v(t) = -10t + 25

Taking the derivative of the distance function of Object B with respect to time, we get:

v(t) = -10t + 50

Comparing the two velocity functions, we can see that Object B was launched with the greatest vertical speed because its velocity function has a higher initial velocity (50 m/s) than that of Object A (25 m/s).

b. The initial height of an object launched can be determined by finding the value of its distance function when t=0.

For Object A, the distance function when t=0 is:

-5(0)^2 + 25(0) + 50 = 50 meters

For Object B, the distance function when t=0 is:

-5(0)^2 + 50(0) + 25 = 25 meters

Therefore, Object A was launched from the greatest height of 50 meters.

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Choice D is correct: This situation satisfies each of the conditions for a binomial variable, so X has a binomial distribution.

A random variable X is said to have a binomial distribution if it satisfies the following conditions:

The variable X represents the number of successes in a fixed number of independent trials.

Each trial has only two possible outcomes: success or failure.

The probability of success is constant for each trial.

The trials are independent.

In this case, Yoshi attempts the same three-point shot until he makes the shot, so the number of attempts is not fixed. However, each attempt can be classified as a success (if he makes the shot) or a failure (if he misses the shot), so the variable X represents the number of successes in a sequence of independent trials with only two possible outcomes. Also, the probability of success is constant for each attempt, and the attempts are independent, so all four conditions for a binomial distribution are satisfied. Therefore, X is a binomial variable.

Choice D is correct: This situation satisfies each of the conditions for a binomial variable, so X has a binomial distribution.

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