At my university 22% of the students enrolled are 'mature'; that is, age 21 or over. a) If I take a random sample of 5 students from the enrolment register what is the probability that exactly two students are mature?6 (5 marks) b) If I take a random sample of 7 students from the enrolment register what is the probability that exactly two students are mature?

Answer:

Step-by-step explanation:

To solve this problem, we can use the grating equation:

m * λ = d * sin(θ)

Where:

m is the order of the fringe

λ is the wavelength of light

d is the slit spacing (distance between adjacent slits)

θ is the angle of the fringe

In this case, we're given:

m = 3 (third-order fringe)

θ = 24.0°

We need to calculate the slit spacing (d) using the information that the grating has 3200 slits per cm. First, we convert the number of slits per cm to the slit spacing in meters:

slits per cm = 3200

slits per m = 3200 * 100 = 320,000

Now we can calculate the slit spacing (d):

d = 1 / (slits per m)

d = 1 / 320,000

Now, let's substitute the given values into the grating equation and solve for λ (wavelength):

m * λ = d * sin(θ)

3 * λ = (1 / 320,000) * sin(24.0°)

λ = (1 / (3 * 320,000)) * sin(24.0°)

Using a calculator, we can calculate the value of λ:

λ ≈ 5.79 × 10^(-7) meters or 579 nm

Therefore, the wavelength of light for which the grating with 3200 slits per cm produces a third-order fringe at a 24.0° angle is approximately 579 nm.

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Answer:

the new ratio of white, red and blue balls is 6:2:1

Step-by-step explanation:

The computation of the new ratio is shown below:

If you have 10 red balls, so the blue balls is in proportion of 3:2 that means 15 would be the blue balls. In addition to this, the while balls is in proportion 8:2 = 4:1 so there is a 40 white balls

Now if 10 white and 10 blue balls would be eliminated

So there is 30 white, 10 red, 5 blue bals

Also

30 = 5 × 6

And,

10 = 5 × 2

so the new ratio of white, red and blue balls is 6:2:1

Given

Annual profits of four years $8 million, $3 million, $2 million, and $9 million.

In the fifth year, it reported a loss of $7 million.

To find:

The mean annual profit.

Solution:

Formula for mean is:

[tex]\text{Mean}=\dfrac{\text{Sum of observations}}{\text{Number of observations}}[/tex]

Annual profits are $8 million, $3 million, $2 million, $9 million and -$7 million. Here negative sign represent loss.

So, the mean annual profit is:

[tex]\text{Mean}=\dfrac{8+3+2+9+(-7)}{5}[/tex]

[tex]\text{Mean}=\dfrac{22-7}{5}[/tex]

[tex]\text{Mean}=\dfrac{15}{5}[/tex]

[tex]\text{Mean}=3[/tex]

Therefore, the  mean annual profit is $3 million.

Answer:

4x - 3.4xy

Step-by-step explanation:

x(4 - 3.4y)

4x - 3.4xy

Based on the formula, we can say that the margin of error will be reduced if the sample size is increased.

To obtain the smallest margin of error with 99% confidence, a sample size of 1689 would be needed.

A margin of error refers to the degree of error that may arise due to chance when attempting to estimate a population parameter such as a mean. It is calculated as the product of a critical value, a standard deviation, and a confidence interval, then divided by the square root of the sample size.

N is the sample size, which refers to the number of items included in the sample from the population. A larger sample size would be beneficial since it lowers the margin of error. When the sample size rises, the standard error of the mean decreases, implying that we are more confident in our estimate of the population mean. A smaller margin of error is desirable since it results in a more precise estimate of the population parameter.

The formula for the margin of error is given by:

Margin of Error = (Critical Value) (Standard Deviation) / sqrt(N).

To obtain the smallest margin of error with 99% confidence, a sample size of 1689 would be needed. The formula for the sample size calculation for this scenario is:

N = [(Zα/2)σ / E]²

Where, Zα/2 = 2.58, σ is the standard deviation, and E is the margin of error.

Using Zα/2 = 2.58 for a 99% confidence interval and the smallest possible margin of error to obtain a sample size for which the margin of error is minimized, we have:

N = [(Zα/2)σ / E]² = [(2.58)σ / E]²

Since the goal is to minimize the margin of error, we use the smallest possible value for E:1.

Therefore, N = [(2.58)σ / 1]²2.

Solving for N:

N = 6.6564σ²

To obtain the smallest margin of error with 99% confidence, we must solve for the smallest possible value of N that satisfies the above equation. We obtain:

N = 6.6564σ²

We don't have any information about the standard deviation, σ, in the given question, so we can't solve for N.

However, based on the formula, we can say that the margin of error will be reduced if the sample size is increased.

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Answer:

0.0129449838188

Answer:

im pretty sure its 69

Step-by-step explanation:

im not joking im pretty sure both angles are 69 because of the triangle formula

During an experiment of acceleration changes, the value that must stay constant during these trials is d. Mass

Inertia, a basic characteristic of all matter, may be measured quantitatively using mass. When a force is applied, an item effectively provides resistance to changes in velocity or position. The change brought about by an applied force is less the more mass an item has. According to Newton's second law of motion an object's acceleration is inversely proportional to its mass and directly proportional to the net force that has been applied to it. It may be expressed mathematically as F = ma.

The goal of this experiment is to see how variations in force impact acceleration. It is crucial to maintain the mass constant throughout the trials in order to isolate the impact of force on acceleration. Any observable variations in acceleration may be entirely attributable to variations in the applied force by maintaining the mass constant.

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Answer:

D 19,905

Step-by-step explanation:

19,905.22×25%=4976.31

19905.22-4976.31=14,928.91

14928.91×8%=1194.31

14928.91-1194.31=13734.6

Answer:

circle area = 50.24 units²

Step-by-step explanation:

circle area = πr² = 3.14(4²) = 3.14(16) = 50.24 units²

There are 28 different bit strings of length 8 that satisfy the condition of having two zeros that are never together.

To count the number of bit strings of length 8 with two zeros that are never together, we can use the concept of combinations.

First, let's consider the possible positions for the two zeros. The zeros cannot be adjacent to each other, which means they must be placed in non-adjacent positions in the string.

Since there are 8 positions in the string, we can choose 2 positions for the zeros in C(8, 2) ways. This gives us the number of ways to select the positions for the zeros.

Once we have chosen the positions for the zeros, the remaining 6 positions must be filled with ones. There is only one way to do this, as all the remaining positions will be filled with ones.

Therefore, the total number of bit strings of length 8 with two zeros that are never together is equal to C(8, 2) = 28.

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Answer:

r=7 my

answer needs to be 20 characters long

Answer:

12 inches

Step-by-step explanation:

Ahmed received a box of gifts. The box is a rectangular prism with the same height and width, and the length

is twice the width. The volume of the box is 3,456 in? What is the height of the box?

Volume of a Rectangular pyramid = Length × Width × Height

From the above question

Height = Width = x

Length = 2 × Width

Length = 2x

Volume = 3,456 cubic inches

Hence,

3,456 = 2x × x × x

3456 = 2x³

x³ = 3456/2

x³ = 1728

Cube root both sides

Cube root(x³) = cube root (1728 cubic Inches)

x = 12 inches

Therefore, the height is 12 inches

Width, Height and Length of rectangular prism are 12, 12 and 24 inch respectively.

Width of rectangular prism = a

Height of rectangular prism = a

Length of rectangular prism = 2a

Volume of the rectangular prism = (l)(b)(h)

Volume of the rectangular prism = (2a)(a)(a)

3,456 = 2a³

a³ = 1,728

a = 12 inch

So,

Width of rectangular prism = 12 inch

Height of rectangular prism = 12 inch

Length of rectangular prism = 2(12)

Length of rectangular prism = 24 inch

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Answer:

Step-by-step explanation:

Look at the graph, look at homework and then look at the number right outside of it.

Answer:

100-47=53 so 53$ profit

Step-by-step explanation:

Answer:

the first plotting was correct.

Step-by-step explanation:

the values will be arranged in an ascending order from their roots which are the first digit

The solution to the problem is `{5, 10, 15, 20, 25, 30}`.

The error message "syntax error, unexpected" is a common message you might encounter when there is a syntax error in your code.

This error message often points to an unexpected symbol or typo in your code. For instance, in the context of a code snippet like this, such an error message could be prompted due to an invalid command or misspelling, or a misused symbol or expression.

Also, another likely cause could be an incorrect use of a function or a non-existent variable or keyword. Hence, to solve the error message, you might need to double-check your code and fix any errors that could cause such an issue. Now, let's answer the question that follows: Given the universal set, S, as;`S = {5, 10, 15, 20, 25, 30}` and the subsets A and B as; A = {5, 10, 15, 20} B = {15, 20, 25, 30}.

To list the elements in the set (AUB) we need to find the union of A and B. The union of two sets A and B is a set of all the elements that are either in A or in B or in both. That is, `AUB = {x: x ∈ A or x ∈ B}`

Therefore, the union of sets A and B is the set `AUB = {5, 10, 15, 20, 25, 30}`

Thus, the list of elements in the set (AUB) is:`{5, 10, 15, 20, 25, 30}` Hence, the solution to the problem is `{5, 10, 15, 20, 25, 30}`.

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Answer:

8/7

Step-by-step explanation:

12*2=24.

7*3=21

24/21 is divisible by 3

=8/7

Answer:

n=2

Step-by-step explanation:

[tex]\frac{3.3.3.3.3.3}{3.3.3.3} = 3^{n}[/tex]

[tex]3^{2}[/tex]= [tex]3^{n}[/tex]

n=2

Answer:

[tex]3^1[/tex]

Step-by-step explanation:

[tex]\frac{3*3*3*3*3}{3*3*3*3} =3^1[/tex]

In the context of analyzing the relationship between studying time and test grades, the domain of the scatterplot would be the number of minutes the students studied (option D).

The domain refers to the set of possible inputs or variables in a given context. In this case, the scatterplot is being created based on the relationship between the time students spent studying and their corresponding test grades. Therefore, the domain in this context would be the number of minutes the students studied (option D).

The domain represents the independent variable, which is the variable that is controlled or manipulated in the analysis. In this scenario, the math teacher wants to analyze the relationship between studying time and test grades, so the number of minutes studied would be the independent variable. The teacher surveys the students to collect data on the time spent studying, and this variable becomes the domain of the scatterplot.

The range, on the other hand, represents the dependent variable, which is the variable that is measured or observed as an outcome or response. In this case, the dependent variable would be the students' test grades. The scatterplot will show how the test grades correspond to the amount of time students studied.

To summarize, in the context of analyzing the relationship between studying time and test grades, the domain of the scatterplot would be the number of minutes the students studied (option D).

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The correct statement regarding the residual plot in this problem, and whether the line of best fit is a good fit, is given as follows:

Yes, the points have no pattern.

For a data-set, the definition of a residual is that it is the difference of the actual output value by the predicted output value, hence it is defined by the subtraction operation as follows:

Residual = Observed - Predicted.

Hence the graph of the line of best fit should have the smallest possible residual values, and no pattern between the residuals.

As there is no pattern between the residuals in this problem, the line is in fact a good fit and the first option is the correct option.

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sec 290° = 1/cos 110°= -2.9238.

To find the value of sec 290° using a reference angle, we first need to determine the reference angle for 290°.

The reference angle is the acute angle formed between the terminal side of the given angle (290°) and the x-axis. To find the reference angle, we subtract the nearest multiple of 180° from the given angle:

Reference angle = 290° - 180° = 110°

Now, we can express sec 290° in terms of the secant of the reference angle (110°). The secant function is defined as the reciprocal of the cosine function:

sec(x) = 1/cos(x)

Therefore, sec 290° can be written as:

sec 290° = 1/cos 110° = -2.9238.

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Answer:

701

Step-by-step explanation:

At the end of 5 years, your savings will have grown to $701.

You will have earned in $201 in interest.

answer.

The maximum possible cost in dollars is $226.76 (approx).

Standard deviation = $88

Let X be the cost of the automobile repair, then X ~ N(367, 88^2) (normal distribution)

Now, we need to find the following probabilities:

(a) P(X > 480)(b) P(X < 240)(c) P(240 < X < 480)(d)

Find X such that P(X < X1) = 0.05, where X1 is the lower 5% point of X(a) P(X > 480)

We need to find P(X > 480)P(X > 480) = P(Z > (480 - 367)/88) [Standardizing the random variable X]P(X > 480) = P(Z > 1.2955)

Using the standard normal table, the value of P(Z > 1.2955) = 0.0983 (approx)

Hence, the required probability is 0.0983 (approx)(b) P(X < 240)

We need to find P(X < 240)P(X < 240) = P(Z < (240 - 367)/88) [Standardizing the random variable X]P(X < 240) = P(Z < -1.4432)

Using the standard normal table, the value of P(Z < -1.4432) = 0.0749 (approx)

Hence, the required probability is 0.0749 (approx)(c) P(240 < X < 480)

We need to find P(240 < X < 480)P(240 < X < 480) = P(Z < (480 - 367)/88) - P(Z < (240 - 367)/88) [Standardizing the random variable X]P(240 < X < 480) = P(Z < 1.2955) - P(Z < -1.4432)

Using the standard normal table, the value of P(Z < 1.2955) = 0.9017 (approx)and the value of P(Z < -1.4432) = 0.0749 (approx)

Hence, the required probability is 0.9017 - 0.0749 = 0.8268 (approx)(d)

Find the maximum possible cost in dollars, if the cost for your car repair is in the lower 5% of automobile repair charges.

This is nothing but finding the lower 5% point of X.We need to find X1 such that P(X < X1) = 0.05.P(X < X1) = P(Z < (X1 - 367)/88) [Standardizing the random variable X]0.05 = P(Z < (X1 - 367)/88)

Using the standard normal table, the value of Z such that P(Z < Z0) = 0.05 is -1.645 (approx)

Hence, we get,-1.645 = (X1 - 367)/88

Solving for X1, we get: X1 = 88*(-1.645) + 367 = $226.76 (approx)

Therefore, the maximum possible cost in dollars is $226.76 (approx).

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A subspace must satisfy three conditions: closure under addition, closure under scalar multiplication, and containing the zero vector. Therefore, the correct statement is B. W is a subspace of P2.

In order to determine whether statement A or B is true, we need to check the subspace criteria.

Let's analyze the statements:

A. W is not a subspace of P2 because 0 € W.

If 0 € W, then W does not contain the zero vector. However, the zero vector is the polynomial 0 + 2(0)x + (0)x^2 = 0, which is an element of W. Thus, statement A is false.

B. W is a subspace of P2.

For W to be a subspace, it needs to satisfy all three subspace criteria. Let's check each criterion:

Since W satisfies all three subspace criteria, statement B is true.

Therefore, the correct statement is B. W is a subspace of P2.

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a) The volume of the solid E in rectangular coordinate system is given by: [tex]$$\iiint_{E}[/tex] dx dy dz = [tex]\int_{-2}^{2} \int_{0}^{\sqrt{z^2}} \int_{-\sqrt{z^2 - y^2}}^{\sqrt{z^2 - y^2}} dx dy dz$$[/tex]

b) The volume of the solid E in cylindrical coordinate system is given by:  [tex]$$\iiint_{E} \rho d\rho d\theta dz = \int_{0}^{2} \int_{0}^{\frac{\pi}{2}} \int_{0}^{\sqrt{4 - z^2}} \rho d\rho d\theta dz$$[/tex]

c) The volume of the solid E is 11π/3.

a) The solid E that lies between the cone z² = x² + y² and the sphere x² + y² + (z + 2)² = 2.

Let the limits of x, y, z be X, Y, Z respectively.

The limits of X:

From the equation, z² = x² + y²

Z² = X² + Y²

X² = Z² - Y²

Let Z = 0, then X² = - Y² which is impossible. Therefore, Y can take any value such that Y < Z.

The limits of Y:

From the equation, z² = x² + y²

Z² = X² + Y²

Y² = Z² - X²

Let Z = 0, then Y² = - X² which is impossible. Therefore, X can take any value such that X < Z.

Limits of Z:

From the equation x² + y² + (z + 2)² = 2z² + 4z + 8 = 2(Z + 1)² + 6

The limits of z are Z < 2 and Z > - 2.

Volume in rectangular coordinate system:

[tex]$$\iiint_{E}[/tex] dx dy dz = [tex]\int_{-2}^{2} \int_{0}^{\sqrt{z^2}} \int_{-\sqrt{z^2 - y^2}}^{\sqrt{z^2 - y^2}} dx dy dz$$[/tex]

b) Volume of solid E in cylindrical coordinate system

Let the limits of ρ, θ, z be R, Θ, Z respectively.

The limits of R:

From the equation, z² = ρ² cos²θ + ρ² sin²θ

Z² = ρ²

Rho² = Z²/ cos²θ + sin²θ

Rho = Z/ cosθ

Let Z = 0, then Rho = 0. Therefore, R can take any value such that 0 ≤ R < 2.

Limits of Θ:

From the equation, z² = ρ² cos²θ + ρ² sin²θ

Z² = ρ² sin²θ

Theta² = tan⁻²(Z²/ ρ²)

Let Z = 0, then Θ = 0. Therefore, Θ can take any value such that 0 ≤ Θ ≤ π/2.

Limits of Z:

From the equation x² + y² + (z + 2)² = 2z² + 4z + 8 = 2(Z + 1)² + 6

The limits of Z are -2 ≤ Z < 2.

Volume in cylindrical coordinate system:

[tex]$$\iiint_{E} \rho d\rho d\theta dz = \int_{0}^{2} \int_{0}^{\frac{\pi}{2}} \int_{0}^{\sqrt{4 - z^2}} \rho d\rho d\theta dz$$[/tex]

c) Evaluation of the volume of solid E:

Using rectangular coordinate system, the volume of solid E is

[tex]$$\iiint_{E} dx dy dz[/tex] = [tex]\int_{-2}^{2} \int_{0}^{\sqrt{z^2}} \int_{-\sqrt{z^2 - y^2}}^{\sqrt{z^2 - y^2}} dx dy dz$$$$[/tex]

[tex]=\int_{-2}^{2} \int_{0}^{\sqrt{z^2}} [x]_{-\sqrt{z^2 - y^2}}^{\sqrt{z^2 - y^2}} dy dz$$$$=\int_{-2}^{2} \int_{0}^{\sqrt{z^2}} 2\sqrt{z^2 - y^2} dy dz$$$$=\int_{-2}^{2} \left[-\frac{1}{2}(z^2 - y^2)^{3/2}\right]_{y=0}^{y=\sqrt{z^2}} dz$$$$=\int_{-2}^{2} \frac{1}{2}z^3 dz = 0$$[/tex]

Therefore, the volume of solid E using rectangular coordinate system is 0.

Using cylindrical coordinate system, the volume of solid E is

[tex]$$\iiint_{E} \rho d\rho d\theta dz = \int_{0}^{2} \int_{0}^{\frac{\pi}{2}} \int_{0}^{\sqrt{4 - z^2}} \rho d\rho d\theta dz$$$$=\int_{0}^{2} \int_{0}^{\frac{\pi}{2}} \left[\frac{\rho^2}{2}\right]_{0}^{\sqrt{4 - z^2}} d\theta dz$$$$=\int_{0}^{2} \int_{0}^{\frac{\pi}{2}} 2 - \frac{z^2}{2} d\theta dz$$$$=\int_{0}^{2} \left[2\theta - \frac{\theta z^2}{2}\right]_{\theta = 0}^{\theta = \frac{\pi}{2}} dz$$$$=\int_{0}^{2} \pi - \frac{\pi z^2}{4} dz = \frac{11\pi}{3}$$[/tex]

Therefore, the volume of solid E using cylindrical coordinate system is 11π/3.

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