A particle is moving along the curve y = 2√5x + 11. As the particle passes through the point (5, 12), its x-coordinate increases at a rate of 5 units per second. Find the rate of change of the distance from the particle to the origin at this instant.

The area of the shaded portion of the diagram is 14.25 mm²

The triangle is inside the semi circle.

The shaded region of the semi circle is outside the triangle.

Therefore,

area of the shaded region = area of the semi circle - area of triangle

Therefore,

area of the semi circle = 1/2 × π × r²

where

Hence,

r = 5 mm

area of the semi circle = 1/2 × 3.14 × 5²

area of the semi circle = 78.50 / 2

area of the semi circle = 39.25 mm²

area of the triangle = 1/2 × b × h

where

Hence,

b = 10mm

h = 5mm

area of the triangle = 1/2 × 10 × 5

area of the triangle = 50 / 2

area of the triangle = 25 mm²

Therefore,

area of the shaded region = 39.25 - 25

area of the shaded region = 14.25 mm²

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Based on the results of the hypothesis test, we can say that a batch containing 1.80% zinc phosphide indicates that there is too little zinc phosphide in this production.

Based on the information provided, the chemical company produces a substance that contains 2% zinc phosphide for controlling rat populations in sugarcane fields. The production must be carefully controlled to ensure that the substance contains exactly 2% zinc phosphide. Records from past production indicate that the actual percentage of zinc phosphide present in the substance is approximately mound-shaped with a mean of 2.0% and a standard deviation of .08%.
Suppose one batch chosen randomly actually contains 1.80% zinc phosphide. This may or may not indicate that there is too little zinc phosphide in this production. To determine whether the batch contains too little zinc phosphide, we can perform a hypothesis test.
The null hypothesis in this case is that the batch contains exactly 2% zinc phosphide, and the alternative hypothesis is that the batch contains less than 2% zinc phosphide. We can use a one-tailed z-test to test this hypothesis.
Calculating the z-score for a batch with 1.80% zinc phosphide, we get:
z = (1.80 - 2.00) / 0.08 = -2.5

Using a standard normal distribution table, we can find that the probability of getting a z-score of -2.5 or lower is approximately 0.006. This means that if the batch truly contains 2% zinc phosphide, there is only a 0.006 probability of getting a sample with 1.80% zinc phosphide or less. Assuming a significance level of 0.05, we reject the null hypothesis if the p-value is less than 0.05. Since the p-value in this case is less than 0.05, we can reject the null hypothesis and conclude that there is evidence that the batch contains less than 2% zinc phosphide.

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The amount and interest after 2 years will be $3788.58 and $88.58, respectively.

Given that:

Investment, P = $3,700

Rate, r = 1.19

Time, n = 2 years

The amount is calculated as,

A = P(1 + r)ⁿ

A = $3700 (1 + 0.0119)²

A = $3700 x 1.0239

A = $3788.58

The amount of interest is calculated as,

I = A - P

I = $3788.58 - $3700

I = $88.58

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Answer:

[tex]\Large \boxed{\boxed{\textsf{$v=15t^2+6$}}}[/tex]

Step-by-step explanation:

If the position of a particle, i.e, the displacement is given by:

[tex]\Large \textsf{$f(t)=5t^3+6t+9$}[/tex]

Then the velocity, is the rate at which the displacement changes over time. This is given by the derivative of the displacement function. Hence velocity:

[tex]\Large \textsf{$v=f'(t)$}[/tex]

To differentiate the function, we can follow this simple rule:

[tex]\Large \boxed{\textsf{For $y=ax^n$, $\frac{dy}{dx}=anx^{n-1}$, where the constant term is excluded}}[/tex]

[tex]\Large \textsf{$\implies f'(t)=15t^2+6$}[/tex]

Therefore, velocity at time t:

[tex]\Large \boxed{\boxed{\textsf{$\therefore v=15t^2+6$}}}[/tex]

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[tex](\stackrel{x_1}{-9}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{-17}~,~\stackrel{y_2}{4}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{4}-\stackrel{y1}{0}}}{\underset{\textit{\large run}} {\underset{x_2}{-17}-\underset{x_1}{(-9)}}} \implies \cfrac{4 }{-17 +9} \implies \cfrac{ 4 }{ -8 } \implies - \cfrac{1 }{ 2 }[/tex]

The system has a nontrivial solution when k=0 or k=64.

The given system can be written as a matrix equation Ax=0, where A is the coefficient matrix and x is the column vector [x1,x2]. Thus,

A = [1 k; k 64] and x = [x1; x2]

For nontrivial solution, the matrix A must be singular, i.e., its determinant must be zero. Therefore,

det(A) = (1)(64) - (k)(k) = 64 - k^2 = 0

Solving the above equation gives k = 8 or k = -8. But k=-8 does not satisfy the given system, so we have k=8. Similarly, k=-8 can be ruled out as it does not satisfy the given system, so we have k=-8. Hence, the values of k for which the system has a nontrivial solution are k=0 and k=64.

To verify the nontrivial solutions, we can substitute k=0 and k=64 in the matrix equation and see that there exists a nontrivial solution (i.e., x is not identically zero).

For k=0, we have A = [1 0; 0 64] and x = [x1; x2]. The equation Ax=0 becomes

[1 0; 0 64][x1; x2] = [0; 0]

which has a nontrivial solution x=[0;1] or x=[1;0].

Similarly, for k=64, we have A = [1 64; 64 64] and x = [x1; x2]. The equation Ax=0 becomes

[1 64; 64 64][x1; x2] = [0; 0]

which has a nontrivial solution x=[-64;1] or x=[1;-1/64].

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Answer:

x > 6.4

Step-by-step explanation:

To solve the inequality 0.3x+1.05>-0.25x+4.57, we can start by simplifying it:

0.3x+1.05 > -0.25x+4.57

0.55x + 1.05 > 4.57

0.55x > 3.52

x > 3.52 / 0.55

x > 6.4

Therefore, the solution to the inequality is x > 6.4.

The number of square centimeters of the cylinder that Mariah paint in terms of π is 96π square centimeters.

In Mathematics and Geometry, the volume of a cylinder can be calculated by using this formula:

Volume of a cylinder, V = πr²h

72π = π(6)²h

Height, h = 2 cm.

In Mathematics and Geometry, the surface area (SA) of a cylinder can be calculated by using this mathematical equation (formula):

SA = 2πrh + 2πr²

Where:

SA = 2π × 6 × 2 + 2π × 6²

SA = 24π + 72π

SA = 96π square centimeters.

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Answer:

Karen takes twenty hours to clean the building

Step-by-step explanation

[tex]∫f(x)dx = 5\\∫f(x)dx = 8\\∫t(x)dx = 5\\∫t(x)dx = -3[/tex]The answers to the integrals are:

[tex]∫(9x)dx = g(x)dx = -3x + C\\∫4g(x)dx = 4(-3)dx = -12x + C[/tex]

Starting with the given information:

[tex]∫f(x)dx = 5\\∫f(x)dx = 8\\∫t(x)dx = 5\\∫t(x)dx = -3[/tex]

We can rearrange these equations to solve for[tex]f(x), t(x),[/tex]and [tex]g(x)[/tex]separately:

[tex]f(x) = 5/dx = 5\\f(x) = 8/dx = 8\\t(x) = 5/dx = 5\\t(x) = -3/dx = -3[/tex]

Thus, we have:

[tex]f(x) = 5\\t(x) = 5\\g(x) = -3[/tex]

Now we can evaluate the given integrals:

[tex]∫(9x)dx = g(x)dx = -3x + C[/tex], where C is the constant of integration

[tex]∫4g(x)dx = 4(-3)dx = -12x + C[/tex], where C is the constant of integration

Therefore, the answers to the integrals are:

[tex]∫(9x)dx = g(x)dx = -3x + C\\∫4g(x)dx = 4(-3)dx = -12x + C[/tex]

Note: the constant of integration C is added to both answers since the integrals are indefinite integrals.

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To find the indefinite integral of sin^3(4θ) cos(4θ) dθ, we can use the substitution u = sin(4θ), which gives us du/dθ = 4cos(4θ), or dθ = du/4cos(4θ).

Substituting this in, we have:

∫ sin^3(4θ) cos(4θ) dθ = ∫ u^3 du/4cos(4θ)

= 1/4 ∫ u^3 sec(4θ) dθ

Using the identity sec^2(4θ) - 1 = tan^2(4θ), we can rewrite sec(4θ) as (tan^2(4θ) + 1)^(1/2), giving us:

1/4 ∫ u^3 (tan^2(4θ) + 1)^(1/2) dθ

Now, we can use the substitution v = tan(4θ), which gives us dv/dθ = 4sec^2(4θ), or dθ = dv/4sec^2(4θ).

Substituting this in, we have:

1/16 ∫ u^3 (v^2 + 1)^(1/2) dv

So, the indefinite integral of sin³(4θ)cos(4θ)dθ is (-1/4)sin²(4θ)cos²(4θ) + C, where C is the constant of integration.

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a)  bn = (-1)^n (4/nπ) for n = 1, 3, 5, ... and bn = 0 for n = 2, 4, 6, ... Using these coefficients, we can sketch the periodic extension of f as a series of odd, triangular waves with decreasing amplitude.

b)  a0 = 0, a1 = 4/π, a2 = 0, a3 = 4/(9π), a4 = 0, a5 = 4/(25π), ... and an = 0 for all other even values of n.

c) an = (2/nπ) (1 - (-1)^n) for n = 1, 2, 3, ...

d)  bn = (2/nπ) (1 - (-1)^n) for n = 1, 3, 5, ...

We can sketch the periodic extension of f as a series of odd, sine waves with decreasing amplitude. In order to sketch the periodic extension of f to which each series converges, we need to first find the Fourier or cosine/sine coefficients of the given functions.

(a) For f(x) = |x| - x, we can see that it is an odd function, since f(-x) = -f(x). Therefore, the Fourier series will only have sine terms. We can find the coefficients using the formula:

bn = (2/L) ∫f(x) sin(nπx/L) dx, where L is the period of the function (in this case, L = 2).

After integrating, we get that bn = (-1)^n (4/nπ) for n = 1, 3, 5, ... and bn = 0 for n = 2, 4, 6, ...

Using these coefficients, we can sketch the periodic extension of f as a series of odd, triangular waves with decreasing amplitude.

(b) For f(x) = 2x^2 - 1, we can see that it is an even function, since f(-x) = f(x). Therefore, the Fourier series will only have cosine terms. We can find the coefficients using the formula:

an = (2/L) ∫f(x) cos(nπx/L) dx, where L is the period of the function (in this case, L = 2).

After integrating, we get that a0 = 0, a1 = 4/π, a2 = 0, a3 = 4/(9π), a4 = 0, a5 = 4/(25π), ... and an = 0 for all other even values of n.

Using these coefficients, we can sketch the periodic extension of f as a series of even, square waves with decreasing amplitude.

(c) For f(x) = e^x, we can see that it is an even function, since e^(-x) = e^x. Therefore, we can represent it as a cosine series using the formula:

a0 = (2/L) ∫f(x) dx from 0 to L, where L is the period of the function (in this case, L = 1).

After integrating, we get that a0 = (e - 1)/2.

We can then find the remaining coefficients using the formula:

an = (2/L) ∫f(x) cos(nπx/L) dx from 0 to L.

After integrating, we get that an = (2/nπ) (1 - (-1)^n) for n = 1, 2, 3, ...

Using these coefficients, we can sketch the periodic extension of f as a series of even, cosine waves with decreasing amplitude.

(d) For f(x) = e^x, we can see that it is an odd function, since e^(-x) = 1/e^x = -e^x/-1. Therefore, we can represent it as a sine series using the formula:

bn = (2/L) ∫f(x) sin(nπx/L) dx from 0 to L, where L is the period of the function (in this case, L = 1).

After integrating, we get that bn = (2/nπ) (1 - (-1)^n) for n = 1, 3, 5, ...

Using these coefficients, we can sketch the periodic extension of f as a series of odd, sine waves with decreasing amplitude.

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To construct a confidence interval for the proportion of all Comcast customers who experienced an internet outage in the past month, we can use the formula:

CI = p ± z*sqrt((p*(1-p))/n)

where p is the sample proportion, z is the z-score for the desired confidence level (86%), and n is the sample size.

First, we need to calculate the sample proportion:

p = 19/108 = 0.176

Next, we need to find the z-score for the 86% confidence level. Using a standard normal distribution table or calculator, we find that the z-score is approximately 1.44.

Now we can plug in the values and calculate the confidence interval:

CI = 0.176 ± 1.44*sqrt((0.176*(1-0.176))/108)

CI = 0.176 ± 0.083

CI = (0.093, 0.259)

Therefore, we can say with 86% confidence that the true proportion of all Comcast customers who experienced an internet outage in the past month is between 0.093 and 0.259.

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The total driving distance can be calculated by adding the number of blocks in each direction: 1 + 1 + 5 + 2 = 9 blocks. If you could fly like a bird, your trip would be 1/4 mile shorter.

In your neighborhood, you need to drive from your house to the grocery store following a path of 1 block south, 1 block east, 5 blocks north, and 2 blocks east. Each block is 1/16 of a mile.

Convert this to miles: 9 blocks * (1/16 mile/block) = 9/16 miles.

If you could fly like a bird, you would take a direct path. To find this distance, use the Pythagorean theorem for a right triangle formed by the east-west and north-south distances.

East-west distance: 1 block east + 2 blocks east = 3 blocks = 3/16 miles.
North-south distance: 5 blocks north - 1 block south = 4 blocks = 4/16 miles.

The direct flying distance can be calculated as:

√[(3/16)^2 + (4/16)^2] = √[(9/256) + (16/256)] = √(25/256) = 5/16 miles.

To find the shorter distance when flying, subtract the direct flying distance from the driving distance:

(9/16) - (5/16) = 4/16 miles, which simplifies to 1/4 mile.

So, if you could fly like a bird, your trip would be 1/4 mile shorter.

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A point on the rim of a wheel with a radius of one meter will travel 219.8 meters if the wheel makes 35 rotations.To find the distance traveled by a point on the rim of a wheel, we need to first calculate the circumference of the wheel. The circumference is equal to the diameter of the wheel multiplied by pi (π).

However, since we are given the radius of the wheel, we can simply multiply the radius by 2 and then by pi to get the circumference.

The formula for calculating the circumference of a circle is:

Circumference = 2 × pi × radius

C = 2 × pi × r

C = 2 × 3.14 × 1 (since the radius is given as one meter)

C = 6.28 meters

Now that we know the circumference of the wheel, we can easily calculate the distance traveled by a point on the rim of the wheel if it makes 35 rotations.

The formula for calculating the distance traveled by a point on the rim of a wheel is:

Distance = Circumference × number of rotations

D = C × n

D = 6.28 × 35

D = 219.8 meters

Therefore, a point on the rim of a wheel with a radius of one meter will travel 219.8 meters if the wheel makes 35 rotations.

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Yes, you can calculate the amount of discount of a $100 item that is 10% off in your head using mental math.

We have,

One way to do this is to recognize that 10% of 100 is 10, so the discount on a $100 item would be $10.

Another way to approach it is to divide the percentage off by 10 to get the dollar amount of the discount.

For example, 10% off is equivalent to a discount of 1/10 of the original price, so for a $100 item, the discount would be $10.

This can be a quick and useful mental math skill to have when shopping or budgeting.

Thus,

Yes, you can calculate the amount of discount of a $100 item that is 10% off in your head using mental math.

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In this problem, we are given the percentage of bags that are underweight, satisfactory, and overweight. We are asked to find the probability of selecting three bags that are either underweight, satisfactory, or overweight.

To find the probability of selecting three bags that are overweight, we need to multiply the probability of selecting one overweight bag by itself three times, since we are selecting three bags. The probability of selecting one overweight bag is 7.5%, or 0.075. Therefore, the probability of selecting three overweight bags is (0.075)^3 = 0.000421875, or approximately 0.042%.

To find the probability of selecting three bags that are satisfactory, we also need to multiply the probability of selecting one satisfactory bag by itself three times. The probability of selecting one satisfactory bag is 90%, or 0.9. Therefore, the probability of selecting three satisfactory bags is (0.9)^3 = 0.729, or approximately 72.9%.

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Complete question:

An automatic machine inserts mixed vegetables into a plastic bag. Past experience shows that some packages were underweight and some were overweight, but most of them had satisfactory weight.

Weight % of Total

Underweight 2.5

Satisfactory 90.0

Overweight 7.5

a) What is the probability of selecting and finding that all three bags are overweight?

b) What is the probability of selecting and finding that all three bags are satisfactory?

When working with a normal distribution and hypothesis testing, it is essential to define the critical region correctly, considering the level of significance and test statistic.

A normal distribution is a continuous probability distribution characterized by its bell-shaped curve, with the mean (µ), median, and mode all equal. In this case, n = 36 refers to the sample size, or the number of observations taken from this distribution.

A critical region is a range of values within a hypothesis test where, if the test statistic falls into this region, the null hypothesis is rejected in favor of the alternative hypothesis. Defining a critical region correctly involves determining the level of significance (α), which is the probability of rejecting the null hypothesis when it's true.

Regarding the circle, it seems unrelated to the given context, but in general, a circle is a 2D shape with all points equidistant from a central point, known as the center. The distance between the center and any point on the circle is called the radius.

Regarding P(-0.09), it appears to refer to a probability value related to a test statistic or a Z-score, which measures how many standard deviations an observation is away from the mean. To find this probability, you can use a Z-table or statistical software.

Understanding the distribution and test statistic, like the Z-score, is vital in interpreting the results of your analysis.

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The arithmetic mean of the number of member in each family is 4.

Arithmetic mean is the mean or the average of the samples. That means, it is the sum of all values divided by the number of values.

In this question, we have 8 values. So, the arithmetic mean (M) is:

[tex]\text{M}=\dfrac{2+2+5+4+8+3+1+7}{8}[/tex]

[tex]\text{M}=\dfrac{32}{8}[/tex]

[tex]\text{M}=4[/tex]    

Thus, The arithmetic mean of the number of member in each family is 4.

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The resulting curve is a limaçon (a type of cardioid) with a loop. It is centered at the origin and has an inner loop with a radius of 4/5 and an outer loop with a radius of 8/5. The curve has symmetry about both the x-axis and the y-axis. Option C is Correct.

To convert the equation R = -8 cos(θ) + 4 sin(θ) to Cartesian coordinates, we can use the following equations:

x = R cos(θ)

y = R sin(θ)

Substituting the given equation, we get:

x = (-8 cos(θ) + 4 sin(θ)) cos(θ)

y = (-8 cos(θ) + 4 sin(θ)) sin(θ)

Simplifying these equations, we get:

x =[tex]-8 cos^2[/tex](θ) + 4 cos(θ) sin(θ)

y = -8 cos(θ) sin(θ) +  [tex]4 sin^2[/tex](θ)

Simplifying further using the identity we get:

x = -8/5 + 4/5 cos(2θ)

y = 4/5 sin(2θ)

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The ordered pairs which are solutions to the system of inequalities given is (10, 2).

Given system of inequalities,

x + 2y ≥ 10

3x - 4y > 12

We have to find the solutions for the system of equations.

Let the equations be,

x + 2y = 10 [equation 1]

3x - 4y = 12 [equation 2]

From [equation 1],

x = 10 - 2y

Substituting in  [equation 2],

3(10 - 2y) - 4y = 12

30 - 6y - 4y = 12

-10y = -18

y = 9/5 = 1.8

x = 10 - 2y = 6.4

The system of equations hold true for (6.4, 1.8).

For (16, 9),

16 + (2 × 9) = 34 ≥ 10 is true.

(3 × 16) - (4 × 9) = 12 not greater than 12.

So this is not true.

For (10, 2),

10 + (2 × 2) = 14 ≥ 10 is true.

(3 × 10) - (4 × 2) = 22 > 12 is true.

Hence the correct ordered pair is (10, 2).

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[0] = { x ∈ Z | x^2 is even } is the set of all even integers, and [1] = { x ∈ Z | x^2 + 1 is even } is the set of all odd integers.

To prove that R is an equivalence relation on Z, we need to show that it satisfies the following three properties:

Reflexivity: For all x in Z, xRx.

Symmetry: For all x, y in Z, if xRy then yRx.

Transitivity: For all x, y, z in Z, if xRy and yRz then xRz.

Reflexivity: For all x in Z, x^2 + x^2 = 2x^2 is even. Therefore, xRx and R is reflexive.

Symmetry: For all x, y in Z, if xRy, then x^2 + y^2 is even. This means that y^2 + x^2 is also even, since even + even = even. Therefore, yRx and R is symmetric.

Transitivity: For all x, y, z in Z, if xRy and yRz, then x^2 + y^2 and y^2 + z^2 are both even. This means that (x^2 + y^2) + (y^2 + z^2) = x^2 + 2y^2 + z^2 is even. Since the sum of two even numbers is even, x^2 + 2y^2 + z^2 is also even, so xRz and R is transitive.

Since R is reflexive, symmetric, and transitive, it is an equivalence relation on Z.

The equivalence classes of R are the subsets of Z that contain all the integers that are related to each other by R. For any integer n in Z, the equivalence class [n] of n is the set of all integers that are related to n by R, i.e., [n] = {x ∈ Z | xRn}.

In this case, if n is even, then [n] contains all even integers because if x is even, then x^2 + n^2 is even. If n is odd, then [n] contains all odd integers because if x is odd, then x^2 + n^2 is even. So the set of equivalence classes of R is:

{ [n] | n ∈ Z }

where [n] = { x ∈ Z | x^2 + n^2 is even }.

For example, [0] = { x ∈ Z | x^2 is even } is the set of all even integers, and [1] = { x ∈ Z | x^2 + 1 is even } is the set of all odd integers.

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Therefore, the probability of rolling an odd number on the die and flipping a heads on the coin is 1/4.

The probability of rolling an odd number on a fair die is 3/6 or 1/2.

The probability of flipping a heads on a fair coin is 1/2.

Since the die roll and the coin flip are independent events, we can multiply their probabilities to find the probability of both events occurring together:

P(rolling an odd number AND flipping a heads) = P(rolling an odd number) x P(flipping a heads)

P(rolling an odd number AND flipping a heads) = (1/2) x (1/2)

= 1/4

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The probability of z being greater than -2.06 is 0.9801. The value of c that satisfies P(0.48 < z < c) = 0.2162 is 1.77.

To find the probability P(z > -2.06) for a standard normal distribution, we can use a standard normal distribution table or a calculator.

Using a standard normal distribution table, we can look up the area to the left of -2.06, which is 0.0199. Since we want the area to the right of -2.06, we can subtract this from 1 to get:

P(z > -2.06) = 1 - 0.0199 = 0.9801

So the probability of z being greater than -2.06 is 0.9801, rounded to four decimal places.

For the second question, we want to find the value of c such that P(0.48 < z < c) = 0.2162, where z is a standard normal random variable.

Using a standard normal distribution table or a calculator, we can find the area to the left of 0.48, which is 0.6844. Since the standard normal distribution is symmetric about zero, the area to the right of c will also be 0.6844. Therefore, we can find c by finding the z-score that corresponds to an area of 0.6844 + 0.2162 = 0.9006 to the left of it.

Looking this up on a standard normal distribution table or using a calculator, we find that the z-score is approximately 1.29. Therefore:

c = 0.48 + 1.29 = 1.77

So the value of c that satisfies P(0.48 < z < c) = 0.2162 is 1.77, rounded to two decimal places.

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The sum of the series of the AP is 4200.

There are 50 term of in AP. The first term is 20 and 60th term is 120. Therefore, the sum of the series can be found as follows:

Using,

aₙ = a + (n - 1)d

where

Therefore,

sum of the series = n / 2 (a + l)

sum of the series = 60 / 2(20 + 120)

sum of the series = 30(140)

sum of the series =  4200

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The dimensions of the cubic shape are:

Length = 10 units

Width = 3 units

Height = 1 units

It should be noted that a geometric body of cubic shape has a volume of 30 cubic units, and we want to know the length, width and height dimensions if we know that the length is greater than the width, but less than the height?l.

We can list out all the possible combinations of l, w, and h that multiply to 30:

1 * 1 * 30 = 30

1 * 2 * 15 = 30

1 * 3 * 10 = 30

1 * 5 * 6 = 30

2 * 3 * 5 = 30

Therefore, the dimensions of the cubic shape are: Length = 10 units, Width = 3 units, Height = 1 unit

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A geometric body of cubic shape has a volume of 30 cubic units, what will be its length, width and height dimensions if we know that the length is greater than the width, but less than the height?

Answer:

The equation of the line is in slope-intercept form, y = mx + b, where m is the slope of the line and b is the y-intercept.

Comparing the given equation y = (-3/2)x - (5/4) with the slope-intercept form, we can see that the y-intercept is -5/4 and the slope of the line is -3/2.

Therefore, the slope of the line is -3/2.

The answer choice which correctly represents the inverse of the function is; Choice A. n(a)=a+15/3.

It follows from the task content that the answer choice which correctly represents the inverse function is to be determined.

Since the given function is; a(n)=3n-15

make n the subject of the formula;

3n = a(n) + 15

n = (a(n) + 15) / 3

Substitute n for n(a) and a(n) for a;

n(a) = ( a + 15 ) / 3

Ultimately, Choice A. n(a)=a+15/3 is correct.

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The equations for the graphs are

Graph of cubic function

The equation of cubic function is

y = a(x - h)^3 + k

For horizontal inflection at (6, 1)

y = a(x - 6)^3 + 1

passing through point (10, 33)

33 = a(10 - 6)^3 + 1

32 = 64a

a = 32/64 = 1/2

hence the equation is: y = 1/2(x - 6)^3 + 1

Square root function

y = a√(x - h) + k

(h, k) is from (5, 6)

y = a√(x - 5) + 6

passing through point (9, -2)

-2 = a√(9 - 5) + 6

-2 = a√(4) + 6

-8 = 2a

a = -4

substituting results to

y = -4√(x - 5) + 6

Graph of cubic function

The equation of cubic function is

y = k - a(x - h)^3

For horizontal inflection at (-0.5, 2)

y = 2 - a(x + 0.5)^3

passing through point (-5, 38.45)

38.45 = 2 - a(-5 + 0.5)^3

38.45 -2 = -a(-4.5)^3

a = -36.45/(-4.5)^3 = 2/5

hence the equation is: y = 2 - 2/5(x + 0.5)^3

Square root function

y = a√(x - h) + k

(h, k) is from (-2, -11)

y = a√(x + 2) - 11

passing through point (2, -1)

-1 = a√(2 + 2) - 11

10 = a√(4)

10 = 2a

a = 5

substituting results to

y = 5√(x + 2) - 11

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The graph of the parabola (x- 5 )²/25 - (y + 3)²/36 = 1 , to see the attachment.

A parabola is a U-shaped curve that is drawn for a quadratic function, f(x) = [tex]ax^2 + bx + c.[/tex] The graph of the parabola is downward (or opens down), when the value of a is less than 0, a < 0.

We have the graph equation is:

[tex](\frac{(x-5)^2}{25} )-(\frac{(y+3)^2}{35} )=1[/tex]

The above expression is a an equation of a conic section

Next, we plot the graph using a graphing tool

To plot the graph, we enter the equation in a graphing tool and attach the display

To see the attachment.

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