How do you do this in math

Answer:

(7, -1)

Step-by-step explanation:

3x + 7y = 14

y = x - 8

3x + 7(x - 8) = 14

3x + 7x - 56 = 14

10x - 56 = 14

Add 56 to both sides.

10x = 70

Divide both sides by 10.

x = 7

3(7) + 7y = 14

21 + 7y = 14

Subtract 21 from both sides.

7y = -7

Divide both sides by 7.

y = -1

(7, -1)

The unique solution of the following differential equations by using Laplace transforms,

(1) y(t) = 1/9 + 4/3t + 1/18[tex]e^{-3t}[/tex] - 1/27t² - 1/54t[tex]e^{-3t}[/tex]

(2) y(t) = (1/18)([tex]e^{-3t}[/tex] - 2t[tex]e^{-3t}[/tex] - 3t + 2)

(1) To solve this differential equation using Laplace transforms, we first take the Laplace transform of both sides, using the fact that L{y'}=sY(s)-y(0) and L{y''}=s²Y(s)-sy(0)-y'(0):

s²Y(s) - 6sY(s) + 9Y(s) = (2/s³) - (6/s-3)³

Simplifying, we get:

Y(s) = (2/s^5) + (6/[tex](s-3)^4[/tex]) / (s-3)²

Using partial fraction decomposition, we get:

Y(s) = (1/30s²) - (1/30s) + (1/18/(s-3)) - (1/90/(s-3)²) + (1/180/(s-3))

Taking the inverse Laplace transform of both sides, we get:

y(t) = (t²/30 - t/30) + (1/18)[tex]e^{(3t)}[/tex] - (1/60)t [tex]e^{(3t)}[/tex] + (1/360) t² [tex]e^{(3t)}[/tex]

Therefore, the unique solution to the differential equation is:

y(t) = (t²/30 - t/30) + (1/18)[tex]e^{(3t)}[/tex] - (1/60)t[tex]e^{(3t)}[/tex] + (1/360) t²[tex]e^{(3t)}[/tex]

(2) Following the same steps as above, we take the Laplace transform of both sides, using the fact that L{y'}=sY(s)-y(0) and L{y''}=s²Y(s)-sy(0)-y'(0):

s²Y(s) + 2sY(s) - 3Y(s) = 1/(s+3)

Simplifying, we get:

Y(s) = 1/(s+3) / (s+1)(s-3)

Using partial fraction decomposition, we get:

Y(s) = (-1/8/(s+1)) + (1/3/(s-3)) + (1/8/(s+3))

Taking the inverse Laplace transform of both sides, we get:

y(t) = (-1/8)[tex]e^{(-t)}[/tex] + (1/3)[tex]e^{(3t)}[/tex] + (1/8)[tex]e^{-3t}[/tex]

Therefore, the unique solution to the differential equation is:

y(t) = (-1/8)[tex]e^{(-t)}[/tex] + (1/3)[tex]e^{(3t)}[/tex] + (1/8)[tex]e^{-3t}[/tex]

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1. The derivative of f(x) = 4 is 0.

2. The derivative of f(x) =63x is 63

3. The derivative of f(x)=7 is 0.

4. The derivative of f(x) is 12x+3

To find the derivatives of f(x) = 4, f(x) = 63x, f(x) = 7, and f(x) = 3(2x²+x) using the limit definition, follow these steps:

1. For f(x) = 4, the derivative, f'(x), is 0 since it is a constant function.

2. For f(x) = 63x, use the limit definition: f'(x) = lim(h→0) [(f(x+h)-f(x))/h]. Plug in f(x) = 63x and simplify: f'(x) = lim(h→0) [(63(x+h)-63x)/h] = lim(h→0) [63h/h] = 63.

3. For f(x) = 7, the derivative, f'(x), is 0 since it is a constant function.

4. For f(x) = 3(2x²+x), apply the limit definition and simplify:

f'(x) = lim(h→0) [(f(x+h)-f(x))/h] = lim(h→0) [(3(2(x+h)²+(x+h))-3(2x²+x))/h] = lim(h→0) [(6x²+6xh+6h²+3h)/h] = lim(h→0) [6x+6x+3] = 12x+3.

In summary, the derivatives are: f'(x) = 0, f'(x) = 63, f'(x) = 0, and f'(x) = 6x+3.

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The birth weight of elephants at the 95th percentile is approximately 322 pounds when rounded to the nearest integer.

To find the birth weight of elephants at the 95th percentile, we need to use the standard Normal distribution table. First, we need to calculate the z-score corresponding to the 95th percentile:

z = invNorm(0.95) = 1.645

Here, invNorm is the inverse Normal distribution function. Using this z-score, we can find the corresponding birth weight using the formula:

x = μ + zσ

where μ is the mean birth weight (240 pounds), σ is the standard deviation (50 pounds), and z is the z-score we just calculated:

x = 240 + 1.645 * 50
x = 317.25

Therefore, the birth weight of elephants at the 95th percentile is approximately 317 pounds (rounded to the nearest integer).

To find the birth weight of elephants at the 95th percentile, we will use the given information: the average birth weight is 240 pounds, and the standard deviation is 50 pounds. We will also use the Z-score for the 95th percentile, which is 1.645.

Now, we can use the formula:
Percentile = Mean + (Z-score * Standard Deviation)

Percentile = 240 + (1.645 * 50)
Percentile ≈ 322.25

The birth weight of elephants at the 95th percentile is approximately 322 pounds when rounded to the nearest integer.

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The width of the rectangle is 0.9 cm to 1 d.p.

A triangle is a three-sided polygon or a three-dimensional object composed of three flat surfaces that intersect at three vertices. Triangles can be classified based on their sides and angles. Equilateral triangles have all three sides equal, isosceles triangles have two sides equal, and scalene triangles have all three sides of different lengths. Triangles can also be classified based on their angles. Right triangles have one 90-degree angle, obtuse triangles have one angle greater than 90-degrees, and acute triangles have all angles less than 90-degrees.

The circumference of the shaded face can be calculated using the formula for circumference of a circle, C = 2πr, where r is the radius of the circle. The radius of the shaded face can be found by subtracting the height of the net (h) from the radius of the cylinder (R). Therefore, the circumference of the shaded face can be calculated as follows:

C = 2π(R-h)
= 2π(2-1)
= 2π
= 6.28

The circumference of the shaded face is 6.28 cm to 1 d.p.

b) To calculate the width, w, of the rectangle, we can use the formula for area of a rectangle, A = lw, where l is the length of the rectangle. The area of the rectangle can be found by adding the area of the two semicircles (πr2) and subtracting the area of the triangular part (½bh). Therefore, the width of the rectangle can be calculated as follows:

A = lw
w = A/l
w = (2πr2+2πr2-½bh)/2(2πr)
w = (4πr2-½bh)/(4πr)
w = (8-1)/(8)
w = 7/8

The width of the rectangle is 0.9 cm to 1 d.p.

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A measurement of how many tasks a computer can accomplish in a certain amount of time is called throughput.

Throughput is a measure of the amount of data or information that can be transmitted through a communication channel or processed by a system in a given period of time. It is usually expressed in bits per second (bps), bytes per second (Bps), or packets per second (pps).

In computing, throughput refers to the rate at which data can be transferred between the CPU, memory, and other components of a computer system. It can also refer to the amount of work a computer system can perform within a given period of time, such as the number of tasks completed per second.

Throughput is an important performance metric in many applications, especially those involving data transfer or real-time processing. A higher throughput generally indicates a more efficient and capable system, while a lower throughput may indicate a bottleneck or performance limitation. Throughput is a measure of the amount of work a computer system can do in a given period of time, typically measured in tasks completed per unit time. It is an important performance metric for computer systems, especially in scenarios where high volume or time-sensitive tasks are being performed

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Ennis can make 4 different triangular borders using the given lengths of wood.

To determine how many different triangular borders Ennis can make, we need to apply the triangle inequality theorem, which states that the sum of any two sides of a triangle must be greater than the third side.

Let's consider each possible combination of three sides from the four given lengths of wood:

8 inches, 10 inches, 12 inches: forms a valid triangle

8 inches, 10 inches, 18 inches: forms a valid triangle

8 inches, 12 inches, 18 inches: forms a valid triangle

10 inches, 12 inches, 18 inches: forms a valid triangle

Therefore, Ennis can make 4 different triangular borders using the given lengths of wood.

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The probability that a randomly selected worker at the fast food restaurant earns more than $6.95/hr is approximately 9.18%.

To calculate the probability that a randomly selected worker earns more than $6.95/hr, we will use the z-score formula to standardize the value and then find the corresponding probability from a standard normal distribution table.

Step 1: Calculate the z-score
z = (X - μ) / σ
where X is the value we're interested in ($6.95/hr), μ is the average hourly wage ($6.35/hr), and σ is the standard deviation ($0.45/hr).

z = (6.95 - 6.35) / 0.45
z = 0.6 / 0.45
z ≈ 1.33

Step 2: Find the probability
Using a standard normal distribution table, we find that the probability of a z-score being less than 1.33 is approximately 0.9082. Since we want the probability that a worker earns more than $6.95/hr, we need to find the complement of this probability.

P(X > 6.95) = 1 - P(X ≤ 6.95)
P(X > 6.95) = 1 - 0.9082
P(X > 6.95) ≈ 0.0918

Therefore, the probability that a randomly selected worker at the fast food restaurant earns more than $6.95/hr is approximately 9.18%.

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Answer:

The y-intercept is (0, 2).

f(x)=15x-6

f(2)= 15*2 -6

=30-6

=24

The probability that the average tail length of the sample is between 3.8 and 3.9 feet is approximately 0.2417.

We have,

First, we need to find the mean and standard deviation of the sample distribution of the mean tail length:

The mean of the sample distribution is equal to the mean of the population, which is 3.75 feet.

The standard deviation of the sample distribution is equal to the standard deviation of the population divided by the square root of the sample size:

σ/√n = 0.6/√36 = 0.1 feet

Now we can use the standard normal distribution to find the probability:

z1 = (3.8 - 3.75) / 0.1 = 0.5

z2 = (3.9 - 3.75) / 0.1 = 1.5

Using a standard normal table or calculator, we can find the probability that z is between 0.5 and 1.5:

P(0.5 ≤ z ≤ 1.5) = P(z ≤ 1.5) - P(z ≤ 0.5) = 0.9332 - 0.6915 = 0.2417

Therefore,

The probability that the average tail length of the sample is between 3.8 and 3.9 feet is approximately 0.2417.

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This means that the probability that x assumes a value between 94.0 and 148.2 is 0.9032 or 90.32% (rounded to four decimal places).

To find the probability that x assumes a value between 94.0 and 148.2, we need to find the area under the normal curve between these two values. We can do this by standardizing the values using the z-score formula and then using a table or calculator to find the area under the standard normal curve.
First, we calculate the z-scores for 94.0 and 148.2:
z1 = (94.0 - 119.8) / 12.5 = -2.05
z2 = (148.2 - 119.8) / 12.5 = 2.25
Next, we look up the area between these two z-scores using a standard normal table or calculator. Using a calculator, we can use the normalcy function:
normalcy(-2.05, 2.25, 0, 1) = 0.9032

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Since Σk=1[infinity] 1/√n^3 converges by the p-series test, we can conclude that the series Σk=1[infinity] 1/ √n^3 +5 converges as well.

We will use the Limit Comparison Test to determine the convergence or divergence of the series Σk=1[infinity] 1/ √n^3 +5.

Let a_n = 1/√(n^3 + 5)

Then, we need to find a series b_n such that:

b_n > 0 for all n

The limit of (a_n/b_n) as n approaches infinity is a positive, finite number.

To find such a series b_n, we can compare a_n to a simpler series that we know converges or diverges. One such series is the series:

Σk=1[infinity] 1/√n^3

which converges by the p-series test with p=3/2.

We know that 0 < a_n < 1/√n^3 for all n, so we can use the inequality:

1/√n^3 + 5 < 1/√n^3

Multiplying both sides by 1/n, we get:

1/n√n^3 + 5/n < 1/n√n^3

1/n^(5/2) + 5/n < 1/n^(5/2)

Let b_n = 1/n^(5/2)

Then, we have:

0 < a_n/b_n < (1/n^(5/2) + 5/n)/1/n^(5/2) = 1 + 5/n^(3/2)

Taking the limit as n approaches infinity, we get:

lim (a_n/b_n) = lim [1/(n^(5/2)√(n^3 + 5))] / (1/n^(5/2))

= lim [(n^(5/2))/(√(n^3 + 5))] = 1

Since 0 < a_n/b_n < 1 + 5/n^(3/2) and lim (a_n/b_n) = 1, we can conclude that the series Σk=1[infinity] 1/ √n^3 +5 and Σk=1[infinity] 1/√n^3 have the same behavior, meaning they both converge or both diverge. Since Σk=1[infinity] 1/√n^3 converges by the p-series test, we can conclude that the series Σk=1[infinity] 1/ √n^3 +5 converges as well.

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Hence ,the solution set is {x | x < 3 or x > 5}, the interval notation would be (-∞, 3) ∪ (5, ∞).

A Rational inequality is a mathematical  statement that includes a fraction in the  variable in the numerator or  denominator, and either a less than, greater than,  less than or equal to, or greater than or equal to symbol.

In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities. The feasible region of a constrained optimization problem is the solution set of the constraints.

Without  the specific  inequality provided, it is difficult to provide a detailed explanation. However, I will give a general explanation on  how to solve a rational inequality and how to determine the solution set.

To solve a rational inequality, follow these steps:

Factor the numerator and denominator of the rational expression.

Determine the critical values of the inequality by setting the denominator equal to zero and solving for the variable.

Create a number line and plot the critical values on it.

Test each interval between the critical values by  choosing  a test value within the interval and determining whether the expression is positive or negative.

Write the  solution set in interval notation based on the  sign of the expression in each interval.

To determine why the solution set includes 3 but does not include 1, you would need to follow the above steps for the specific rational inequality provided. The critical values would be  the values of the variable that  make the denominator equal to zero. If one of the critical values is 1, that would mean  that the expression is undefined at x=1, and therefore it  cannot be included in the solution set.

Once  you have found  the critical values and tested the intervals, you can  write the solution set in interval notation. For example, if the solution set is {x | x < 3 or x > 5}, the interval notation would be (-∞, 3) ∪ (5, ∞).

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This is integral to find the length of the arc of the parabola y = x² + 1 inscribed in the disc of equation x² + (y – 1)² = 1.

To set up an integral to find the length of the arc of parabola y = x2 + 1 inscribed in the disc of equation x2 + (y – 1)2 = 1, we can use the formula for arc length:

[tex]L =\int_{[a,b]} \sqrt{[1 + (dy/dx)2] }dx[/tex]

where a and b are the x-coordinates of the points where the parabola intersects the circle, and dy/dx is the derivative of y with respect to x.

First, we need to find the x-coordinates of the points of intersection. We can solve for x in the equation of the circle:

[tex]x^2 + (y - 1)^2 = 1\\x^2 + y^2 - 2y + 1 = 1\\x^2 + y^2 - 2y = 0\\x^2 + (y - 1)^2 - 1 = 0\\x^2 + (x^2 + 2y - 1) - 1 = 0\\2x^2 + 2y -2 = 0x^2 + y - 1 = 0[/tex]

Substituting y = x² + 1, we get:

x² + x² + 1 – 1 = 0
2x² = 0
x = 0

So the parabola intersects the circle at (0,1) and (0,-1).

Next, we need to find the derivative of y with respect to x:

dy/dx = 2x

Substituting into the formula for arc length, we get:

[tex]L = \int_{[-1,1]} \sqrt{[1 + (2x)^2}dx[/tex]

This is integral to find the length of the arc of the parabola y = x² + 1 inscribed in the disc of equation x² + (y – 1)² = 1. We do not ask to evaluate this integral.

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The maximum revenue is $6,075.

The elasticity of demand when the hourly instruction rate is $65 can be determined using the formula E = (dq/dp)*(p/q). To increase your revenue, you should lower the hourly instruction rate. To maximize revenues, you should change the instruction rate to $45.


1. Calculate q when p is $65: q = 270(65) - 65d => q = 17550 - 65d
2. Calculate derivative dq/dp: dq/dp = -d
3. Calculate E: E = (-d)*(65/(17550-65d))
4. Set E = -1 (for maximum revenue) and solve for d: -1 = (-d)*(65/(17550-65d))
5. Solve for d: d = 3
6. Substitute d in the demand equation to find p: 270p - 3p = 17550 => p = $45
7. Calculate the maximum revenue: q = 270(45) - 3(45) => q = 135
8. Maximum revenue: $45 * 135 = $6,075

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The probability that a Democrat opposes stronger gun control laws is 0.16.

To find the probability that a Democrat opposes stronger gun control laws, we need to look at the table provided. In the table, the row for Democrats shows two values: 0.25 for "Favor" and 0.16 for "Oppose." These values represent the proportion of Democrats who favor and oppose stronger gun control laws, respectively.

Since the question asks for the probability that a Democrat opposes stronger gun control laws, we focus on the value of 0.16, which represents the proportion of Democrats who oppose stronger gun control laws.

Therefore, the probability that a Democrat opposes stronger gun control laws is 0.16.

Therefore, the probability that a Democrat opposes stronger gun control laws is 0.16.

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Euler's method is a numerical method used to approximate solutions to first-order ordinary differential equations (ODEs). It is based on the idea of walking along tangent lines of nearby solutions for short periods of time. The given statement is true.

The basic idea of Euler's method is to approximate the solution to an ODE at discrete time steps using a simple iterative formula that involves the slope of the solution at each time step. At each time step, the slope of the solution is approximated by the slope of the tangent line to the solution at that point. The method then takes a small step along this tangent line to approximate the solution at the next time step.

This process is repeated over and over again, with each step approximating the solution at the next time point. While the method is not exact, it can provide a useful approximation of the true solution if the time steps are small enough.

In summary, Euler's method is based on the idea of approximating the solution to an ODE by walking along tangent lines of nearby solutions for short periods of time.

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Equation {−7(4x−2) +7x} has the simplified form is (−21x + 14)

A simple equation is a mathematical statement that equates two expressions using an equal sign, and which can be solved in a straightforward manner without using complex mathematical operations. In other words, it is an equation that involves only one variable and has a degree of 1 (linear).

To simplify −7(4x−2) +7x, we can first use the distributive property to remove the parentheses:

⇒ −7(4x−2) +7x

⇒ −28x + 14 + 7x

Next, we can combine like terms by adding the x terms and the constant terms separately:

⇒ −28x + 14 + 7x

⇒ (−28x + 7x) + 14

⇒ −21x + 14

Therefore, the simplified form of {−7(4x−2) +7x} is {−21x + 14}.

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Answer:

C. Ten Thousandths

Step-by-step explanation:

three is four spots to the right of the decimal. This means that it is in the ten thousandths place value.

Decimal (.) Tenths (9) Hundredths (2) Thousandths (1) Ten Thousandths (3)

The statistical question is solved and

a) The null hypothesis is (H0) and alternative hypothesis is (Ha)

b) The z-test statistic is approximately 1.747.

c) The P-value for the z-test is 0.1614.

Given data,

(a)

The null hypothesis (H0): The mean femininity score for male hotel managers is equal to the mean femininity score for men in general (μ = 5.19).

The alternative hypothesis (Ha): The mean femininity score for male hotel managers is different from the mean femininity score for men in general (μ ≠ 5.19).

(b)

To calculate the z-test statistic, we'll use the formula:

z = (y - μ) / (σ / √n)

where:

y = sample mean femininity score (y = 5.29)

μ = population mean femininity score (μ = 5.19)

σ = standard deviation of the population (σ = 0.78)

n = sample size (n = 148)

Substituting the given values:

z = (5.29 - 5.19) / (0.78 / √148)

Calculating the expression:

z ≈ 1.747

Therefore, the z-test statistic is approximately 1.747.

(c)

To find the P-value for the z-test, we need to determine the probability of observing a z-value as extreme as 1.747 or more extreme in a two-tailed distribution.

Using a standard normal distribution table or a statistical calculator, we find that the P-value for a z-value of 1.747 is approximately 0.0807.

Since this is a two-tailed test, we multiply the P-value by 2:

P-value = 2 * 0.0807 ≈ 0.1614

Hence , the P-value for the z-test is approximately 0.1614.

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Answer:7

Step-by-step explanation:

the range, in statistics, is the difference between the highest and lowest value. 7-0=7.

with stats, it important to learn the

a) The coefficient a of the term [tex]x^4[/tex] in the expansion of[tex](x^2 + 4)^8[/tex]is 17920.

b) The coefficient a of the term [tex]x^8y^2[/tex] in the expansion of [tex](x - 4y)^{10[/tex] is

2949120.

We can use the Binomial Theorem, which states that the coefficient of

the term[tex]x^r[/tex] in the expansion of[tex](a + b)^n[/tex] is given by the expression:

[tex]C(n, r) \times a^{(n-r)} \times b^r[/tex]

where C(n, r) is the binomial coefficient, given by:

C(n, r) = n! / (r! × (n-r)!)

So in our case, we have:

n = 8

r = 4

a =[tex]x^2[/tex]

b = 4

Plugging these values into the formula, we get:

[tex]C(8, 4) \times (x^2)^{(8-4)} \times4^4\\= C(8, 4) \times x^8 \times 256\\= 70 \times x^8 \times 256\\= 17920x^8[/tex]

b.) We can again use the Binomial Theorem. This time, we have:

n = 10

r = 8

a = x

b = -4y

(Note that we use -4y for b, since the term involves a negative power of y.)

Plugging these values into the formula, we get:

[tex]C(10, 8) \times x^{(10-8)} \times (-4y)^8\\= C(10, 8) \times x^2 \times 65536y^8\\= 45 \times x^2 \times 65536y^8\\= 2949120x^2y^8[/tex]

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Design, the main effects of A, B, C, and D are not confounded with blocks because each block contains exactly one run for each level of each factor.

The 24 runs into four blocks, we can use a balanced incomplete block design (BIBD) with parameters (v, b, r, k) = (24, 4, 6, 2).

This means that there are 24 runs, divided into 4 blocks, each block contains 6 runs, and each pair of runs appears together in 2 blocks.

The runs can be divided into blocks:

Block 1:

ABCD, ABDC, ACBD, ADBC, ADBC, ACDB

Block 2:

BACD, BADC, BCAD, BDAC, BDCA, BCDA

Block 3:

CABD, CADB, CBAD, CDAB, CDBA, CBDA

Block 4:

DABC, DACB, DBAC, DCAB, DCBA, DBCA

The two-factor interactions are confounded with blocks because each pair of runs appears together in exactly two blocks.

Specifically, the AB, AC, AD, BC, BD, and CD interactions are confounded with blocks.

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The inverse function is a scaling of the original function by a factor of 1/6.

What is function?

A function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output.

The inverse of f(x) = 6x can be found by solving for x in terms of y and then replacing y with x:

y = 6x

x = y/6

Therefore, the inverse function is:

f^-1(x) = x/6

Alternatively, we can write it as:

f^-1(x) = (1/6) x

Note that the inverse function is a scaling of the original function by a factor of 1/6.

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The dimensions that minimize the cost of the material are a height of approximately 3.132 m and a radius of approximately 0.508 m.

Let's start by setting up some notation for the dimensions of the cylindrical container. Let the height of the container be h, and let the radius of the top and bottom be r. Then, the volume of the container is given by:

[tex]V =\pi r^2h[/tex]

We want to minimize the cost of the material used to make the container. The cost is composed of two parts: the cost of the material used for the top and bottom, and the cost of the material used for the side. Let's compute these separately.

The cost of the material used for the top and bottom is given by the area of two circles with radius r, multiplied by the cost per square meter:

[tex]C1 = 2\pi r^2 * 20[/tex]

The cost of the material used for the side is given by the area of the side of the cylinder, which is a rectangle with height h and length equal to the circumference of the base (which is 2πr), multiplied by the cost per square meter:

C2 = 2πrh * 15

The total cost is the sum of these two costs:

[tex]C = C1 + C2 = 2\pi r^2 * 20 + 2\pi rh * 15[/tex]

We want to minimize this cost subject to the constraint that the volume is 10 [tex]m^3[/tex]:

[tex]V = \pi r^2h = 10[/tex]

We can use the volume equation to eliminate h, obtaining:

[tex]h = 10/(\pi r^2)[/tex]

Substituting this expression for h into the cost equation, we obtain:

[tex]C = 2\pi r^2 * 20 + 2\pi r * 15 * 10/(\pi r^2)[/tex]

Simplifying, we have:

[tex]C = 40\pi r^2 + 300/r[/tex]

To minimize this function, we take its derivative with respect to r and set it equal to zero:

[tex]dC/dr = 80\pi r - 300/r^2 = 0[/tex]

Solving for r, we obtain:

[tex]r = (300/(80\pi ))^{(1/3)} = 0.508 m[/tex]

To find the corresponding value of h, we can use the volume equation:

[tex]h = 10/(\pi r^2)[/tex] ≈ 3.132 m

Therefore, the dimensions that minimize the cost of the material are a height of approximately 3.132 m and a radius of approximately 0.508 m.

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If the three-vertices of a rectangular-piece of material are ​(-2.2​,-2.3​), ​(-2.2​,1.5​) and ​(1.5​,1.5​), then the fourth-vertex is (1.5, -2.3).

A "Rectangle" is defined as a quadrilateral shape which has "four-sides" and "four-angles", where the opposite sides are parallel and of equal length, and the four angles are all right angles.

Let the coordinates of fourth-vertex be = (x,y).

Since it's a rectangular-piece of material, the "opposite-sides" of  rectangle must be parallel and have same-length.

The three vertices of th rectangular piece are :

⇒ Vertex 1: (-2.2, -2.3),

⇒ Vertex 2: (-2.2, 1.5),

⇒ Vertex 3: (1.5, 1.5)

We see that first two vertices have the same "x-coordinate" of -2.2, and the last two vertices have same "y-coordinate" of 1.5.

So, the "fourth-vertex" should have the same x-coordinate as Vertex 3, and the same y-coordinate as Vertex 1.

Therefore, coordinates of fourth-vertex is (1.5, -2.3).

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The approximation for f(1.4) using Euler's method with two steps of equal size is -0.632.

Euler's method is a numerical method for approximating the solutions to differential equations. It works by approximating the derivative at each step and using it to estimate the next value of the function.

In this case, we are given the differential equation dy/dx = x-y-1 and the initial condition f(1)=-2. We want to find an approximation for f(1.4) using Euler's method with two steps of equal size, starting at x=1.

To use Euler's method, we first need to determine the step size, which is the distance between x-values at each step. Since we have two steps of equal size, the step size is (1.4-1)/2 = 0.2.

Next, we use the initial condition to find the first approximation:

f(1.2) ≈ f(1) + f'(1)*0.2

= -2 + (1 - (-2) - 1)*0.2

= -1.2

Now, we can use this approximation to find the second approximation:

f(1.4) ≈ f(1.2) + f'(1.2)*0.2

= -1.2 + (1.2 - (-1.2) - 1)*0.2

= -0.632

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a. All members of the Avery Fitness Club in Carrollton, GA. Subset of members chosen for the survey.

b. List of all members from which the sample will be drawn. A probability sample of Stratified sampling will be used to ensure the representation of different member categories.

c. 100 members to ensure a representative sample while keeping costs and time constraints in mind.

a. The population for the survey is all members of the Avery Fitness Club in Carrollton, GA. The sample is a subset of the population selected for the survey. The sampling frame is a list of all the members of the Avery Fitness Club who are eligible to be selected for the sample. A population is the entire group of people or objects that the researcher wants to study, while a sample is a smaller group selected from the population. A sampling frame is a list of all the individuals or objects that the sample can be drawn.

b. For this survey, a Simple Random Sample (SRS) would be the best choice. This is because each member of the population has an equal chance of being selected for the sample, and this helps to minimize bias. Other options such as Systematic, Stratified, or Cluster Samples may introduce bias or complexity in the sampling process that might not be necessary for this survey.

c. The sample size for the survey should be determined based on the desired level of precision, the margin of error, and confidence level. For example, if we want a 95% confidence level and a margin of error of 5%, we would need a sample size of 246 members of the Avery Fitness Club. This ensures that the sample is large enough to accurately represent the population, while also minimizing the potential for sampling error.

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The container with dimensions 6 meters by 6 meters by 6  meters has

the minimum cost among all containers with a volume of 324  cubic

meters.

Let's first determine the dimensions of the square wooden base.

Let the side length of the square base be x meters. Then the height of

the container would also be x meters, since the container is made of a

square base and the sides are made of cardboard.

Therefore, the volume of the container can be expressed as[tex]V = x^2 \times x = x^3[/tex] cubic meters.

We want to find the dimensions of the cheapest container with a volume

of 324 cubic meters. Therefore, we need to minimize the cost of the

container, which is a function of the surface area of the cardboard sides.

The surface area of the cardboard sides is given by A = 4xh = 4x^2

square meters, where h is the height of the container.

Let's use the fact that the cost of wood is three times the cost of

cardboard to express the cost of the container in terms of x:

[tex]C(x) = 3x^2 + 4x^2 = 7x^2[/tex]

where the first term represents the cost of the wooden base and the

second term represents the cost of the cardboard sides.

Now we can express the cost of the container in terms of its volume:

[tex]C(V) = 7(V^{(2/3)})[/tex]

We want to find the value of x that minimizes C(V) subject to the

constraint that V = 324.

To do this, we can use the method of Lagrange multipliers:

[tex]L(x, \lambda) = 7(x^{(2/3)}) + \lambda(324 - x^3)[/tex]

Taking the partial derivative of L with respect to x and setting it equal to zero, we get:

[tex](14/3)x^{(-1/3)} - 3\lambda x^2 = 0[/tex]

Taking the partial derivative of L with respect to λ and setting it equal to zero, we get:

[tex]324 - x^3 = 0[/tex]

Solving for x, we get:

[tex]x = (324/3)^{(1/3)}[/tex] = 6 meters

Therefore, the dimensions of the cheapest container with a volume of

324 cubic meters are 6 meters by 6 meters by 6 meters. To verify that

this gives a minimum cost, we can take the second derivative of C(V)

with respect to V and evaluate it at V = 324:

C''(324) = -98/81 < 0

Since the second derivative is negative, this confirms that C(V) has a

local maximum at V = 324, and hence a local minimum at x = 6.

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