Show that the surface area of the solid region bounded by the three cylinders x^2 + y^2 = 1, y^2 + z^2 = 1 and x^2 + z^2 = 1 is 48 – 24√2.

The translating point will be (x, y) --> (8-x, -5-y).

We have to translate a point 8 units right and 5 units down followed by a 180 degree rotation.

Now, the rule for 180 rotation is

(x, y)  --> (x, -y)

and, to shift 8 unit right apply (8-x)

and, to shift 5 unit down apply (5-y)

Then, the translating point will be (x, y) --> (8-x, -5-y).

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The approach to showing uniform continuity will depend on the specific function and domain given.

Without a given function and domain, I cannot provide a specific answer. However, I can provide a general approach to determining whether a function is uniformly continuous on a given domain.

To show that a function is uniformly continuous on a domain, we need to show that for any ε > 0, there exists a δ > 0 such that for any x, y in the domain with |x - y| < δ, we have |f(x) - f(y)| < ε.

One approach to showing uniform continuity is to use the theorem that a continuous function on a closed and bounded interval is uniformly continuous (the Extreme Value Theorem and Corollary). This means that if the domain of the function is a closed and bounded interval, and the function is continuous on that interval, then it is uniformly continuous on that interval.

Another approach is to use the Intermediate Value Theorem. If we can show that the function satisfies the conditions of the Intermediate Value Theorem on the given domain, then we can conclude that the function is uniformly continuous on that domain. The Intermediate Value Theorem states that if f is continuous on a closed interval [a, b], and if M is a number between f(a) and f(b), then there exists a number c in [a, b] such that f(c) = M.

To use the Intermediate Value Theorem to show uniform continuity, we need to show that for any ε > 0, there exists a δ > 0 such that for any x, y in the domain with |x - y| < δ, we have |f(x) - f(y)| < ε/2. Then, using the Intermediate Value Theorem, we can show that for any M such that |M - f(x)| < ε/2, there exists a number c in the domain such that f(c) = M. Combining these two results, we can show that for any ε > 0, there exists a δ > 0 such that for any x, y in the domain with |x - y| < δ, we have |f(x) - f(y)| < ε.

Overall, the approach to showing uniform continuity will depend on the specific function and domain given.

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The integral test states that if a series is a sum of terms that are positive and decreasing, and if the terms of the series can be expressed as the values of a continuous and decreasing function, then the series converges if and only if the corresponding improper integral converges.

Let's apply the integral test to the given series. We need to find a continuous, positive, and decreasing function f(x) such that the series is the sum of the values of f(x) for x ranging from 2 to infinity.

For the first series, we have:

∑n=2∞n(lnn)p

Let f(x) = x(lnx)p. Then f(x) is continuous, positive, and decreasing for x ≥ 2. Moreover, we have:

f'(x) = (lnx)p + px(lnx)p-1

f''(x) = (lnx)p-1 + p(lnx)p-2 + p(lnx)p-1

Since f''(x) is positive for x ≥ 2 and p > 0, f(x) is concave up and the trapezoidal approximation underestimates the integral. Therefore, we have:

∫2∞f(x)dx = ∫2∞x(lnx)pdx

Using integration by substitution, let u = lnx, then du = 1/x dx. Therefore:

∫2∞x(lnx)pdx = ∫ln2∞u^pe^udu

Since the exponential function grows faster than any power of u, the integral converges if and only if p < -1.

For the second series, we have:

∑n=2∞1/n(ln⁡n)²

Let f(x) = 1/(x(lnx)²). Then f(x) is continuous, positive, and decreasing for x ≥ 2. Moreover, we have:

f'(x) = -(lnx-2)/(x(lnx)³)

f''(x) = (lnx-2)²/(x²(lnx)⁴) - 3(lnx-2)/(x²(lnx)⁴)

Since f''(x) is negative for x ≥ 2, f(x) is concave down and the trapezoidal approximation overestimates the integral. Therefore, we have:

∫2∞f(x)dx ≤ ∑n=2∞f(n) ≤ f(2) + ∫2∞f(x)dx

where the inequality follows from the fact that the series is the sum of the values of f(x) for x ranging from 2 to infinity.

Using the comparison test, we have:

∫2∞f(x)dx = ∫ln2∞(1/u²)du = 1/ln2

Therefore, the series converges if and only if p > 1.

In summary, the series ∑n=2∞n(lnn)p converges if and only if p < -1, and the series ∑n=2∞1/n(ln⁡n)² converges if and only if p > 1. For values of p such that -1 ≤ p ≤ 1, the series diverges.
To find the values of p for which the series converges or diverges using the integral test, we will first write the series and then perform the integral test.

The given series is:

∑(n=2 to infinity) [1/n(ln(n))^p]

Now, let's consider the function f(x) = 1/x(ln(x))^p for x ≥ 2. The function is continuous, positive, and decreasing for x ≥ 2 when p > 0.

We will now perform the integral test:

∫(2 to infinity) [1/x(ln(x))^p] dx

To evaluate this integral, we will use the substitution method:

Let u = ln(x), so du = (1/x) dx.

When x = 2, u = ln(2).
When x approaches infinity, u approaches infinity.

Now the integral becomes:

∫(ln(2) to infinity) [1/u^p] du

This is now an integral of the form ∫(a to infinity) [1/u^p] du, which converges when p > 1 and diverges when p ≤ 1.

So, for the given series:

- It converges when p > 1.
- It diverges when p ≤ 1.

In conclusion, using the integral test, the series ∑(n=2 to infinity) [1/n(ln(n))^p] converges for values of p > 1 and diverges for values of p ≤ 1.

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Answer:

11

Step-by-step explanation:

Volume of right cone = (1/3) · π · r² · h

V = 968π

h = 24 units

Let's solve

968π = (1/3) · π · r² · 24

2904π =  π · r² · 24

121π = π · r²

121 = r²

r = 11

So, the radius is 11 units

We can express this as a percentage and round it to two decimal places:

P(red or yellow golf ball) = 16/22 * 100%

= 72.73% (rounded to two decimal places)

To find the probability of taking out a red or a yellow golf ball, we need to add the probability of taking out a red golf ball and the probability of taking out a yellow golf ball. We can find the probability of taking out a red golf ball by dividing the number of red golf balls by the total number of golf balls in the box:

P(red golf ball) = 9 / (9 + 6 + 7) = 9 / 22

Similarly, we can find the probability of taking out a yellow golf ball:

P(yellow golf ball) = 7 / (9 + 6 + 7) = 7 / 22

To find the probability of taking out either a red or a yellow golf ball, we can add these probabilities:

P(red or yellow golf ball) = P(red golf ball) + P(yellow golf ball)

= 9/22 + 7/22

= 16/22

Finally, we can express this as a percentage and round it to two decimal places:

P(red or yellow golf ball) = 16/22 * 100%

= 72.73% (rounded to two decimal places)

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To calculate the deposit sample mean, we need to add up all the bank deposits and divide by the number of respondents. From the table, the total bank deposits is $45,000 and there are 10 respondents. So the deposit sample mean is:

Deposit sample mean = Total bank deposits / Number of respondents
Deposit sample mean = $45,000 / 10
Deposit sample mean = $4,500

To calculate the deposit sample standard deviation, we need to first find the differences between each respondent's bank deposit and the sample mean. We then square these differences, add them up, divide by the number of respondents minus one (known as the degrees of freedom), and then take the square root. Here are the steps:

Step 1: Find the differences between each respondent's bank deposit and the sample mean:

Respondent 1: $3,000 - $4,500 = -$1,500
Respondent 2: $5,000 - $4,500 = $500
Respondent 3: $4,500 - $4,500 = $0
Respondent 4: $6,000 - $4,500 = $1,500
Respondent 5: $3,500 - $4,500 = -$1,000
Respondent 6: $5,500 - $4,500 = $1,000
Respondent 7: $6,500 - $4,500 = $2,000
Respondent 8: $4,000 - $4,500 = -$500
Respondent 9: $4,500 - $4,500 = $0
Respondent 10: $4,500 - $4,500 = $0

Step 2: Square each difference:

Respondent 1: (-$1,500)^2 = $2,250,000
Respondent 2: $500^2 = $250,000
Respondent 3: $0^2 = $0
Respondent 4: $1,500^2 = $2,250,000
Respondent 5: (-$1,000)^2 = $1,000,000
Respondent 6: $1,000^2 = $1,000,000
Respondent 7: $2,000^2 = $4,000,000
Respondent 8: (-$500)^2 = $250,000
Respondent 9: $0^2 = $0
Respondent 10: $0^2 = $0

Step 3: Add up the squared differences:

$2,250,000 + $250,000 + $0 + $2,250,000 + $1,000,000 + $1,000,000 + $4,000,000 + $250,000 + $0 + $0 = $11,000,000

Step 4: Divide by the degrees of freedom (number of respondents minus one):

$11,000,000 / 9 = $1,222,222.22

Step 5: Take the square root:

Deposit sample standard deviation = √$1,222,222.22 = $1,105.54

Therefore, the deposit sample mean is $4,500 and the deposit sample standard deviation is $1,105.54.

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Answer: 720

Step-by-step explanation:

there are 6 letters in "choice"

and they must be permuted into 6 word letters:

6P6 = 720 distinct possibilities

The measure of the central angle m ∠JIK is 160°.

Given that the circle I, in which IJ is the radius of 9 units, the area of shaded sector = 36π, we need to find the m ∠JIK, the central angle.

Area of the sector = central angle / 360° × π × radius²

∴ 36π = m ∠JIK / 360° × π × 9²

m ∠JIK = 360° × 4 / 9

m ∠JIK = 160°

Hence, the measure of the central angle m ∠JIK is 160°.

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The constant of proportionality is equal to the unit rate in a proportional relationship. The unit rate for Bolin's tai chi classes is 16.

In Mathematics and Geometry, a proportional relationship refers to a type of relationship that produces equivalent ratios or unit rates, and it can be modeled or represented by the following mathematical equation:

y = kx

Where:

In Mathematics, the unit rate is sometimes referred to as unit price or unit ratio and it can be defined as the price that is being charged by a seller for the sale of a single unit of product or quantity, especially in a proportional relationship:

Constant of proportionality, k = y/x

Constant of proportionality, k = 16.

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The average sales for that period is 1375.

The average, also known as the mean, is a statistical measure that represents the central tendency of a set of values. It is calculated by summing up all the values in a dataset and dividing the sum by the total number of values.

Mathematically, the average (mean) is calculated as:

Average = (Sum of all values) / (Total number of values)

To calculate the average sales for the given period, you'll need to follow these steps:

1. Add up the daily sales figures: 1000 + 1200 + 1300 + 2000 = 5500

2. Count the number of days in the period: 4 days

3. Divide the total sales by the number of days to find the average: 5500 / 4 = 1375

So, the average sales for that period is 1375.

Therefore, the correct answer is: B) 1,375

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The modification to a research study that will result in a narrower confidence interval is option D is correct choice

The modification to a research study that will result in a narrower confidence interval is option D: decreasing the confidence level and increasing the sample size. By decreasing the confidence level, we are willing to accept a lower level of certainty in our results, which can lead to a narrower interval. Increasing the sample size also leads to a narrower interval as it reduces the variability in our data and increases the precision of our estimates.

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The probability that a data value is between 60 and 36 is 95.44%.

We have,

Mean = 34

Standard deviation = 2

So, P( 30 < x < 36)

= P (30 - 34/2) - P(36-34/2)

= P(-2) - P(2)

=  0.9772498 -0.0227501

= 0.9544

= 95.44%

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Answer: a. true

b. true

c. false

d. true

Step-by-step explanation:

Answer:  B

Step-by-step explanation:

If you fold that bottom side up.  the door is not on the closer side to the window. so B is wrong because the door is near the window.

The experimental probability of rolling a number greater than 4 is 18/60

It will be given by the number of times that the outcome was greater than 4 (so a 5 or a 6) over the total number of trials.

We can see that the total number of trials is 60, and we have:

Adding that: 10 + 8 = 18

Then the experimental probability of a number greater than 4 is:

E = 18/60

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We can conclude that the system represented by the given Gauss-Jordan elimination has infinitely many solutions, and the values of the variables can be expressed in terms of a free variable x4.

The Gauss-Jordan elimination is a method used to solve a system of linear equations. The final step of the method is to transform the augmented matrix of the system into reduced row echelon form, which allows for easy identification of the solution(s) of the system.

In the given final step of the Gauss-Jordan elimination, the augmented matrix of the system is represented as:

11 0 0 51

0  1 0 21

0  0 1 -3

0  0 0  0

The augmented matrix is in reduced row echelon form, where the leading coefficients of each row are all equal to 1, and there are no other non-zero elements in the same columns as the leading coefficients. The last row of the matrix corresponds to the equation 0 = 0, which represents an identity that does not provide any new information about the system.

The system represented by this matrix is:

11x1 + 51x4 = 0

x2 + 21x4 = 0

x3 - 3x4 = 0

We can see that the third row of the matrix corresponds to an equation of the form 0x1 + 0x2 + 0x3 + 0x4 = 0, which indicates that the variable x4 is a free variable. This means that the system has infinitely many solutions, and the value of x4 can be chosen arbitrarily.

The values of x1, x2, and x3 can be expressed in terms of x4 using the equations given by the first three rows of the matrix. For example, we can solve for x1 as follows:

11x1 + 51x4 = 0

x1 = -51/11 x4

Similarly, we can solve for x2 and x3:

x2 = -21 x4

x3 = 3 x4

Therefore, the general solution of the system is:

x1 = -51/11 x4

x2 = -21 x4

x3 = 3 x4

x4 is a free variable

In summary, we can conclude that the system represented by the given Gauss-Jordan elimination has infinitely many solutions, and the values of the variables can be expressed in terms of a free variable x4.

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The symbol ∪ represents the "union of events" in the context of probability and set theory.

The symbol ∪ indicates the union of events. This option corresponds to choice (d) in your given list. The union of events refers to the occurrence of at least one of the events in question. In other words, it combines the outcomes of two or more events into a single set, without any repetitions. This concept is essential in understanding probability theory, as it helps to analyze the likelihood of different events happening together or separately.

This means that it represents the set of all outcomes that belong to either one or both of the events being considered. For example, if event A represents rolling an even number on a die and event B represents rolling a number greater than 4, then the union of events A and B would be the set of outcomes {2, 4, 5, 6}. It is important to note that the union of events is different from the intersection of events, which represents the set of outcomes that belong to both events being considered. The sample space, on the other hand, represents the set of all possible outcomes of an experiment. Finally, the symbol ∑ represents the sum of probabilities of events, not the symbol ∪.

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The solution to the integral equation y(x) = 2 + x [t − ty(t)] dt is: y(x) = 1 + e⁻ˣ

Note that this solution satisfies the initial condition y(0) = 2.

To solve the given integral equation y(x) = 2 + x [t − ty(t)] dt, we need to first find the value of y(x) that satisfies this equation. We can obtain an initial condition for y(x) by setting x=0 in the equation and solving for y(0). Then, we can use a method such as separation of variables or substitution to find the general solution for y(x).

Let's start by finding the initial condition for y(x). Setting x=0 in the integral equation, we get:

y(0) = 2 + 0 [t − t y(t)] dt

y(0) = 2

So, we know that y(0) = 2. This will be useful when we find the general solution for y(x).

Now, let's use substitution to solve the integral equation. Let u = y(x), du/dx = y'(x), and v = t - y(t). Then, we have:

y(x) = 2 + x [t − ty(t)] dt

u = 2 + x [v] dt

du/dx = v + x dv/dx

Substituting du/dx and v in terms of u and x, we get:

v = t - u

du/dx = t - u + x (dv/dx)

du/dx + u = t + x (dv/dx)

We can use the integrating factor method to solve this first-order linear differential equation. The integrating factor is eˣ, so we have:

eˣ du/dx + eˣ u = teˣ + x eˣ (dv/dx)

(d/dx)(eˣ u) = (teˣ)' = eˣ

eˣ u = eˣ + C

u = 1 + Ce⁻ˣ

Substituting u = y(x) and using the initial condition y(0) = 2, we get:

y(x) = 1 + Ce⁻ˣ (general solution)

y(0) = 2 = 1 + C (using initial condition)

C = 1

Therefore, the solution to the integral equation y(x) = 2 + x [t − ty(t)] dt is:

y(x) = 1 + e⁻ˣ

Note that this solution satisfies the initial condition y(0) = 2.

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Answer:

77 cm^2

Step-by-step explanation:

rectangular prism or cuboid

Right rectangular prism Solve for surface area▾

A = 77

L = 2

w = 3

h = 6.5

A=2(wl+hl+hw) = 2.(3.2+6.5.2+6.5.3)=77

chegg

The new dimensions of the shape that is being dilated by the scale factor of 3 would be = 180 by 270.

To calculate the new dimensions of a shape, the formula for a scale factor can be used.

Scale factor = Bigger dimensions/smaller dimensions

Scale factor = 3

Length of bigger dimension = 60

width = 90

Dilated length = 60×3 = 180

width of dilated shape = 90×3= 270

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The expected profit is $67.5K

To determine what ABC should do to maximize expected profits, we can use a decision tree to analyze the different possible outcomes and their probabilities.

First, let's define the events and their probabilities:

H: the show is a hit (P(H) = 0.25)

F: the show is a flop (P(F) = 0.75)

PH: the market research firm predicts a hit (P(PH|H) = 0.9, P(PH|F) = 0.2)

PF: the market research firm predicts a flop (P(PF|H) = 0.1, P(PF|F) = 0.8)

Using these probabilities, we can construct the following decision tree:

               / PH: P = 0.225 (0.25 * 0.9)

              /

             /

            /

    H: P = 0.25

           \

            \

             \ PF: P = 0.025 (0.25 * 0.1)

              \

               \

                \

                 \

                  \

                   \ PH: P = 0.15 (0.75 * 0.2)

                    \

                     \

                      F: P = 0.75

                     /

                    /

           PF: P = 0.6 (0.75 * 0.8)

starting from the top of the tree, we can calculate the expected profits for each decision:

If ABC launches the show without doing the market research, the expected profit is:

E1 = P(H) * $400K + P(F) * (-$100K) = $75K

If ABC does the market research and it predicts a hit, the expected profit is:

E2 = P(H and PH) * $400K - $40K + P(F and PH) * (-$40K) = $89K

If ABC does the market research and it predicts a flop, the expected profit is:

E3 = P(H and PF) * $400K - $40K + P(F and PF) * (-$100K - $40K) = -$52K

Therefore, the decision that maximizes expected profits is to do the market research and launch the show only if the market research predicts a hit.

The expected profit in this case is:

E = P(H and PH) * $400K - $40K + P(F and PH) * (-$40K) = 0.225 * $400K - $40K + 0.15 * (-$40K) = $67.5K

Therefore, the expected profit is $67.5K.

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The local truncation error incurred in the approximation of y(2.01) using the next term in the corresponding Taylor series is O(0.0001)

We can use the Taylor series method of order three to estimate y(2.01) in one step. Let's first write the Taylor series expansion of y(x) about x=2 up to the third derivative:

[tex]y(x) = y(2) + (x-2)y'(2) + \frac{(x-2)^{2} }{2!} y''(2) + \frac{(x-2)^{3} }{2!} y'''(2) + O((x-2)^{4} )[/tex]

where  [tex]y'(x) = 3x^{2} y(x), y''(x) = 6xy(x) + 3x^{2} y'(x), y'''(x) = 9x^{2} y'(x) + 18xy'(x) + 6x^{2} y''(x).[/tex]

(a) To estimate y(2.01) in one step, we need to evaluate the above expression at x=2.01. Using y(2) = 3 and [tex]y'(2) = 3(2)^{2} (3) = 36[/tex], we get:

[tex]y(2.01) = y(2) + (2.01-2)y'(2) + \frac{(2.01-2)^{2} }{2!} y''(2)[/tex]

[tex]= 3 + 0.01(36) + \frac{(0.01)^{2} }{2!} (6(2)(3) + 3(2)^{2} (3)(3))[/tex]

=3.1089

Therefore, y(2.01) =3.1089.

(b) The local truncation error is given by the next term in the Taylor series expansion, which is O((x-2)⁴) in this case. Evaluating this term at x=2.01, we get:

O((2.01-2)⁴) = O(0.0001)

Therefore, the local truncation error incurred in the approximation of y(2.01) using the next term in the corresponding Taylor series is O(0.0001).

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The rule for the composed transformation formed by the translation 8 units right and 5 units down followed by a 180 degree rotation is (x, y) --> (8-x, -5-y)

The rule for the transformation formed by the translation 8 units right and 5 units down followed by a 180-degree rotation can be determined by considering the effect of each transformation separately and then composing them.

First, let's consider the effect of the translation. A translation moves every point in the plane a certain distance in a certain direction. In this case, we are translating 8 units to the right and 5 units down. So, if we have a point (x, y), the translated point will be (x+8, y-5).

Next, let's consider the effect of the 180-degree rotation. A rotation of 180 degrees flips a figure around a line of symmetry, which in this case would be the point where the horizontal line passing through the midpoint of the translation intersects the vertical line passing through the midpoint of the translation. This point is (4, -2.5).

Thus, if we start with a point (x, y), the effect of the translation is to move it to (x+8, y-5), and the effect of the rotation is to flip it around the point (4, -2.5). Therefore, the rule for the composed transformation is:

(x, y) --> (x+8, y-5) --> (8-x, -5-y)

In other words, to apply this transformation to a point, we first translate it 8 units right and 5 units down, and then we reflect it across the point (4, -2.5).

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Complete question is:

What is the rule for the transformation formed by the translation 8 unitys right and 5 units down followed by a 180 degree rotation , assuming the initial point as (x,y)?

The answer is not one of the choices provided.

The distribution of sample means follows a normal distribution with a mean equal to the population mean (11) and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

So, for a sample size of 35, the distribution of sample means is normal with a mean of 11 and a standard deviation of 4.7/sqrt(35) = 0.795.

We need to find the probability that the mean blood pressure of the 35 people will be less than 122. We can standardize the distribution of sample means to a standard normal distribution with mean 0 and standard deviation 1 using the z-score formula:

z = (x - mu) / (sigma / sqrt(n))

where x is the sample mean, mu is the population mean, sigma is the population standard deviation, and n is the sample size.

Substituting the given values, we get:

z = (122 - 11) / (4.7 / sqrt(35)) = 37.98

We can then use a standard normal distribution table or calculator to find the probability of z being less than 37.98. Since the standard normal distribution is symmetric, we can also find this probability as 1 minus the probability of z being greater than 37.98.

Using a standard normal distribution table or calculator, we get:

P(z < 37.98) = 1 (to a very high degree of precision)

Therefore, the probability that the mean blood pressure of 35 people will be less than 122 is essentially 1, or 100%. The answer is not one of the choices provided.

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The 40% clearance discount is already factored into the clearance price of $30, so applying the 20% coupon only reduces the price further by 20% of $30, or $6. Therefore, you would pay $24 for the item with both discounts applied.

Let's break down the discounts and calculate the final price of the item using the terms "normal", "price", and "quantitatively".
The normal price of the item: $50

First, apply the 40% clearance discount:
40% off the normal price = 0.4 * $50 = $20

Subtract the clearance discount from the normal price:
New discounted price = $50 - $20 = $30

Now, apply the 20% off coupon to the discounted price:
20% off the new discounted price = 0.2 * $30 = $6

Quantitatively, the calculation would be:
Normal price = $50
Clearance price (40% off) = $30
Coupon discount (20% off clearance price) = 0.20 x $30 = $6
Final price = $30 - $6 = $24

Subtract the coupon discount from the discounted price:
Final price = $30 - $6 = $24

So, you should pay $24 for the item. It is not the same as taking 60% off the normal price because the discounts are applied sequentially, not combined. Quantitatively, you can see that taking 60% off the normal price would result in a $30 discount ($50 * 0.6), while the actual total discount here is $26 ($20 + $6).

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The surface area of the solid formed by the net = 150.72 in²

From the figure we can observe that the solid formed by the net is the net of a cylinder.

The cylinder bases are the 2 circles and the curved surface of the cylinder is the rectangle.

The surface area of the cylinder = Area of the 2 circles + area of the rectangle

Here, the diameter of circle is 4 in

So, the radius of circle = ½ × 4

                                     = 2 in

So, the area of the 2 circles would be,

2(πr²)

= 2 × 3.14 × 2²

= 25.12 in²

Here the width of the rectangle is 10 in. and the length is nothing but the circumference of the circle.

so, length L =  πd

                   = π × 4

                   = 12.56 in

Now the area of rectangle would be,

L × W

= 12.56 × 10

= 125.6 in²

The total surface area of net would be,

Area of the 2 circles + area of the rectangle

= 25.12 + 125.6

= 150.72 in²

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