Find the missing dimension for the triangle. The area is 256. 5 cm sq and the base is 27 cm

Answer:

To make the polynomial 9x^10 + ax^b + 100 a perfect square trinomial, we need to add a constant term to it such that it becomes a square of a binomial.

Let's first write the square of a binomial in general form:

(a + b)^2 = a^2 + 2ab + b^2

If we compare this general form with our polynomial, we can see that the first term, 9x^10, is equal to (3x^5)^2, which means that we can write our polynomial as:

(3x^5)^2 + ax^b + 100 = (3x^5 + c)^2

Expanding the right-hand side of this equation, we get:

(3x^5 + c)^2 = 9x^10 + 6cx^15 + c^2

Comparing the coefficient of x^15 on both sides, we get:

6c = 0

Since c cannot be zero (otherwise we would end up with the original polynomial), this means that we must have:

c = 0

Therefore, we can write our polynomial as:

(3x^5)^2 + ax^b + 100 = (3x^5)^2

Expanding the right-hand side, we get:

(3x^5)^2 = 9x^10

Therefore, we must have:

a = 0

b = 10

So the values of a and b that make the polynomial 9x^10 + ax^b + 100 a perfect square trinomial are a = 0 and b = 10.

The agent should sample at least 109 two-bedroom homes to estimate the mean selling price within $10,000 with an 89.9% level of confidence.

To estimate the required sample size, we need to use the formula:

n = (Zα/2 * σ / E)²

where Zα/2 = the critical value of the standard normal distribution for the given confidence level. For an 89.9% level of confidence, the value of Zα/2 is 1.645.

σ = the population standard deviation (unknown)

E = the margin of error (maximum distance between the sample mean and the true population mean)

To estimate σ, we can use the range method, which assumes that the population standard deviation is approximately equal to the range divided by 4:

σ ≈ (highest value - lowest value) / 4

In this case, σ ≈ ($1,000,000 - $50,000) / 4 = $237,500

Substituting the values into the formula,

n = (Zα/2 * σ / E)²

n = (1.645 * $237,500 / $10,000)²

n ≈ 109

Therefore, the agent should sample at least 109 two-bedroom homes to estimate the mean selling price within $10,000 with an 89.9% level of confidence.

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It is not possible for Zachary to buy a burger, a souvenir, and a pass with the given prices and have $4.50 left over.

Let's represent the cost of the souvenir as x.

Then, according to the problem:

The cost of the burger is 1/6 of the cost of the souvenir, which is (1/6)x.

The cost of the pass is 4/3 times the cost of the souvenir, which is (4/3)x.

The total cost of the burger, souvenir, and pass is equal to the amount Zachary brought to the state fair, which is $38.25.

Zachary had $4.50 left over after buying these items, so the cost of the burger, souvenir, and pass must be $33.75.

Putting all this information together, we can write an equation:

(1/6)x + x + (4/3)x = 33.75

Simplifying the left side of the equation:

(7/6)x = 33.75

Multiplying both sides by 6/7:

x = 30

Therefore, the cost of the souvenir is $30, the cost of the burger is (1/6) * 30 = $5, and the cost of the pass is (4/3) * 30 = $40.

To check that these values are correct, we can add them up:

30 + 5 + 40 = 75

And we can subtract the total cost from the amount Zachary brought:

38.25 - 75 = -36.75

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a) The apple sample has a greater typical value or median than the orange sample.

b) The orange sample has a greater variability or spread than the apple sample.

a) The median of the apple sample is 150 grams, while the median of the orange sample is 130 grams. Therefore, the apple sample has a greater typical value or median.

b) The interquartile range (IQR) of the apple sample is 8 grams, while the IQR of the orange sample is 111 grams. Therefore, the orange sample has a greater variability or spread.

c) It is possible that the heaviest piece of fruit in the two samples was an orange, as the orange sample has a larger range of weights than the apple sample. However, without knowing the maximum weight in each sample, we cannot say for certain whether the heaviest fruit was an orange or an apple.

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Answer:

Step-by-step explanation:

(3√27)^4 = (3^1/3 * 3^3/2)^4 = (3^5/2)^4 = 3^10 = 59049

So the equivalent expression is not listed among the given options.

Answer:

To find two numbers that add up to +29 and multiply to +100, you can use algebra. Let's call the two numbers "x" and "y". We know that:

x + y = 29

xy = 100

We can use the first equation to solve for one of the variables in terms of the other. For example, we can solve for "y" in terms of "x" by subtracting "x" from both sides:

y = 29 - x

Now we can substitute this expression for "y" into the second equation:

x(29 - x) = 100

Expanding the left-hand side of the equation gives:

29x - x^2 = 100

Rearranging and simplifying gives a quadratic equation:

x^2 - 29x + 100 = 0

This quadratic can be factored as:

(x - 4)(x - 25) = 0

So the two numbers that add up to +29 and multiply to +100 are +4 and +25.

The current ratio of length to width for US paper money is approximately 2.61 to 6.14 inches. This means that US paper money is roughly rectangular in shape, with a length that is about 2.61 times greater than its width.

The current size of US paper money is standardized by the Bureau of Engraving and Printing (BEP). According to the BEP, the current size of a US paper bill is 2.61 inches wide and 6.14 inches long. This size has remained the same since the 1920s, although earlier bills were larger.

The rectangular shape of US paper money makes it easy to handle and store, and the standardized size ensures that it can be easily recognized and processed by vending machines, bank machines, and other automated devices.

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The distance between the bakery and the library is about 6.7 units.

To discover the distance among points using coordinate subtraction, we want to use the distance formula.

The distance formula is derived from the Pythagorean theorem, which states that during a proper triangle, the forecourt of the period of the hypotenuse( the longest aspect) is identical to the sum of the places of the lengths of the opposite two aspects.

the distance components is as follows

[tex]distance =\sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)[/tex]

in which [tex]( x_1, y_1)[/tex] and [tex]( x_2, y_2)[/tex] are the coordinates of the two factors.

In this example, let's anticipate that the equals of the bakery( b) are( 3, 5) and the equals of the library( c) are( 9, 2). applying the space formula, we get

[tex]distance = \sqrt{((9 - 3)^2 + (2 - 5)^2)[/tex]

[tex]distance = \sqrt{x(6^2 + (-3)^2)[/tex]

distance = [tex]\sqrt{( 36 9)[/tex]

distance = [tex]\sqrt{(45)[/tex]

distance ≈ 6.7082

hence, the distance between the bakery and the library is about 6.7 units.

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The mean of the puppy weights changes by 0.14 pounds.

To determine how the mean of the puppy weights changes with the removal of the outlier, follow these steps:

1. First, identify the outlier in the given data set: Puppy Weights: 2.9, 3.0, 3.1, 3.1, 3.2, 3.3, 3.3, 3.4, 4.4. The outlier is 4.4 as it is significantly higher than the rest of the values.

2. Calculate the mean with the outlier: Add all the weights and divide by the total number of puppies (9). (2.9+3.0+3.1+3.1+3.2+3.3+3.3+3.4+4.4)/9 = 29.7/9 = 3.3

3. Calculate the mean without the outlier: Remove the outlier (4.4) from the sum and divide by the new total number of puppies (8). (29.7 - 4.4)/8 = 25.3/8 = 3.1625

4. Calculate the change in mean by subtracting the mean without the outlier from the mean with the outlier, and round to the nearest hundredth of a pound: 3.3 - 3.1625 = 0.1375, which rounds to 0.14.

With the removal of the outlier, the mean of the puppy weights changes by 0.14 pounds.

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Answer:

(3-x)^2=y

Step-by-step explanation:

To find the rate of change of the temperature difference between the two spacecraft, we need to first find the temperature at each spacecraft's position at time t=4.

For the first spacecraft, rt sin(t) = r4sin(4) and t=4, so its position is (4sin(4), 4, 0). Using the temperature function, we have T(4sin(4), 4, 0) = (4sin(4))(4)(5-2) = 48.08.

For the second spacecraft, rz cos(t) = r3cos(4) and t=-4/3, so its position is (3cos(4), -4/3, 7). Using the temperature function, we have T(3cos(4), -4/3, 7) = (3cos(4))(-4/3)(5-2) = -9.09.

Therefore, the temperature difference D between the two spacecraft at time t=4 is D = 48.08 - (-9.09) = 57.17.

To find the rate of change of D with respect to time, we use the Chain Rule. Let x = 4sin(t) and y = 4, so D = T(x, y, 0) - T(3cos(t), -4/3, 7). Then,

dD/dt = dD/dx * dx/dt + dD/dy * dy/dt

We already know that D = 48.08 - 9.09 = 57.17, so dD/dx = dT/dx = y(5-2x) = 4(5-2(4sin(4))) = -31.64.

We also have dx/dt = 4cos(4) and dy/dt = 0, since y is constant.

To find dD/dy, we take the partial derivative of T with respect to y, holding x and z constant: dT/dy = x(5-2y) = (4sin(4))(5-2(4)) = -28.16.

Putting it all together, we get:

dD/dt = dD/dx * dx/dt + dD/dy * dy/dt
= (-31.64)(4cos(4)) + (-28.16)(0)
= -126.56

Therefore, the rate of change of the temperature difference between the two spacecraft at time t=4 is -126.56.
Given the paths of the two spacecraft: r1(t) = (t sin(t), t, 0) and r2(t) = (t cos(t), -t, 7), and the temperature function T(x, y, z) = x * y * z^2, we want to determine the rate of change of the temperature difference D at time t=4 using the Chain Rule.

First, let's find the temperature for each spacecraft at time t:

T1(t) = T(r1(t)) = (t sin(t)) * t * 0^2
T1(t) = 0

T2(t) = T(r2(t)) = (t cos(t)) * (-t) * 7^2
T2(t) = -49t^2 cos(t)

Now, find the temperature difference D(t) = T2(t) - T1(t) = -49t^2 cos(t)

Next, find the derivative of D(t) with respect to t:

dD/dt = -98t cos(t) + 49t^2 sin(t)

Now, we need to evaluate dD/dt at t=4:

dD/dt(4) = -98(4) cos(4) + 49(4)^2 sin(4) ≈ -104.32

Thus, the rate of change of the temperature difference D at time t=4 is approximately -104.32 (in decimal notation, rounded to two decimal places).

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Answer:

I believe the answer is D.

Step-by-step explanation:

18 divided by 6 is 3, meaning y is constantly moving 3 spaces up as x moves to the right 1 on a graph.

The lateral surface area of a cylinder is given by the formula:

Lateral surface area = 2πrh

where r is the radius of the cylinder and h is the height of the cylinder.

In this case, the height of the cylinder is 8 ft and the radius of the cylinder is 4 ft (half of the diameter). So, we can plug in these values to get:

Lateral surface area = 2π(4 ft)(8 ft)

Lateral surface area = 64π square feet

Rounding to the nearest tenth, we get:

Lateral surface area ≈ 201.1 square feet (rounded to one decimal place)

Therefore, the lateral surface area of the cylinder is approximately 201.1 square feet.

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The investor invested $500,000 in Swan Peak and $600,000 in Riverside Community.

Let's denote the amount invested in Swan Peak as x and the amount invested in Riverside Community as y.

According to the given information:

1. The return on investment in Swan Peak was a 110% increase, which means the total return was 100% + 110% = 210% of the initial investment.

2. The return on investment in Riverside Community was 50% over the initial investment, which means the total return was 100% + 50% = 150% of the initial investment.

We are also given that the investor earned $1 million in profits.

Based on the above information, we can set up the following equations:

1.1 million = 2.1x + 1.5y  (equation 1)    [This equation represents the total profits earned by the investor.]

x + y = 1.1 million  (equation 2)    [This equation represents the total amount invested.]

To solve these equations, we can use substitution or elimination method. Let's use the elimination method:

Multiply equation 2 by 2.1 to make the coefficients of x in both equations equal:

2.1x + 2.1y = 2.31 million   (equation 3)

Now, subtract equation 1 from equation 3 to eliminate x:

(2.1x + 2.1y) - (2.1x + 1.5y) = 2.31 million - 1.1 million

0.6y = 1.21 million

Divide both sides by 0.6:

y = 2.01 million / 0.6

y ≈ 3.35 million

Substitute the value of y into equation 2:

x + 3.35 million = 1.1 million

x ≈ 1.1 million - 3.35 million

x ≈ -2.25 million

Since the amount invested cannot be negative, we discard the negative value.

Therefore, the investor invested approximately $500,000 in Swan Peak (x) and approximately $600,000 in Riverside Community (y).

Hence, the investor invested $500,000 in Swan Peak and $600,000 in Riverside Community.

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The surface area of the cylinder is 96cm²

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

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

Surface area of the cylinder = The area around the cylinder + 2 * The area of the top

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

Surface area of the cylinder = 64 + 2 * 16

Evaluate

Surface area of the cylinder = 96

Hence, the surface area of the cylinder is 96cm²

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The direction of fastest increase in temperature at the point (1, 3, 1) is given by the unit vector (-5/sqrt(10) , -3/sqrt(10) , -9/sqrt(10)).

To find the direction of fastest increase in temperature at the point (1, 3, 1), we need to find the gradient of the temperature function T(x, y, z) at that point.

The gradient of a function is a vector that points in the direction of steepest increase, and its magnitude is the rate of change in that direction. So, we can find the gradient vector ∇T(x, y, z) as follows:

∇T(x, y, z) = ( ∂T/∂x , ∂T/∂y , ∂T/∂z )

=[tex]( -20xe^(-2x^2-y^2-3z^2) , -2ye^(-2x^2-y^2-3z^2) , -6ze^(-2x^2-y^2-3z^2) )[/tex]

Therefore, at the point (1, 3, 1), the gradient of T(x, y, z) is:

∇T(1, 3, 1) = [tex]( -20e^(-8) , -6e^(-8) , -18e^(-8) )[/tex]

To find the direction of fastest increase, we need to normalize this vector to a unit vector. The magnitude of the gradient vector is:

|∇T(1, 3, 1)| = sqrt( (-[tex]20e^(-8))^2 + (-6e^(-8))^2 + (-18e^(-8))^2 )[/tex]

= sqrt( 640e^(-16) )

= 8e^(-8) sqrt(10)

So, the unit vector in the direction of fastest increase is:

( -20e^(-8) / (8e^(-8) sqrt(10)) , -6e^(-8) / (8e^(-8) sqrt(10)) , -18e^(-8) / (8e^(-8) sqrt(10)) )

= ( -5/sqrt(10) , -3/sqrt(10) , -9/sqrt(10) )

Therefore, the direction of fastest increase in temperature at the point (1, 3, 1) is given by the unit vector (-5/sqrt(10) , -3/sqrt(10) , -9/sqrt(10)).

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Answer: Nori

Step-by-step explanation:

2% of 5000 = 100

1% of 1000 = 10

The conclusion is YEN ≈ 63.43°.and  YN = 4√5 cm.

We can begin by drawing a diagram of the triangle LNM with the given measurements:

     N

     |\

     | \

    y|  \ x

     |   \

     |____\

     L  8cm M

Since X is the midpoint of MN, we know that MX = NX = 6/2 = 3cm. Similarly, Y is the midpoint of LN, so LY = NY = 8/2 = 4cm.

To find YN, we can use the Pythagorean theorem:

 Y________N

  |\     |

  | \    |

  |  \   | 6cm

  |   \  |

  |    \ |

 L|_____Y\|

     4cm

YN² = YL² + LN²

YN² = 4² + 8²

YN² = 80

YN = √80 = 4√5 cm

Therefore, YN = 4√5 cm.

To find YẼN, we need to find the angle YLN. Since Y is the midpoint of LN, YL is half the length of LN, which is 8cm. So YL = 4cm. We can use trigonometry to find the angle YLN:

tan(YLN) = opposite/adjacent

tan(YLN) = YL/LN

tan(YLN) = 4/8

tan(YLN) = 0.5

YLN ≈ 26.57°

Since LÑN = 90°, we know that YEN is the complement of YLN:

YEN = 90° - YLN

YEN ≈ 63.43°

Therefore, YEN ≈ 63.43°.

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The Stafford loan will have a lower balance by $272.54 at the time of repayment.

For the Stafford Loan, the principal is $10,720 and the interest rate is 5.95% compounded monthly, so the monthly interest rate is 0.0595/12 = 0.00495833.

After 12 months, the balance of the loan will be:

$10,720(1 + 0.00495833)¹² = $11,204.60

After the time of six-month grace period, the balance will be:

$11,204.60(1 + 0.00495833)⁶ = $11,689.66

For the PLUS Loan, the principal is $11,443.63 and the interest rate is 6.55% compounded monthly, so the monthly interest rate is 0.0655/12 = 0.00545833.

After 12 months, the balance of the loan will be:

$11,443.63(1 + 0.00545833)¹² = $11,962.20

Therefore, the Stafford Loan will have a lower balance at the time of repayment by:

$11,962.20 - $11,689.66 = $272.54

Hence, the Stafford loan will have a lower balance by $272.54 at the time of repayment.

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The matrix that represents a dilation by a factor of 3 followed by a reflection over the horizontal axis is given as follows:

[tex]\left[\begin{array}{cc}3&0\\0&-3\end{array}\right][/tex]

A dilation can be defined as a transformation that multiplies the distance between every point in an object and a fixed point, called the center of dilation, by a constant factor called the scale factor.

The scale factor for this problem is given as follows:

k = 3.

Hence the matrix is:

[tex]\left[\begin{array}{cc}3&0\\0&3\end{array}\right][/tex]

For the reflection over the horizontal axis, we have that the y-coordinate is multiplied by -1, hence the matrix rule is:

[tex]\left[\begin{array}{cc}3&0\\0&-3\end{array}\right][/tex]

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This will result in Circle 1 having the same center and radius as Circle 2, thus proving that the circles are similar.

To prove that Circle 1 and Circle 2 are similar, we can apply the following transformations to Circle 1:

1. Translation: Translate Circle 1 by moving its center from (-3, 5) to (7, 5). This is a horizontal translation of 10 units to the right.

2. Dilation: Dilate Circle 1 with a scale factor of 0.4, which will reduce its radius from 10 units to 4 units (the same as Circle 2).

These transformations will result in Circle 1 having the same center and radius as Circle 2, thus proving that the circles are similar.

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The utility maximizing units of consumption are approximately 1 or 15 units, depending on other factors such as budget constraints and the specific preferences of the individual.

To find the utility maximizing units of consumption, we need to calculate the first derivative of the utility function (U) with respect to q and set it equal to zero. Here's the utility function:

U = 18q + 7q^2 - (1/3)q^3

Now, we'll find the first derivative (dU/dq):

dU/dq = 18 + 14q - q^2

To find the utility maximizing units, set dU/dq to zero and solve for q:

0 = 18 + 14q - q^2

Rearrange the equation:

q^2 - 14q + 18 = 0

Now, we'll solve for q using the quadratic formula:

q = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -14, and c = 18. Plug these values into the formula:

q = (14 ± √((-14)^2 - 4 * 18)) / 2

q = (14 ± √(196 - 72)) / 2

q = (14 ± √124) / 2

The two possible solutions for q are:

q1 ≈ 1.27
q2 ≈ 14.73

Since the individual consumes discrete units, the utility maximizing consumption will be the whole number closest to these values.

Therefore, the utility maximizing units of consumption are approximately 1 or 15 units, depending on other factors such as budget constraints and the specific preferences of the individual.

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The measure of minor arc MPQ in a circle with central angle <MQP measuring 44 degrees is 316 degrees.

To find the measure of minor arc MPQ, we need to first find the measure of central angle <MNQ that intercepts this arc. Since minor arc MPQ and minor arc MP are adjacent, their sum equals the measure of minor arc MPNQ,

<MPQ+arc MP = <MPNQ

Substituting the measure of minor arc MP as 44 degrees, we get,

ZMPQ+ 44 360

Solving for MPQ, we get,

ZMPQ = 360-44

<MPQ = 316 degrees

Therefore, the measure of minor arc MPQ is 316 degrees.

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The probability that all four shirts selected from the box of 24 will be regular size is 0.1367 or 13.67%.

Probability refers to the likelihood or chance of an event occurring. In this scenario, we are interested in finding the probability that all four shirts selected from the box of 24 will be regular size.

To determine this probability, we first need to find the total number of ways in which we can select four shirts from the box. This is known as the sample space and can be calculated using the combination formula, which is nCr = n!/(r!(n-r)!). In this case, we have 24 shirts and are selecting four, so the sample space is 24C4 = (24!)/(4!(24-4)!) = 10,626.

Next, we need to find the number of ways in which we can select four regular shirts from the 15 available. This can be calculated using the same combination formula, which is 15C4 = (15!)/(4!(15-4)!) = 1,455.

Finally, we can calculate the probability of selecting all four regular shirts by dividing the number of favorable outcomes (i.e., selecting four regular shirts) by the total number of possible outcomes (i.e., the sample space). So, the probability of selecting all four regular shirts is 1,455/10,626 = 0.1367 or 13.67%.

Therefore, the probability that all four shirts selected from the box of 24 will be regular size is 0.1367 or 13.67%.

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Answer:

Obtuse

Step-by-step explanation:

Right angles are 90° so incorrect

Acute angles are less than 90°, also incorrect

Obtuse angles are between 90 and 180. And this is the range which 137 is found

So angle S is obtuse

Answer:

B) Obtuse

Step-by-step explanation:

Let's look at the definitions for each answer choice:

Acute: An acute angle is an angle that measures less than 90°.

Obtuse: An obtuse angle is an angle that measures more than 90°.

Right: A right angle is an angle that measures exactly 90°.

Given that Angle S is 137°, we can classify this angle as an obtuse angle, as 137>90.

Hope this helps! :)

The answer is reasonable because it is positive and also the equation is true . it would take Lara about 5.3 hours to clean the tank by herself.

Let's denote the time it takes for Lara to clean the tank alone as "L". We can use the formula for the combined work rate of two people, which is:

(1/7) + (1/L) = (1/3)

Multiplying both sides by the least common denominator, 21L, gives:

3L + 21 = 7L

Subtracting 3L from both sides, we get:

21 = 4L

Dividing both sides by 4, we get:

L = 5.25 hours (to the nearest tenth)

The answer is reasonable because it is positive, and it is also less than 7 hours, which is Kevin's time. When substituted back into the original equation, we get:

(1/7) + (1/5.25) = (1/3)

0.1429 + 0.1905 = 0.3333

0.3334 ≈ 0.3333

The equation is true, so the answer is reasonable. Therefore, it would take Lara about 5.3 hours to clean.

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The correct expression is 4,000 times. 1. 4 to the sixth power.

The formula for the future value of an investment with annual compounding interest is:

A = P(1 + r)ⁿ

A = future value

P = principal amount

r = annual interest rate expressed as a decimal

n = number of years.

In this case, Jaleesa deposited $4,000 at an annual interest rate of 4% (0.04 as a decimal) and the investment is compounded annually for 6 years. So the expression that can be used to find the value of her investment at the end of 6 years is:

A = 4,000(1 + 0.04)⁶

Simplifying the expression, we get:

A = 4,000(1.04)⁶

Therefore, the correct expression is 4,000 times. 1. 4 to the sixth power.

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The parametric equations to plot points. The sixth point is (0, 1).

The parametric equations given are:

x = sin t

y = 1 − cos t

To plot points for these equations, we can choose some values of t and substitute them in the equations to get the corresponding values of x and y. Here are some points we can plot:

When t = 0, x = sin 0 = 0 and y = 1 − cos 0 = 1.

So the first point is (0, 1).

When t = π/4, x = sin (π/4) = √2/2 and y = 1 − cos (π/4) = 1 − √2/2.

So the second point is (√2/2, 1 − √2/2).

When t = π/2, x = sin (π/2) = 1 and y = 1 − cos (π/2) = 0.

So the third point is (1, 0).

When t = π, x = sin π = 0 and y = 1 − cos π = 2.

So the fourth point is (0, 2).

When t = 3π/2, x = sin (3π/2) = −1 and y = 1 − cos (3π/2) = 0.

So the fifth point is (−1, 0).

When t = 2π, x = sin 2π = 0 and y = 1 − cos 2π = 1.

So the sixth point is (0, 1).

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If Ali took 6 pencils from a new box of pencils and there are 18 pencils left in the box, then there were 24 pencils in the brand new box.

To see why, you can add the number of pencils Ali took to the number of pencils left in the box:

6 + 18 = 24

Therefore, there were 24 pencils in the brand new box before Ali took 6 of them.

The length of the variable x, obtained from the formula for the surface area of a solid about 10.63 inches

What is the surface area of a solid object?

The surface area of a solid object is the area of the outside surface of the  object.

The dimensions of of the open top box with the square corners 2 in by 2 in cut from the corners are;

Length of the box, L = x - 4 inches

Width of the box, W = x - 4 inches

Height of the box, H = 2 inches

The surface area of the box is therefore;

Volume, A = L × W + 2 × L × H + 2 × W × H

Plugging in the expressions for the length, width and height of the box, the surface area = (x - 4) × (x - 4) + 2 × 2 × (x - 4) + 2 × 2 × (x - 4) = 9·x² - 40·x + 16

The surface area of the box, A = 9·x² - 40·x + 16

Second part;

When the surface area = 609 in², we get;

A = 609 = 9·x² - 40·x + 16

9·x² - 40·x + 16 - 609 = 9·x² - 40·x - 593 = 0

x = (20 + √(5737))/9 ≈ 10.63, and x = (20 - √(5737))/9 ≈ -6.19

The length x ≈ 10.63 inches

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