How can i solve this?

Question

American Airlines requires that the total outside dimensions (length+width+height) of a checked bag not exceed 62 inches. Suppose you want to check a bag whose height is same as its width. What is the largest volume bag of this shape that you can check on an American Air Flight?

Answer 1

Let x is width and height. Then length is (62 - 2x)., then volume V(x) = x²(62 - 2x) = 62x² - 2x³;

V'(x) = 124x - 6x² = 0; x = 0 or x = 62/3; V''(x) = 124 - 12x; V''(62/3) = 124 - 248 = - 124 < 0. So, we have maximum volume, max V = V(62/3) = (62/3)²(62 - 124/3) = (62/3)³ = 427.11 in³.

Answer 2

Word Problem Factual Information:

American Airlines requires that the total outside dimensions (length+width+height) of a checked bag not exceed 62 inches.

Suppose you want to check a bag whose height is same as its width.

What is the largest volume bag of this shape that you can check on an American Air Flight.

Solving for:

The largest volume bag that you can check on an American Air Flight.

Step 1: Important Factual Information

1. Let the length of the bag be (x) inches.
2. Since the height and width are the same,
3. They are both 1/2 (62 - x) inches.

Step 2:

Write the volume V(x) of the bag as the product of its length, width, and height:

V(x) = V(x) = x times 1/2(62-x) x 1/2 (62-x)

V(x) = 1/4x times (62-x)^2

Step 3:

Find the derivative V(x) to find the critical points:

V (x) = 1/4 times (62-x)^2 + x times 2(62-x) times (-1)

V(x) = 1/4 times (62 - x) times (62 - x) - 2x

V(x) = 1/4 times (62 - x) times (62 - 3x)

Step 4:

Set the derivative equal to zero to find the critical points:

V (x) = 0

x = 62/3

Step 5:

Since x must be between 0 and 62 inches,

1. the citical point x = 62/3 is the only one that is within the interval.

Step 6:

Calculate the volume at x = 62/3:

V (62/3) = 1/4 times 62/3 times (62 - 62/3 )^2

62/3 = 20.66666667^3 = 8826.962963 roundoff = 8826.96

V (62/3) = 8826.96 in^3

Solution:

Therefore, the largest volume bag that can be checked on an American Airlines flight is 8826.96 in^3

I hope the mathematical calculations (step by step approach) was helpful. Please let me know if you require any more assistance. If anyone in my neighborhood is interested in setting up an in-person math tutoring session. I look forward to hearing from them. Have an amazing day. Doris H.