i cant figure this last problem out, i would really appreciate the help,
A certain airline requires that the total outside dimensions (length + width + height) of a carry-on bag not exceed 37 inches. Suppose you want to carry on a bag whose length is twice its height. What is the largest volume bag of this shape that you can carry on a flight? (Round your answer to the nearest integer.)
A certain airline requires that the total outside dimensions (length + width + height) of a carry-on bag not exceed 37 inches. Suppose you want to carry on a bag whose length is twice its height. What is the largest volume bag of this shape that you can carry on a flight? (Round your answer to the nearest integer.)
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So the dimensions of the bag are going to be x by 2x by 37 - 3x.
The volume of this bag is V = 2x^2 (37 - 3x) = 74 x^2 - 6x^3
dV/dx = 148x - 18x^2
To find the extreme points, we must solve dV/dx = 0
148x - 18x^2 = 0
2x(74 - 9x) = 0
x = 0 or x = 74/9
So the largest volume is
148(74/9) - 18 (74/9)^2 = 405224/243 ≈ 1668 in^3
The volume of this bag is V = 2x^2 (37 - 3x) = 74 x^2 - 6x^3
dV/dx = 148x - 18x^2
To find the extreme points, we must solve dV/dx = 0
148x - 18x^2 = 0
2x(74 - 9x) = 0
x = 0 or x = 74/9
So the largest volume is
148(74/9) - 18 (74/9)^2 = 405224/243 ≈ 1668 in^3