Question

A parcel delivery service will deliver a package only if the length plus girth​ (distance around) does not exceed 84 inches. ​(A) Find the dimensions of a rectangular box with square ends that satisfies the delivery​ service's restriction and has maximum volume. What is the maximum​ volume? ​(B) Find the dimensions​ (radius and​ height) of a cylindrical container that meets the delivery​ service's restriction and has maximum volume. What is the maximum​ volume?

Answer 1

Answer:

A. 14x14x28

B. The maximum volume is 5488 cuibic inches

Step-by-step explanation:

The problem states that the box has square ends, so you can express volume with:

$v=x^{2} y$

Using the restriction stated in the problem to get another equation you can substitute in the one above:

$4x+y=84$

Substituting y whit this equation gives:

$v=x^{2} (84-4x)\\\\v=84x^{2} -4x^{3}$

Now find the limit of x:

$\frac{84x^{2}-4x^{3}}{dx}=168x-12x^{2}\\\\x=\frac{168}{12}=14$

Find the length:

$y=84-4(14)=28$

You can now calculate the maximum volume:

$v=(14)^{2}(28)= 5488$