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?
Accepted Solution
A:
Answer:A. 14x14x28B. The maximum volume is 5488 cuibic inchesStep-by-step explanation:The problem states that the box has square ends, so you can express volume with:[tex]v=x^{2} y[/tex]Using the restriction stated in the problem to get another equation you can substitute in the one above:[tex]4x+y=84\\\\[/tex]Substituting y whit this equation gives:[tex]v=x^{2} (84-4x)\\\\v=84x^{2} -4x^{3}[/tex]Now find the limit of x:[tex]\frac{84x^{2}-4x^{3}}{dx}=168x-12x^{2}\\\\x=\frac{168}{12}=14[/tex]Find the length:[tex]y=84-4(14)=28[/tex]You can now calculate the maximum volume:[tex]v=(14)^{2}(28)= 5488[/tex]