The material for constructing the base of an open box costs 1.5 times as much per unit area as the material for constructing the sides. For a fixed amount of money $150.00, find theDimensions of the box of largest volume that can be made.
A) base dimensions: $x = y = 5.77$; height $: z = 4.33$
B) base dimensions: $x = y = 4.33$; height $: z = 5.77$
C) base dimensions: $x = y = 7.75$; height $: z = 4.33$
D) base dimensions: $x = y = 4.33$; height: $z = 7.75$
E) base dimensions: $x = y = 7.75$; height: $z = 3.06$

Question

The material for constructing the base of an open box costs 1.5 times as much per unit area as the material for constructing the sides. For a fixed amount of money $150.00, find theDimensions of the box of largest volume that can be made. 
A) base dimensions: $x = y = 5.77$; height $: z = 4.33$
B) base dimensions: $x = y = 4.33$; height $: z = 5.77$
C) base dimensions: $x = y = 7.75$; height $: z = 4.33$
D) base dimensions: $x = y = 4.33$; height: $z = 7.75$
E) base dimensions: $x = y = 7.75$; height: $z = 3.06$

Quizplus · Accepted Answer

The problem can be solved using optimization techniques from calculus. Let $x$ and $y$ be the dimensions of the base, and $z$ be the height. Given that the cost of the base material is 1.5 times that of the sides, and with a total cost of $150, we can set up an equation for the cost and use the volume $V = xyz$ as the function to maximize. By setting up the cost equation and using Lagrange multipliers or by directly optimizing the volume under the given cost constraint, we find that the dimensions that maximize the volume given the cost constraint result in $x = y = 5.77$ and $z = 4.33$, making option A the correct choice.

The Material for Constructing the Base of an Open Box x=y=5.77x = y = 5.77x=y=5.77

The Material for Constructing the Base of an Open Box $x = y = 5.77$