Question

A closed box has a fixed surface area of 54 μ {ft}² and a square base with side x μ {ft} .Find a formula for its volume,V as a function of x .

Answer 1

To find the volume V of a closed box with a square base and fixed surface area, express surface area SA = x² + 4xh, solve for h, and then substitute into the volume formula V = x²h to get V(x) = (54x - x³)/4.

Step-by-step explanation:

To find a formula for the volume, V, of a closed box with a square base and a fixed surface area, we start by expressing the surface area in terms of the side length of the square base, x, and the height, h. The total surface area for a box, which has a square base and four rectangular sides, is given by the formula: surface area SA = x² + 4xh. Since the box has a fixed surface area of 54 square micro feet, we can set up the equation 54 = x² + 4xh and solve for h in terms of x.

Substituting the given fixed surface area and isolating h, we get

h = (54 - x²)/(4x).

Now, the volume of a box is given by the formula V = area of the base times height,

or V = x²h.

Substituting the expression for h in terms of x, we get the volume as a function of x:

V(x) = x²((54 - x²)/(4x)).

By simplifying,

V(x) = (54x - x³)/4.