An open top box with a square base has a surface area of 100 square inches. Express the volume of the box as a function of the length of the edge of the base. What is its domain?

Answer: d) {eq}V= 25x - frac {1}{2}x^3 {/eq} and domain is ({eq}0, 5 sqrt {2} {/eq}) Explanation: Let the edge length of the square base be x. Given that the surface area of the box is 100 square inches. Surface area of the box = Area of the base + Area of the 4 sides = {eq}x^2 + 4xz = 100 {/eq} (z is the height of the box) Now, we need to express the volume of the box as a function of x. Volume of the box = Area of the base ? height = {eq}x^2z {/eq} From the surface area equation, we can find the height of the box as: {eq}z = \frac{100 - x^2}{4x} {/eq} Substitute z in the volume equation: {eq}V = x^2\left(frac{100 - x^2}{4x} right) {/eq} Simplify the above equation: {eq}V = 25x - frac {1}{2}x^3 {/eq} The volume of the box can be expressed as a function of the length of the edge of the base as {eq}V= 25x - frac {1}{2}x^3 {/eq} and its domain is ({eq}0, 5 sqrt {2} {/eq}) because the length of the edge of the base cannot be negative and it should be less than or equal to the diagonal of the square base which is {eq}5 sqrt {2} {/eq} (using Pythagoras theorem).