Question

In constructing an open rectangular box from 27 ft of material, what dimensions will result in a box of maximum volume? Let x be the length of the box, let y be the width of the box, and let z be the height of the box. What is the equation that represents the total surface area of the box? Assume that the box is open on top. Enter your answer in the answer box and then click Check Answer. ?

Answer 1

The equation that represents the total surface area of an open rectangular box is A = xy + 2xz + 2yz. To find the dimensions that result in a box of maximum volume, we can take the derivative of the volume function with respect to x and set it equal to 0. Solving this equation will give us the value of x that maximizes the volume.

Step-by-step explanation:

The total surface area of an open rectangular box can be represented by the equation: A = xy + 2xz + 2yz. To find the dimensions that will result in a box of maximum volume, we need to maximize the volume function. Since the volume of a rectangular box is given by V = xyz, we can use the constraint that the total material used is 27 ft to express one of the variables in terms of the other two.

Let's solve for y in terms of x and z: y = (27 - 2xz)/(2x + 2z). Substituting this into the volume function, we get: V = x(27 - 2xz)xz/(2x + 2z). To find the dimensions that maximize the volume, we can take the derivative of V with respect to x and set it equal to 0. Solving this equation will give us the value of x that maximizes the volume.

Once we have the value of x, we can substitute it back into the equation for y to find the corresponding value of y. The value of z can be found by using the constraint equation: 27 = 2x + 2y + z.

Answer 2

To maximize the volume of an open rectangular box made from 27 ft of material, express the surface area as S = xy + 2xz + 2yz, with the constraint 2x + 2y + 4z = 27, and then use calculus to find the dimensions that give the maximum volume.

Step-by-step explanation:

To determine the dimensions of an open rectangular box made from 27 ft of material that results in the maximum volume, we need to express the surface area and the volume of the box in terms of the variables x (length), y (width), and z (height). Since the box is open on top, the surface area (S) is given by the formula S = xy + 2xz + 2yz. However, we also have a constraint on the material available, which translates to the perimeter of the base plus twice the height being equal to 27 ft: 2x + 2y + 4z = 27. From this equation, we can express one variable in terms of the others and substitute it into our surface area equation. Then, we can use calculus to maximize the volume function, V = xyz.