A 160‐inch strip of metal 20 inches wide is to be made into a small open trough by bending up two sides on the long side, at right angles to the base. The sides will be the same height, x. If the trough is to have a maximum volume, how many inches should be turned up on each side?
To find the value of x, we need to maximize the volume of the trough. We can set up an equation for the volume of the trough and then find the value of x that maximizes this equation.
Let's first visualize the trough. We have a 160-inch strip of metal that will be bent up along two sides, perpendicular to the base. The width of the strip is 20 inches. We need to determine how many inches of metal should be turned up on each side (height of the trough), which we'll call x.
If we fold up x inches on each side, the base of the trough will be 160 - 2x inches long, as we use 2x inches of the metal for the sides. The width and height of the trough will be 20 and x inches, respectively. Therefore, the volume of the trough can be calculated using the formula: Volume = length × width × height.
Substituting the values, the volume of the trough can be expressed as V = (160 - 2x) × 20 × x.
To find the value of x that maximizes V, we need to find the maximum of this equation. Let's differentiate V with respect to x and set the derivative equal to zero. Then we can solve for x.
dV/dx = (160 - 2x) × 20 + (20x) × (-2) = 3200 - 40x - 40x = 3200 - 80x
Setting dV/dx = 0, we have:
3200 - 80x = 0
80x = 3200
x = 3200/80
x = 40
Therefore, to maximize the volume of the trough, we should fold up 40 inches on each side.