Question

A manufacturing firm wants to package its product in a cylindrical container 1 ft. high with surface area 12π^2 ft. What should the radius of the circular top and bottom​ be? (Hint: The surface area consists of the circular top and bottom and a rectangle that represents the side cut open vertically and​ unrolled.)

Answer 1

To summarize:

The radius of the circular top and bottom of the cylindrical container should be 2π ft.

Explanation:

To find the radius of the circular top and bottom of the cylindrical container, we can use the formula for the surface area of a cylinder. The surface area of a cylinder is given by the formula A = 2πrh + 2πr^2, where A is the surface area, r is the radius of the circular top and bottom, and h is the height of the cylinder.

In this problem, we are given that the height of the cylinder is 1 ft and the surface area is 12π^2 ft. Substituting these values into the formula, we have:

12π^2 = 2π(1)(r) + 2πr^2

Simplifying the equation, we get:

12π^2 = 2πr + 2πr^2

Dividing both sides of the equation by 2π, we have:

6π = r + r^2

Now, we have a quadratic equation. To solve for r, we can rearrange the equation as:

r^2 + r - 6π = 0

We can solve this equation by factoring or using the quadratic formula. Factoring the equation, we have:

(r - 2π)(r + 3π) = 0

Setting each factor equal to zero, we find two possible values for r:

r - 2π = 0, which gives r = 2π

r + 3π = 0, which gives r = -3π

Since a negative radius does not make sense in this context, we discard the solution r = -3π.

Therefore, the radius of the circular top and bottom of the cylindrical container should be 2π ft.