Question

Manufacturing to Specifications. A manufacturing firm wants to package its product in a cylindrical container 3 ft high with surface area 8π ft2. What should the radius of the circular top and bottom of the container be? (Hint: The surface area consists of the circular top and bottom and a rectangle that represents the side cut open vertically and unrolled.)

Accepted Answer

Identify the components of the surface area of the cylinder. The surface area consists of two circular bases (top and bottom) and one rectangular side (the lateral surface). Write the formula for the surface area of a cylinder: $$S = 2\pi r^2$ + 2\pi r h$$, where $$r is $$the radius and $$h is $$the height. Substitute the given height $$h = 3 ft $$and the total surface area $$S = 8\pi ft$^2$$ into the formula: $$8\pi = 2\pi r^2$ + 2\pi r (3). $$Simplify the equation by dividing both sides by $$2\pi to $$make it easier to solve: $$4 = r^2$ + 3r. $$Rearrange the equation into standard quadratic form: $$r^2 + 3r - 4 = 0$, then solve for r$ using the quadratic formula or factoring.

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Key Concepts

Surface Area of a Cylinder

Formulating Equations from Geometric Constraints

Solving Quadratic Equations