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.)

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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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Surface Area of a Cylinder

The surface area of a cylinder includes the areas of two circular bases and the rectangular side (lateral surface). It is calculated as 2πr² for the top and bottom circles plus 2πrh for the side, where r is the radius and h is the height.

Formulating Equations from Geometric Constraints

To solve the problem, translate the given surface area and height into an equation involving the radius. This involves setting the total surface area equal to the sum of the areas of the two circles and the rectangle, allowing you to solve for the unknown radius.

Solving Quadratic Equations

After forming the equation, you may need to rearrange it into a quadratic form to solve for the radius. Understanding how to solve quadratic equations using factoring, completing the square, or the quadratic formula is essential to find the correct radius value.