Radius of a Can A can of Blue Runner Red Kidney Beans has surface area 371 cm². Its height is 12 cm. What is the radius of the circular top? Round to the nearest hundredth.

Verified step by step guidance

1. Recall the formula for the surface area of a cylinder: \(\text{Surface Area} = 2\pi r^2 + 2\pi r h\), where \(r\) is the radius and \(h\) is the height.
2. Substitute the given values into the formula: surface area \(= 371\) cm\(^2\) and height \(h = 12\) cm, so the equation becomes \$371 = 2\pi r^2 + 2\pi r (12)$.
3. Simplify the equation by factoring out \$2\pi r\(: \)371 = 2\pi r^2 + 24\pi r$.
4. Rewrite the equation as a quadratic in terms of \(r\): \$2\pi r^2 + 24\pi r - 371 = 0$.
5. Use the quadratic formula \(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 2\pi\), \(b = 24\pi\), and \(c = -371\) to solve for \(r\), then round your answer to the nearest hundredth.

Key Concepts

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 + 2πrh, where r is the radius and h is the height. Understanding this formula is essential to relate the given surface area to the radius and height.

Solving Quadratic Equations

When the surface area formula is set equal to the given value, it forms a quadratic equation in terms of the radius. Solving this quadratic equation, either by factoring, completing the square, or using the quadratic formula, helps find the radius of the can.

Rounding to a Specific Decimal Place

After calculating the radius, the result must be rounded to the nearest hundredth. This involves identifying the second decimal place and rounding the number accordingly, ensuring the final answer is precise and meets the problem's requirements.