Final answer:
To determine the radius of the cylinder given the surface area, we use the provided surface area formula and the surface area value to set up and solve a quadratic equation. We solve for the radius 'x' ensuring to take the positive value since the radius cannot be negative.
Step-by-step explanation:
To find the radius of a cylinder with a given surface area, we can start by using the provided formula for the surface area:
A = 6.28x² + 92.1x.
Here, the variable 'x' represents the radius of the cylinder. We are given that the surface area A is 1138.72 cm². Therefore, we set the equation equal to 1138.72 and solve for x (the radius).
The equation to solve is:
6.28x² + 92.1x - 1138.72 = 0
This is a quadratic equation in the standard form ax² + bx + c = 0, where a = 6.28, b = 92.1, and c = -1138.72.
We can solve this quadratic equation using the quadratic formula, which is:
x = √{-b ± √{b² - 4ac} / 2a}.
After calculating the values under the radical sign and simplifying, we obtain two potential values for x (the radius). We must then choose the positive solution, as the radius of a cylinder cannot be negative.