To find the radius of the circular top of the can of beans, we can use the formula for the surface area of a cylinder. The surface area of a cylinder is given by the formula:
SA = 2πrh + πr^2,
where SA is the surface area, r is the radius of the circular top, and h is the height of the cylinder.
In this case, we know that the surface area of the can is 339 cm^2 and the height is 18 cm. Plugging these values into the formula, we get:
339 = 2π(18)r + πr^2.
Simplifying this equation, we have:
339 = 36πr + πr^2.
To solve for the radius, we need to rearrange the equation into the form πr^2 + 36πr - 339 = 0.
Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Since factoring might not be straightforward in this case, let's use the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / (2a),
where a = 1, b = 36π, and c = -339.
Plugging in these values, we get:
r = (-36π ± √((36π)^2 - 4(1)(-339))) / (2(1)).
Simplifying further, we have:
r = (-36π ± √(1296π^2 + 1356)) / 2.
Now, this equation gives us two possible solutions for the radius. We'll consider both:
1. r = (-36π + √(1296π^2 + 1356)) / 2.
2. r = (-36π - √(1296π^2 + 1356)) / 2.
These two values represent the possible radii of the circular top of the can of beans.