To find the radius of the cylindrical container, we need to consider the formula for the surface area of a cylinder. The surface area (SA) of a cylinder can be calculated using the formula:
SA = 2 * π * r^2 + 2 * π * r * h
Where:
r is the radius of the circular top and bottom,
h is the height of the cylinder,
and π (pi) is a mathematical constant approximately equal to 3.14159.
We know the total surface area is 6π square feet and the height (h) is 2 feet. Let’s plug these values into the formula and solve for the radius (r).
Given:
SA = 6π
h = 2
Using the surface area formula:
6π = 2 * π * r^2 + 2 * π * r * 2
6π = 2πr^2 + 4πr
Now, let's solve this equation for r:
First, we can simplify by dividing each term by 2π to reduce the equation:
3 = r^2 + 2r
We now have a quadratic equation:
0 = r^2 + 2r - 3
This is in the standard quadratic form of ax^2 + bx + c = 0, where:
a = 1, b = 2, and c = -3.
Next, we will solve for r using the quadratic formula:
r = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values from our equation, we get:
r = (-2 ± √(2² - 4*1*(-3))) / (2*1)
r = (-2 ± √(4 + 12)) / 2
r = (-2 ± √16) / 2
r = (-2 ± 4) / 2
This equation has two solutions:
r1 = (-2 + 4) / 2 = 2 / 2 = 1
r2 = (-2 - 4) / 2 = -6 / 2 = -3
Since a radius can't be negative, we discard the negative solution. Therefore, the radius of the circular top and bottom of the cylindrical container is 1 foot.