Question

select the correct answer.
franco has enough paint to cover an area of approximately 226 square feet. he needs to paint the entire surface of a cylinder that is 5 feet long. what is the maximum possible radius of the cylinder that can be covered by the paint?
a. 14.4 feet
b. 4 feet
c. 3.8 feet
d. 9 feet

Explanation:

Step1: Recall total surface area formula

The total surface area of a cylinder is $A = 2\pi r^2 + 2\pi r h$, where $r$ is radius, $h$ is height (length) of the cylinder.

Step2: Plug in known values

We know $A = 226$ sq. ft, $h = 5$ ft, and use $\pi \approx 3.14$. Substitute into the formula:
$226 = 2(3.14)r^2 + 2(3.14)r(5)$
Simplify the equation:
$226 = 6.28r^2 + 31.4r$
Rearrange to standard quadratic form:
$6.28r^2 + 31.4r - 226 = 0$

Step3: Solve quadratic equation

Use the quadratic formula $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a=6.28$, $b=31.4$, $c=-226$.
First calculate discriminant:
$\sqrt{b^2 - 4ac} = \sqrt{(31.4)^2 - 4(6.28)(-226)}$
$\sqrt{985.96 + 5674.88} = \sqrt{6660.84} \approx 81.61$
Now solve for positive $r$ (radius can't be negative):
$r = \frac{-31.4 + 81.61}{2(6.28)} = \frac{50.21}{12.56} \approx 3.998 \approx 4$

Answer:

B. 4 feet