Question

3
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. 9 feet
b. 4 feet
c. 14.4 feet
d. 3.8 feet

Explanation:

Step1: Write the formula for the surface area of a cylinder

The surface area formula of a cylinder is \(S = 2\pi r^{2}+2\pi rh\), where \(r\) is the radius and \(h\) is the height (or length in this case). Given \(S = 226\) square feet and \(h = 5\) feet. So, \(226=2\pi r^{2}+2\pi r\times5\).

Step2: Simplify the equation

First, factor out \(2\pi r\): \(226 = 2\pi r(r + 5)\). Since \(\pi\approx3.14\), the equation becomes \(226=6.28r(r + 5)=6.28r^{2}+31.4r\). Rearrange it to the quadratic form \(6.28r^{2}+31.4r-226 = 0\).

Step3: Use the quadratic formula

For a quadratic equation \(ax^{2}+bx + c=0\) (\(a = 6.28\), \(b = 31.4\), \(c=-226\)), the quadratic formula is \(r=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).

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We take the positive root \(r=\frac{-31.4 + 81.7}{12.56}=\frac{50.3}{12.56}\approx4\)

Answer:

B. 4 feet