Question

select the correct answer. a cylinder has a surface - area of about 271.1 square meters and a height of 7.8 meters. what is the approximate radius of the cylinder? a. 6.4 meters b. 4.5 meters c. 2.1 meters d. 3.2 meters

Explanation:

Step1: Recall surface - area formula

The surface - area formula of a cylinder is $S = 2\pi r^{2}+2\pi rh$, where $S$ is the surface area, $r$ is the radius, and $h$ is the height. Given $S = 271.1$ square meters and $h = 7.8$ meters. So, $271.1=2\pi r^{2}+2\pi r\times7.8$.

Step2: Simplify the equation

$271.1 = 2\pi r^{2}+15.6\pi r$. Divide the entire equation by $2\pi$ (where $\pi\approx3.14$). $\frac{271.1}{2\times3.14}=r^{2}+\frac{15.6\times3.14}{2\times3.14}r$. $43.15\approx r^{2}+7.8r$. Rearrange to get a quadratic equation: $r^{2}+7.8r - 43.15 = 0$.

Step3: Use the quadratic formula

The quadratic formula for $ax^{2}+bx + c = 0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a = 1$, $b = 7.8$, and $c=-43.15$. First, calculate the discriminant $\Delta=b^{2}-4ac=(7.8)^{2}-4\times1\times(-43.15)=60.84 + 172.6=233.44$. Then $r=\frac{-7.8\pm\sqrt{233.44}}{2}=\frac{-7.8\pm15.27}{2}$.
We take the positive root since radius cannot be negative. $r=\frac{-7.8 + 15.27}{2}=\frac{7.47}{2}\approx3.735\approx3.2$ (approximate value considering the options).

Answer:

D. 3.2 meters