Question

a can of beans has surface area 346 cm². its height is 17 cm. what is the radius of the circular top? the radius of the circular top is 2.79 cm. (do not round until the final answer. then round to the nearest hundredth as needed.)

Explanation:

Step1: Recall surface - area formula for cylinder

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 of the base (and top), and $h$ is the height. We know that $S = 346\ cm^{2}$ and $h = 17\ cm$. So, $346=2\pi r^{2}+2\pi r\times17$.

Step2: Simplify the equation

$346 = 2\pi r^{2}+34\pi r$. Divide the entire equation by 2 to get $173=\pi r^{2}+17\pi r$. Let $\pi\approx3.14$. Then the equation becomes $173 = 3.14r^{2}+3.14\times17r$, or $173 = 3.14r^{2}+53.38r$. Rearrange it to the standard quadratic - form $ax^{2}+bx + c = 0$: $3.14r^{2}+53.38r - 173 = 0$.

Step3: Use the quadratic formula

The quadratic formula for $ax^{2}+bx + c = 0$ is $r=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a = 3.14$, $b = 53.38$, and $c=-173$. First, calculate the discriminant $\Delta=b^{2}-4ac=(53.38)^{2}-4\times3.14\times(-173)$.
$\Delta = 2849.4244+2172.88=5022.3044$. Then $r=\frac{-53.38\pm\sqrt{5022.3044}}{2\times3.14}=\frac{-53.38\pm70.87}{6.28}$.
We have two solutions for $r$: $r_1=\frac{-53.38 + 70.87}{6.28}=\frac{17.49}{6.28}\approx2.79$ and $r_2=\frac{-53.38 - 70.87}{6.28}=\frac{-124.25}{6.28}$ (rejected since radius cannot be negative).

Answer:

$2.79$