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 cm. (do not round until the final answer. then round to the nearest hundredth as needed.)

Explanation:

Step1: Recall surface - area formula

The surface - area formula for 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. We know that $S = 346$ cm² 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 $\Delta=b^{2}-4ac=(53.38)^{2}-4\times3.14\times(-173)$.
$\Delta = 2849.4244+2177.68=5027.1044$.
Then, $r=\frac{-53.38\pm\sqrt{5027.1044}}{2\times3.14}=\frac{-53.38\pm70.89}{6.28}$.
We have two solutions for $r$:
$r_1=\frac{-53.38 + 70.89}{6.28}=\frac{17.51}{6.28}\approx2.79$ and $r_2=\frac{-53.38 - 70.89}{6.28}=\frac{-124.27}{6.28}$ (rejected since radius cannot be negative).

Answer:

$2.79$