Question

a can of beans has surface area 348 cm². its height is 19 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 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 circular base (and top), and $h$ is the height. We know that $S = 348$ $cm^{2}$ and $h = 19$ $cm$. So, $348=2\pi r^{2}+2\pi r\times19$.

Step2: Simplify the equation

First, factor out $2\pi r$ from the right - hand side: $348 = 2\pi r(r + 19)$. Divide both sides by $2\pi$: $\frac{348}{2\pi}=r(r + 19)$. Since $\frac{348}{2\pi}=\frac{174}{\pi}\approx55.4$. The equation becomes $55.4=r^{2}+19r$. Rearrange it to the standard quadratic form $r^{2}+19r - 55.4 = 0$.

Step3: Use the quadratic formula

The quadratic formula for a quadratic equation $ax^{2}+bx + c = 0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. For $r^{2}+19r - 55.4 = 0$, $a = 1$, $b = 19$, and $c=-55.4$. First, calculate the discriminant $\Delta=b^{2}-4ac=(19)^{2}-4\times1\times(-55.4)=361 + 221.6 = 582.6$. Then $r=\frac{-19\pm\sqrt{582.6}}{2}$. We take the positive root because the radius cannot be negative. $r=\frac{-19+\sqrt{582.6}}{2}\approx\frac{-19 + 24.14}{2}=\frac{5.14}{2}=2.57$.

Answer:

$2.57$