Question

A can of beans has surface area 312 cm^2. Its height is 11 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.)

Answer 1

Answer: The radius of the circular top is 3.45cm

Explanation

• Surface area (SA): 312cm².

,

• Height (h): 11cm.

A can of beans can be approximated to a cylinder figure. Then, the cylinder has a formula for its surface area (SA), which is:

$SA=2\pi rh+2\pi r^2$

where r represents the radius and h represents the height.

As we have the value for SA and h we can replace them in the formula:

$312=2\pi r(11)+2\pi r^2$

Simplifying the expression by multiplying the parenthesis:

$312=22\pi r+2\pi r^2$

As we have a second-degree polynomial, we can use the General Quadratic Formula to find the solution for r. In order to do so, we have to set the equation to 0 as follows:

$0=ax^2+bx+c$

where a, b and c are used in the General Quadratic Formula:

$r_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

In our case, our equation is:

$0=2\pi r^2+22\pi r-312$

where a = 2π, b = 22π, and c = -312.

Then, replacing these values in the General Quadratic formula we get:

$r_{1,2}=\frac{-22\pi\pm\sqrt{(22\pi)^2-4(2\pi)(-312)}}{2(2\pi)}$

Simplifying the terms inside the square root and the denominator:

$r_{1,2}=\frac{-22\pi\pm\sqrt{(22\pi)^2-2496\pi}}{4\pi}$$r_{1,2}=\frac{-22\pi\pm\sqrt{12618}}{4\pi}$

Finally, finding both solutions:

$r_1=\frac{-22\pi+112.33}{4\pi}\approx3.44$$r_2=\frac{-22\pi-112.33}{4\pi}=-14.44$

As we cannot have a negative radius, then the correct answer is that the radius measures 3.44cm