Question

question #8
the lateral surface area of a right circular cone is represented by the equation: s = πr√(r² + h²) where r is the radius of the circular base, and h is the height of the cone. if the lateral surface area of a large cone - shaped tunnel is 562.35 square centimeters and its radius is 8.3 centimeters, find its height to the nearest hundredth of a centimeter.

64.6 cm.
178.8 cm.
19.91 cm.
67.24 cm.

Explanation:

Step1: Substitute given values into formula

Given $s = 562.35$, $r=8.3$ and $s=\pi r\sqrt{r^{2}+h^{2}}$, we have $562.35=\pi\times8.3\sqrt{8.3^{2}+h^{2}}$.

Step2: Isolate the square - root term

First, divide both sides by $\pi\times8.3$. $\frac{562.35}{\pi\times8.3}=\sqrt{8.3^{2}+h^{2}}$. Calculate $\frac{562.35}{\pi\times8.3}\approx\frac{562.35}{3.14\times8.3}\approx\frac{562.35}{26.062}\approx21.58$. So, $21.58=\sqrt{68.89 + h^{2}}$.

Step3: Square both sides

$(21.58)^{2}=68.89 + h^{2}$. Then $465.7364=68.89+h^{2}$.

Step4: Solve for $h^{2}$

Subtract 68.89 from both sides: $h^{2}=465.7364 - 68.89=396.8464$.

Step5: Solve for $h$

Take the square - root of both sides: $h=\sqrt{396.8464}\approx19.92\approx19.91$ (rounded to the nearest hundredth).

Answer:

C. 19.91 cm