Question

geology a tiankeng is a sinkhole with nearly vertical walls. the tianpingmiao tiankeng is approximately cylindrical with a diameter of 180 meters and a depth of 420 meters.
a. if the top of the tiankeng is open and plants can grow on the bottom and sides, what is the surface area available for plants? round to the nearest square meter
b. what is the volume of water that could fill the tianpingmiao tiankeng? round to the nearest cubic meter

Explanation:

Step1: Calculate the radius

Given diameter $d = 180$ m, radius $r=\frac{d}{2}=\frac{180}{2}=90$ m.

Step2: Calculate the lateral - surface area

The formula for the lateral - surface area of a cylinder is $A_{l}=2\pi rh$. Substitute $r = 90$ m and $h = 420$ m. $A_{l}=2\pi\times90\times420=75600\pi$ $m^{2}$.

Step3: Calculate the base - area

The formula for the area of the base of a cylinder is $A_{b}=\pi r^{2}$. Substitute $r = 90$ m. $A_{b}=\pi\times90^{2}=8100\pi$ $m^{2}$.

Step4: Calculate the total surface area for plants

The total surface area $A = A_{l}+A_{b}=75600\pi + 8100\pi=83700\pi\approx83700\times3.14159\approx262941$ $m^{2}$.

Step5: Calculate the volume of the cylinder

The formula for the volume of a cylinder is $V=\pi r^{2}h$. Substitute $r = 90$ m and $h = 420$ m. $V=\pi\times90^{2}\times420=\pi\times8100\times420 = 3402000\pi\approx3402000\times3.14159\approx10684292$ $m^{3}$.

Answer:

a. $262941$
b. $10684292$