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 surface - area formula for the open - top cylinder

The surface area $A$ of an open - top cylinder is $A=\pi r^{2}+2\pi rh$, where $r$ is the radius and $h$ is the height. Given $d = 180$ meters, so $r=\frac{d}{2}=90$ meters and $h = 420$ meters.
$A=\pi\times90^{2}+2\pi\times90\times420$.

Step2: Calculate each term

First term: $\pi\times90^{2}=\pi\times8100\approx3.14\times8100 = 25434$.
Second term: $2\pi\times90\times420=2\times3.14\times90\times420=2\times3.14\times37800 = 237384$.

Step3: Sum up the terms

$A=25434 + 237384=262818\approx262818$ square meters.

Step4: Calculate the volume formula for the cylinder

The volume $V$ of a cylinder is $V=\pi r^{2}h$. Substitute $r = 90$ meters and $h = 420$ meters.
$V=\pi\times90^{2}\times420=\pi\times8100\times420=3.14\times8100\times420=3.14\times3402000 = 10682280\approx10682280$ cubic meters.

Answer:

a. $262818$ square meters
b. $10682280$ cubic meters