Question

Volume of Water in a Reservoir A reservoir shaped like a hemispherical bowl of radius 8 \(\mathrm{m}\) is filled with water to a depth of 4 \(\mathrm{m}\) (a) Find an estimate \(S\) of the water's volume by approximating the water with eight circumscribed solid cylinders. (b) It can be shown that the water's volume is \(V=(320 \pi) / 3 \mathrm{m}^{3}\) . Find the error \(|V-S|\) as a percentage of \(V\) to the nearest percent.

Short answer

The estimated volume \(S\) is obtained by summing up the volumes of the eight solid cylinders. The error percentage is calculated by comparing \(S\) to the actual volume \(V\). The final error percentage should be rounded to the nearest percent.

Step-by-step solution

Approximate Hemisphere with Cylinders

We begin by approximating the hemisphere with eight solid cylinders. The total radius is 8m, and the water depth is 4m. Thus, we can imagine each cylinder having a height of \(4 / 8 = 0.5 m\). The volume of each cylinder can be calculated with the formula for the volume of a cylinder, \(V = \pi r^2 h\). However, the radii of the eight cylinders are not the same. To calculate them, we have to use the equation of a circle in a vertical plane: \(r^2 = R^2 - (R - 0.5x)^2\) where \(x\) is a number from 0 to 8, corresponding to the height of each cylinder within the hemisphere.

Calculate Cylinder Volumes

Using the formula derived in step 1, the radius of each cylinder is calculated and then substituted into the volume formula. Then, the volumes of the eight cylinders are added up to provide an estimate, \(S\) of the water's volume.

Calculate Error Percentage

We then calculate the error percentage by comparing \(S\) to the actual volume \(V = (320 \pi) / 3 m^3\). The error is calculated by the formula: \(|(V - S) / V| * 100\%\). We then round this error to the nearest percent.