Question

10. Figure 5.39 shows the level curves indicating the varying depth( in feet) of a 25 ft by 50 ft swimming pool. Use a Riemann sum toestimate, to the nearest 100 ft3 the volume of waterthat the pool contains.

Answer 1

The water is going to fill the entire length and width of the swimming pool. It is the depth that varies across the tiles. By tiles I mean the 50 little squares of dimensions 5' x 5' drawn across the surface of the pool.

The depth off the pool varies from 4' to 11'. Create a table as follows:



The first row in the table is the depth.
The second row is the count of the number of tiles that has depth 4', 5', etc.
The third row is the product of the first two rows.

I have a filled in a few values and you can complete the rest.
Use the first table to find the LOWER Riemann sum. That is, if a tile has, for example, a 9' and a 10' curve passing through it, then for the lower Riemann sum, count it towards 9'. Add up all the products in the third row and then multiply it by the area of the tile, which is 5' x 5' = 25 ft^2 to get the lower Riemann sum for the volume.

Create a similar table for the UPPER Riemann sum.

Take the average of the two sums to get the best estimate for the volume of water in the pool.

Note: In each table, the second row should total 50 because there are a total of 50 tiles.



When counting Find the lower Riemann sum