Math 171 Group Worksheet 5on Chapter 5 ideas

Sample Solution

To determine the volume of Crater Lake, we partitioned the map into square that were 1km by 1km indimension of 1,000 meters by 1,000 meters. For each square that contained part of the lake, we determinedwhat proportion of the square was underwater. We also determined the range of depths that appeared inthe square. We visually recorded the shallowest color (Orange) to deepest color (Blue).

For the overapproximation, we took the deepest color and recorded the largest value for that color as thedepth over the entire region. For the underapproximation, we took the shallowest color and recorded thesmallest value for that color as the depth over the entire region.The volume of each square was computed using the formula:

Volume = Coverage · Depth · Area

To find the overall volume, we added up the volume of each piece.

The underapproximation summed up to be 7.9 billion cubic meters.

The overapproximation summed up to be 25.8 billion cubic meters.

The average would give a better result at 16.9 million cubic meters.

A quick internet search returns that the true volume is 17.6 million cubic meters.

The following pages is the spreadsheet used to determine the above values.

1

Locatiooo% cov Volume(Under) Volume(Over)in cubic meters in cubic meters

D1 D 0.3 Orange 0 Yellow 250 0.00E+00 7.50E+07E1 E 0.5 Orange 0 Yellow 250 0.00E+00 1.25E+08F1 F 0.3 Orange 0 Green 400 0.00E+00 1.20E+08G1 G 0.6 Orange 0 Green 400 0.00E+00 2.40E+08H1 H 0.5 Orange 0 Green 400 0.00E+00 2.00E+08 Color Min MaxB2 B 0.1 Orange 0 Yellow 250 0.00E+00 2.50E+07 Orange 0 50C2 C 0.6 Orange 0 Green 400 0.00E+00 2.40E+08 Yellow 50 250D2 D 1 Orange 0 Blue 550 0.00E+00 5.50E+08 Green 250 400E2 E 1 Yellow 50 Blue 550 5.00E+07 5.50E+08 Blue 400 550F2 F 1 Yellow 50 Blue 550 5.00E+07 5.50E+08G2 G 1 Blue 400 Blue 550 4.00E+08 5.50E+08H2 H 0.9 Orange 0 Blue 550 0.00E+00 4.95E+08I2 I 0.4 Orange 0 Green 400 0.00E+00 1.60E+08A3 A 0.1 Orange 0 Orange 50 0.00E+00 5.00E+06B3 B 0.7 Orange 0 Yellow 250 0.00E+00 1.75E+08C3 C 1 Yellow 50 Blue 550 5.00E+07 5.50E+08D3 D 1 Blue 400 Blue 550 4.00E+08 5.50E+08E3 E 1 Blue 400 Blue 550 4.00E+08 5.50E+08F3 F 1 Yellow 50 Blue 550 5.00E+07 5.50E+08G3 G 1 Green 250 Blue 550 2.50E+08 5.50E+08H3 H 1 Blue 400 Blue 550 4.00E+08 5.50E+08I3 I 1 Yellow 50 Blue 550 5.00E+07 5.50E+08J3 J 0.5 Orange 0 Green 400 0.00E+00 2.00E+08A4 A 0.3 Orange 0 Yellow 250 0.00E+00 7.50E+07B4 B 1 Orange 0 Green 400 0.00E+00 4.00E+08C4 C 1 Yellow 50 Blue 550 5.00E+07 5.50E+08D4 D 1 Green 250 Blue 550 2.50E+08 5.50E+08E4 E 1 Green 250 Blue 550 2.50E+08 5.50E+08F4 F 1 Blue 400 Blue 550 4.00E+08 5.50E+08G4 G 1 Blue 400 Blue 550 4.00E+08 5.50E+08H4 H 1 Blue 400 Blue 550 4.00E+08 5.50E+08I4 I 1 Yellow 50 Blue 550 5.00E+07 5.50E+08J4 J 0.5 Orange 0 Green 400 0.00E+00 2.00E+08A5 A 0.6 Orange 0 Yellow 250 0.00E+00 1.50E+08B5 B 0.2 Orange 0 Orange 50 0.00E+00 1.00E+07C5 C 0.9 Orange 0 Yellow 250 0.00E+00 2.25E+08D5 D 1 Yellow 50 Green 400 5.00E+07 4.00E+08E5 E 1 Yellow 50 Green 400 5.00E+07 4.00E+08F5 F 1 Green 250 Blue 550 2.50E+08 5.50E+08G5 G 1 Blue 400 Blue 550 4.00E+08 5.50E+08H5 H 1 Blue 400 Blue 550 4.00E+08 5.50E+08I5 I 1 Yellow 50 Green 400 5.00E+07 4.00E+08J5 J 0.2 Orange 0 Orange 50 0.00E+00 1.00E+07A6 A 0.8 Orange 0 Yellow 250 0.00E+00 2.00E+08B6 B 0.8 Orange 0 Yellow 250 0.00E+00 2.00E+08

Min Max

Color Depth Chart

C6 C 1 Orange 0 Green 400 0.00E+00 4.00E+08D6 D 1 Yellow 50 Blue 550 5.00E+07 5.50E+08E6 E 1 Yellow 50 Blue 550 5.00E+07 5.50E+08F6 F 1 Green 250 Blue 550 2.50E+08 5.50E+08G6 G 1 Blue 400 Blue 550 4.00E+08 5.50E+08H6 H 1 Blue 400 Blue 550 4.00E+08 5.50E+08I6 I 0.9 Orange 0 Green 400 0.00E+00 3.60E+08J6 J 0.1 Orange 0 Orange 50 0.00E+00 5.00E+06A7 A 0.4 Orange 0 Yellow 250 0.00E+00 1.00E+08B7 B 1 Orange 0 Green 400 0.00E+00 4.00E+08C7 C 1 Yellow 50 Blue 550 5.00E+07 5.50E+08D7 D 1 Blue 400 Blue 550 4.00E+08 5.50E+08E7 E 1 Blue 400 Blue 550 4.00E+08 5.50E+08F7 F 1 Blue 400 Blue 550 4.00E+08 5.50E+08G7 G 1 Green 250 Blue 550 2.50E+08 5.50E+08H7 H 1 Yellow 50 Blue 550 5.00E+07 5.50E+08I7 I 0.4 Orange 0 Yellow 250 0.00E+00 1.00E+08B8 B 0.2 Orange 0 Yellow 250 0.00E+00 5.00E+07C8 C 0.6 Orange 0 Green 400 0.00E+00 2.40E+08D8 D 0.7 Orange 0 Green 400 0.00E+00 2.80E+08E8 E 0.8 Orange 0 Green 400 0.00E+00 3.20E+08F8 F 1 Yellow 50 Green 400 5.00E+07 4.00E+08G8 G 0.9 Yellow 50 Green 400 4.50E+07 3.60E+08H8 H 0.5 Orange 0 Green 400 0.00E+00 2.00E+08I8 I 0.1 Orange 0 Green 400 0.00E+00 4.00E+07E9 E 0.2 Orange 0 Yellow 250 0.00E+00 5.00E+07F9 F 0.5 Orange 0 Yellow 250 0.00E+00 1.25E+08G9 G 0.1 Orange 0 Yellow 250 0.00E+00 2.50E+07

Under Approx Over ApproxTotal 7.90E+09 2.58E+10

7.9 billion 25.8 billioncubic meters of watbic meters of wa

True value 17.6 billioncubic meters of water

cubic meters of water

Average1.69E+10

16.9 billions