The trough shown in the figure is 5 feet long, and its vertical cross sections are inverted...
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Transcript of The trough shown in the figure is 5 feet long, and its vertical cross sections are inverted...
The trough shown in the figure is 5 feet The trough shown in the figure is 5 feet long, and its vertical cross sections are long, and its vertical cross sections are inverted isosceles triangles with base 2 inverted isosceles triangles with base 2 feet and height 3 feet. Water is being feet and height 3 feet. Water is being siphoned out of the trough at the rate of 2 siphoned out of the trough at the rate of 2 cubic feet per minute. At any time cubic feet per minute. At any time tt, let , let hh be the depth and be the depth and VV be the volume of be the volume of water in the trough.water in the trough.
a) Find the volume of water in the trough when it is full.a) Find the volume of water in the trough when it is full.
b) What is the rate of change in b) What is the rate of change in hh at the instant at the instant when the trough is ¼ full by volume?when the trough is ¼ full by volume?
c) What is the rate of change in the area of c) What is the rate of change in the area of the surface of the water (shaded in the figure) the surface of the water (shaded in the figure) at the instant when the trough is ¼ full by at the instant when the trough is ¼ full by volume?volume?
a) Find the volume of water in the trough when it is full.a) Find the volume of water in the trough when it is full.
3ft15
5322
12
1
lhbV
b) What is the rate of change in b) What is the rate of change in hh at the instant when the at the instant when the trough is ¼ full by volume?trough is ¼ full by volume?
2
3
5
53
2
2
1
3
2
2
1
h
hh
lhhV
hb
bhh
b
3
2
323
2
b) What is the rate of change in b) What is the rate of change in hh at the instant when the at the instant when the trough is ¼ full by volume?trough is ¼ full by volume?
min/ft5
33
102-
(given) min/ft2
3
10
3
hdt
dhdt
dhh
dt
dVdt
dhh
dt
dV
h
h
2
33
5
4
15 2
minft/4.15
623
5
3
dt
dh
[ 0 , 3 ][ 0 , 3 ]
[ -2 , 0 ][ -2 , 0 ]
Solving for dh/dt graphically...Solving for dh/dt graphically...
25
3V h
10
3
dV dhh
dt dt
102
3
dhhdt
dt
dh
h
5
3
Solving for dh/dt numerically...Solving for dh/dt numerically...
c) What is the rate of change in the area of the surface of the water c) What is the rate of change in the area of the surface of the water (shaded in the figure) at the instant when the trough is ¼ full by (shaded in the figure) at the instant when the trough is ¼ full by volume?volume?
h
h
b
blS
3
10
3
25
5
min/ft3
415
20
5
2
3
103
10
2
dt
dh
dt
dS