Word problem help!

P

perfectstorm

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Oct 25, 2009
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I know that V= 1/3pr^2h and the SA=pr^2+prl and I'd probably use dy/dt = 5 and dx/dt = 1. It is in the shape of a triangle so x^2+y^2=r^2, but other than that I'm unsure how to set it up.

A tank in the shape of a right circular cone is being filled with water at a rate of 5 ft^3 per minute, but water is also flowing out at a rate of 1 ft^3 per minute. The tank is 60 feet deep and 40 feet across the top.

A) How quickly is the depth of the water changing when the water is 42 ft deep?

B) If the tank is empty, how long will it take for the tank to be 1/2 full?

C) How deep is the water when the tank is 1/2 full? How quickly is the depth of the water changing when the tank is 1/2 full?
 
\(\displaystyle I'll \ do \ A \ for \ you. \ Given: \ Right \ Circular \ Cone, \ \frac{dV}{dt} \ = \ 5-1 \ = \ 4ft.^{3}/min, \ h \ =60ft., \ r \ = \ 20ft.\)

\(\displaystyle Find \ \frac{dh}{dt} \ when \ h \ = \ 42ft.\)

\(\displaystyle \frac{h}{r} \ = \ \frac{60}{20}, \ r \ = \ \frac{h}{3}, \ \frac{dr}{dt} \ = \ \frac{1}{3}\frac{dh}{dt}\)

\(\displaystyle \frac{60}{20} \ = \ \frac{42}{r}, \ r \ = \ 14ft.\)

\(\displaystyle V_{cone} \ = \ \frac{\pi r^{2}h}{3}, \ \frac{dV}{dt} \ = \ \frac{\pi}{3}[2rh\frac{dr}{dt}+r^{2}\frac{dh}{dt}]\)

\(\displaystyle Ergo, \ 4 \ = \ \frac{\pi}{3}[2(14)(42)(1/3)\frac{dh}{dt}+196\frac{dh}{dt}]\)

\(\displaystyle 4 \ = \ \frac{\pi}{3}[392\frac{dh}{dt} +196\frac{dh}{dt}] \ = \ \frac{\pi}{3}[588\frac{dh}{dt}] \ = \ 196 \pi\frac{dh}{dt}.\)

\(\displaystyle Therefore, \ \frac{dh}{dt} \ = \ \frac{4}{196 \pi} \ = \ \frac{1}{49 \pi} \ ft. \ per \ min. \ rising \ when \ h \ = \ 42ft.\)
 
Wow thank you! I actually understood that. For part B would 1/2 = the volume or the height?

I went with V= 1/2, h=0

1/2 = p/3 [(2)(20)(0)dr/dt + 400dh/dt]
1/2 = p/3 [0 + 400 dh/dt]
400p/3= 133.33p dh/dt
1/2=133.33 pdh/dt = 66.665p min.
 
\(\displaystyle B) \ V \ = \ \frac{\pi r^ {2}h}{3} \ = \ \frac{\pi(20^{2})(60)}{3} \ = \ 8000\pi cu.ft.\ (full \ tank)\)

\(\displaystyle Tank \ half \ full \ = \ 4000\pi cu.ft..\)

\(\displaystyle Now, \ if \ it \ takes \ one \ minute \ to \ fill \ the \ tank \ with \ 4 \ cu.ft \ of \ water, \ then \ how \ long \ will \ it\)

\(\displaystyle \ take \ to \ fill \ the \ tank \ with \ 4000 \pi cu. \ ft. \ of \ water?\)

\(\displaystyle Can \ you \ take \ it \ from \ here?\)
 
16 hours and 40 min. Where did you get the 1 min to fill the tank with 4 cu. ft of water from? And yeah I think I can manage C. thanks a lot for your help
 
\(\displaystyle What \ is \frac{dV}{dt}?\)

\(\displaystyle Your \ answer \ is \ wrong, \ you \ forgot \ about \ \pi.\)
 
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