If a cylindrical tank with radius 5 meters is being filled with water at a rate of 3 cubic meters per minute, how fast is the height of the water increasing?

opplettmcgx

opplettmcgx

Answered question

2023-03-15

If a cylindrical tank with radius 5 meters is being filled with water at a rate of 3 cubic meters per minute, how fast is the height of the water increasing?

Answer & Explanation

Lena Navarro

Lena Navarro

Beginner2023-03-16Added 9 answers

The answer is d h d t = 3 25 π m min
With related rates, we need a function to relate the 2 variables, in this case it is clearly volume and height. The formula is:
V = π r 2 h
There is radius in the formula, but in this problem, radius is constant so it is not a variable. We can substitute the value in:
V = π ( 5 m ) 2 h
We must implicitly differentiate wrt (with respect to) time because the rate in this problem is tied to time:
d V d t = ( 25 m 2 ) π d h d t
In the problem, we are given 3 m 3 min which is d V d t . So we substitute this in:
d h d t = 3 m 3 min ( 25 m 2 ) π = 3 25 π m min
In general
- discover a formula to connect the two variables
- To get rid of the constant variables, substitute values
- implicitly differentiate wrt time (most often the case)
- substitute the given rate
- and solve for the desired rate.

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