im working on advanced problems and the question is following:
A swimming pool is 20 ft wide, 40 ft long and 3 ft deep at the shallow end, and 9ft deep at its deepest point.
1) Express the volume of the water in the pool as a function of height h of the water above the deepest point (Hint: The volume will be a piecewise-defined function)
2) Determine the domain and range of the function found in part (1)
how should I approach this type of problems? I haven't learned about this topic yet so I'm not sure if I'm on the right track
Volume (at full capacity) = V
V = 0.5 (length)(deep end depth + shallow end depth)(width)
I'm stuck after this..
The answer is given for 1) its piece-wise defined function(sorry I don't know how to do latex)
how do I get this answer? step by step explanation would be very much appreciated thank you
Best Answer
Presumably the bottom varies uniformly between 3-9'.
For $h\in[0,6]$ the length of the surface water line is $h {40 \over 6}$. For $h \in [6,9]$ the length of the surface water line is $40$. Let $l(h) = \begin{cases} h {40 \over 6}, & h\in[0,6) \\ 40, & h \in [6,9] \end{cases}$.
Then the volume of water is given by $v(h) = 20 \int_0^h l(x) dx$.