[Math] Related rates calculus problem finding how fast a water level is rising

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A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at is deepest point. A cross-section is shown in the figure. If the pool is being filled at a rate of 0.7 ft3/min, how fast is the water level rising when the depth at the deepest point is 5 ft? (Round your answer to five decimal places.)

The pool is an irregular shape and I have no formula to find the volume or height. I don't know where to start this since I do not have related rates formula. Can someone please help?

cross-section_of_pool

Best Answer

Shifting the pool bottom to left as shown for easier calculation of area /volume. Basis is that triangle area removed/added with same base and height has same area and, so the volume.

enter image description here Representing time rate with respect to time by dots $$ 20 \;dV = x\; dz , \quad 20 \;\dot V= x \;\dot z $$ Equation of slant floor $$x=12+\dfrac{11 z}{3}$$ $$ 20 \dot V= (12+\dfrac{11 z}{3})\dot z $$ Given that $$ z=5, \dot V= 0.7 $$ $$ \dot z = \dfrac{20\times 0.7}{12+55/3} = 0.461538 $$ $ ft^3 $ per minute.

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