The pool - optimization

A block-shaped pool with a volume of 200 m³ is to be built in the recreation area. Its length should be 4 times the width, while the price of 1 m² of the pool bottom is 2 times cheaper than 1 m² of the pool wall. What dimensions must the pool have to make the construction the cheapest?

Final Answer:

a =  20 m
b =  5 m
c =  2 m

Step-by-step explanation:

$$\begin{gathered} V=200{\text{m}}^3 \\ a=4b \\ p=ab/2+2(a+b)\cdot c \\ V=abc \\ 200=4b^{2}c \\ c=50/b^2 \\ p=4b^2/2+2(4b+b)\cdot 50/b^2 \\ p=2b^2+500b/b^2 \\ p=2b^2+500/b \\ {p}^{\prime }=4b-500/b^2 \\ {p}^{\prime }=0 \\ 4b^3=500 \\ b=\sqrt[3]{500/4}=5\text{m} \\ a=4\cdot b=4\cdot 5=20\text{m} \end{gathered}$$

$b=5=5\text{m}$

$c=50/b^2=50/5^2=2\text{m}$

Try another example