Question concerning a swimwing pool's volumes, and minim size of sides for the given volume Question: a volume of the swimming pool must be $25\,\mathrm m^3$. the base is squared. What are the sides of the swimming pool, that we can use the minimum amount of materials? My working We know that the base is squared and the volume formula is $V=a\cdot b\cdot h$. So here $$V = x\cdot x\cdot h = 25 = x^2h$$ Now I thought to find the $x$ value I calculate the area of one of the 5 sides. so: $A = x \cdot h$ and from the volume formula: $h = 25/x^2$ so $A = x \cdot \dfrac{25}{x^2}$ and after this I am confused. Any help would be appreciated. Thanks you Correct answer: $3.7$, $3.7$ and $1.8$

You are supposed to minimize the area (v.g. $A=\frac{25}x$). One alternative is using calculus, another is observing if you can use a known inequality such as $\frac{25}x>0$ and observing that $\frac{25}x\to0$ when $x\to\infty$. This will make a very wide yet zero-depth pool, which is probably not the answer needed.

You have forgotten the pool's floor, now, adding all four sides of the pool and its floor you have: $$A=4xh+x^2,$$ and you can replace $h$ so you get: $$A=\frac{100}x+x^2$$ A know inequality is not likely here, so use calculus to get $x=\sqrt[3]{50}\simeq3.7$.