Q: A grain silo consists of a cylindrical main section and a hemispherical roof. If the total volume of the silo (including the part inside the roof section) is 16,000 ft3 and the cylindrical part is 30 ft tall, what is the radius of the silo, correct to the nearest tenth of a foot?
I started working from this equation for volume (cylinder and half of a sphere)…
V=(pi)(r^2)(h)+(2(pi)r^3)/(3)
Plugged in the given values and solved for the radius but haven't gotten the correct answer yet.
Can anyone help?
Best Answer
$$V=h\pi r^2+\frac23\pi r^3=16000$$ $$30\pi r^2+\frac23\pi r^3=16000$$ $$\pi r^2\left(\frac23 r+30\right)=16000$$ This is a cubic equation in $r$. If you want to avoid the tedious Cardano's formula for solving such equations, you need numerical methods, which are relevant here because the solution only needs to be accurate to a tenth of a place.
Say we guess that $r=10$ feet, which gives us a volume of 11519 cubic feet. If we try $r=12$ we get 17191 cubic feet. Since these two values are on opposite sides of the required 16000, we can try values between these two limits to find $r$.
These last two evaluations give $11.6<r<11.65$, i.e. the radius of the silo is 11.6 feet to one decimal place. (The exact answer is $11.6161\dots$ feet.)