KCC's Quizzes AQQ262 about conical bottle height

1. The height of the bottle is about 10.22

2. The volume of the liquid is about half of the volume of the bottle.

To calculate the height H of the bottle, you write down the equation for the liquid volume V in the two cases:

V=(PI/3)*(H-8) * (R² + r1² +R*r1) (volume of a truncated cone of height = H-8)

V=(PI/3)*(H-2)*r2² (volume of a cone of height = H-2)

Now, for geometric reasons, you have r1= 8*R/H and r2= R* (H-2)/H, so, by using the fact that volume V is the same, you get the following equation:

(H-8) * (R² + 64*R²/H² + 8*R²/H) = (H-2)³ * R² /H²)

After some calculation, you get to:

H² - 2*H - 84 = 0

You keep just the positive root, which is about 10.22 (ok: is >8)

With H=10.22 the volume of the liquid is about half of the bottle volume. You can see that by evaluating V vs the volume of the cone bottle. 

Bingo gpiarino ! You get it, congratulations! For question 2 one can also just say the volume cannot be determined with the data given (in fact one needs to give R).

Bingo gpiarino ! You get it, congratulations! For question 2 one can also just say the volume cannot be determined with the data given (in fact one needs to give R).