# 034 Review Problem - Sphere dropped into a cone

**Problem 34**

The inside of a vase is an inverted cone 2.983 in. across the top and 5.016 in. deep. If a heavy sphere 2.498 in. in diameter is dropped into it when the vase is full of water, how much water will overflow?

**Solution 34**

$\tan \theta = \dfrac{1.4915}{5.016}$

$\theta = 16.56^\circ$

$y = \dfrac{r}{\sin \theta} = \dfrac{1.249}{\sin 16.56^\circ}$

$y = 4.382 ~ \text{in.}$

$5.016 - h = y - r$

$5.016 - h = 4.382 - 1.249$

$h = 1.883 ~ \text{in.}$

Amount of water that overflows

$V = \text{segment of a sphere}$

$V = \frac{1}{3}\pi h^2(3r - h) = \frac{1}{3}\pi(1.883^2)[ \, 3(1.249) - 1.883 \, ]$

$V = 6.92 ~ \text{in.}^3$ *answer*

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