Spherical tank

The tank of a water tower is a sphere of radius 35ft. If the tank is filled to one quarter of full, what is the height of the water?

Correct result:

h =  22.8446 ft

Solution:

$$\begin{gathered} r=35\text{ ft } \\ V={\displaystyle \frac{4}{3}}\cdot \pi \cdot r^3={\displaystyle \frac{4}{3}}\cdot 3.1416\cdot 35^3\doteq 179594.38{\text{ft}}^3 \\ {V}_1={\displaystyle \frac{1}{4}}\cdot V={\displaystyle \frac{1}{4}}\cdot 179594.38\doteq 44898.595{\text{ft}}^3 \\ {V}_2={\displaystyle \frac{\pi \cdot h^2}{3}}(3r-h) \\ {V}_1=V_2 \\ {\displaystyle \frac{\pi \cdot r^3}{3}}={\displaystyle \frac{\pi \cdot h^2}{3}}(3r-h) \\ {r}^3=h^2(3r-h) \\ 42875=(105-h)h^2 \\ {h}^3-105h^2+42875=0 \\ {h}_1=-18.6231 \\ {h}_2=22.8446 \\ {h}_3=100.778 \\ 0<h<2r \\ h=h_2=22.8446=22.8446\text{ ft } \\ \text{ Verifying Solution: } \\ {V}_2={\displaystyle \frac{\pi \cdot h^2}{3}}\cdot (3\cdot r-h)={\displaystyle \frac{3.1416\cdot 22.8446^2}{3}}\cdot (3\cdot 35-22.8446)\doteq 44898.5017{\text{ft}}^3 \end{gathered}$$

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Math student

ca you show me the step by step of the height

4 months ago  2 Likes Like

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