Answer to Question #147078 in Geometry for solid mensuration

Question

An inverted regular square pyramid has an altitude of 12m and with top
edge of 3m is fully–filled with water.

6. How much water is to be removed so that water is 7m deep?

Answer 1

Given $a=3\ m, H=12\ m$

$$V=\dfrac{1}{3}a^2H$$ $$V=\dfrac{1}{3}(3\ m)^2(12\ m)=36\ m^3$$

The volume of the water is 36 m³.

$$\dfrac{a}{H}=\dfrac{a_1}{H_1}=const$$

Then

$$a_1=a\cdot\dfrac{H_1}{H}$$

$$a_1=3\ m\cdot\dfrac{7\ m}{12\ m}=\dfrac{7}{4}\ m$$

$$V_1=\dfrac{1}{3}a_1 ^2H_1$$ $$V_1=\dfrac{1}{3}(\dfrac{7}{4}\ m) ^2\cdot 7 m=\dfrac{343}{48} \ m^3$$

How much water is to be removed so that water is 7m deep?

$$V-V_1=36\ m^3-\dfrac{343}{48} \ m^3=\dfrac{1385}{48} \ m^3$$

$\dfrac{1385}{48}$ m³ is to be removed so that water is 7m deep.