Question

1. A hemispherical bowl with diameter 5cm is completely filled with water. If the water is completely poured into a right circular conical vessel of height 10cm, find the radius of base of cone..​

Answer 1

Radius of cone = 1.768 cm

Explanation:

`\text{Volume of a sphere } $V_(sphere) = (4)/(3)\pi r^3\\\\\text {where \; \\r = radius of the sphere}\\\\\text{For the hemispherical bowl the volume is 1/2 of the volume }\\\\So we have volume of a hemisphere of radius r, V_(hemisphere)= (1)/(2) \cdot (4)/(3)\pi r^3 = (2)/(3)\pi r^3$\\\\We have diameter = 5cm so radius 2.5cm\\\\Volume of hemi-sphere of radius 2.5 cm = (2)/(3)\pi 2.5^3 = (2)/(3) 15.625 \pi = (31.25)/(3)\pi`

`\text{Volume of cone of height h and radius r } = $(1)/(3)\pi r^2h`

`\text{Given h = 10cm this works out to }\\\\V_(cone) = (1)/(3)\pi r^2 10 = (10)/(3)\pi r^2`

Setting both volumes equal to each other gives

`(31.25)/(3)\pi = (10)/(3)\pi r^2`

Crossing out
`(\pi)/(3)` on both sides gives us

`31.25 = 10r^2\\\\r^2 = (31.25)/(10) = 3.125\\\\r = 1.768 cm`