K, Q1.A sphere of radius 5cm is lowered into a rectangular tank 50cm long and 20cm wide containing water to a depth of 10cm.The sphere floats with half of its surface above the level of the surface of the water
Calculate (a)the volume of the sphere, (b)the volume of water in the tank (cccc)the surface area of the sphere above the water.=
sphere: v = 4/3 pi r^3 = 500pi/3 = 524 cm^3
The diameter of the sphere is equal to the depth of the water, so it will be completely submerged as the water level rises.
To solve this problem, we'll break it down step by step.
(a) To find the volume of the sphere, we can use the formula:
Volume of a sphere = (4/3) * π * r^3
Given that the radius of the sphere is 5 cm, we can substitute this value into the formula:
Volume of sphere = (4/3) * π * (5^3)
= (4/3) * π * 125
= 4 * π * 41.67
= 166.8π cm^3 (rounded to one decimal place)
Therefore, the volume of the sphere is approximately 166.8π cm^3.
(b) To find the volume of water in the tank, we need to calculate the volume of the rectangular tank. The formula for volume of a rectangular tank is:
Volume of rectangular tank = length * width * depth
Given that the length is 50 cm, the width is 20 cm, and the depth of the water is 10 cm, we can substitute these values into the formula:
Volume of tank = 50 * 20 * 10
= 10,000 cm^3
Therefore, the volume of water in the tank is 10,000 cm^3.
(c) To calculate the surface area of the sphere above the water, we need to find the area of the spherical cap. The formula for the surface area of a spherical cap is:
Surface area of spherical cap = 2πr(h+r)
Where r is the radius of the sphere and h is the height of the cap.
Given that the radius of the sphere is 5 cm and the height of the cap is also 5 cm (half the diameter of the sphere), we can substitute these values into the formula:
Surface area of sphere cap = 2π * 5 * (5+5)
= 10π * 10
= 100π cm^2
Therefore, the surface area of the sphere above the water is 100π cm^2.