A right cylindrical oil tank is 15 feet tall and its circular bases have diameters of 4 feet each. When the tank is lying flat on its side (not on one of the circular ends), the oil inside is 3 feet deep. How deep, in feet, would the oil have been if the tank had been standing upright on one of its bases? Express your answer as a decimal to the nearest tenth.
We will need to use Calculus to solve this.....
Looking at the circular cross section of the cylinder, it can be expressed as a circle with a radius of 2 centered at the origin......its equation is :
x^2 + y^2 = 4
y = √ [ 4 - x^2 ]
The area of the circle covered by the oil is given by
Area of circle - area not covered by the oil =
pi * (2)^2 - ∫ √[ 4 - x^2] - 1 dx ≈ 10.1096 ft^2
So...the volume of the oil is
Area covered by oil * cylinder height = 10.196 * 15 = 151.644 ft^2
So....the height of the oil when the tank is upright can be found as
151.644 = pi * (2)^2 * height
151.644 = 4 pi * height divide both sides by 4 pi
151.644 / [ 4 pi ] = height ≈ 12.1 ft