Consider a spherical storage tank containing oil. The tank has a diameter of 8 ft. You are asked

Question

to calculate the height h to which a dipstick 10 ft long would be wet with oil when immersed in
the tank when it contains 6 3 ft of oil.
The equation that gives the height, h , of the liquid in the spherical tank for the given
volume and radius is given by
    2 2 f (h)  2h h 9h  7.6394 cos(h )
(a) By using an appropriate method that you have learned in this course, show that this
function has at least one root
(b) Find the positive lowest root of the equation by using
(i) Newton Raphson method
(ii) Secant method
For part (b), choose suitable initial guess(es) to perform the computation until
2 1 10 . a   

Answer 1

When the tank has 6.3 ft of oil, the dipstick will be wet by 6.3 feet if it is inserted vertically to the very bottom of the tank.

The radius of the surface of water at height x in a sphere of radius R is given by r(x)=sqrt(R^2-(x-R)^2)
By integrating r(x) with from 0 to height h, the volume of water is
V(h)=∫r(x)dx from 0 to h
= π(x³R-x^4/4)/3

Which gives you the volume in terms of h.
To find the height in terms of volume, you can solve the problem using Newton's method, secant method or any other iterative method.

Answer 2

Correction:
V(h) should read:
V(h)=π(3h²R-h³)/3
=πh²(R-h/3)

Answer 3

then the initial value for h, what?