Question

You are designing a spherical tank to hold water for a small village in a developing country. The volume of liquid it can hold can be computed as V = πh2[3R − h]3 where V = volume (m3), h = depth of water in tank (m), and R = the tank radius (m).

Answer 1

A.
`9\pi h^2 - \pi h^3 -90 = 0\\\\`

B. h1 = 2.0813772719

h2 = 2.0272465228

h3 = 2.0269057423

Step-by-step explanation:

V = πh^2 x [3R − h]/3

Given v=30, R=3

`30 = \pi h^2 [(9-h)/(3) ]\\\\9\pi h^2 - \pi h^3 -90 = 0\\\\`

B. An initial guess within the interval [0; 6] is selected,

initial guess h0 = 1.5

Newton method x₂ = x₁ - f'(x₁)/f(x₁)

h₂ = h₁ - f'(x₁)/f(x₁)

`h_2 = h_1 - f'(x_1)/f(x_1)\\\\h_2 = h_1 - (18\pi h-3\pi h^2)/(9\pi h^2 - \pi h^3 -90)`

h1 = 2.0813772719

h2 = 2.0272465228

h3 = 2.0269057423