Friday 23 March 2018 Assignment Two First Year Class Question 1 The water from a

Question

Friday 23 March 2018 Assignment Two First Year Class Question 1 The water from a vertical cylindrical tank is drained by opening a valve at the base. The water will flow faster initially when the tank is full but slows down as the tank drains. Initially the depth of water is 144 metres. It turns out that the rate at which the water levels drops is proportional to the square root of the water's depth, y metres. t is measured in minutes and that the depth is given by: (i) (ii) (iii) (iv) y (12-0.5t)2 After 6 minutes what is the depth of the remaining water in the tank? At what time does the water in the tank have a depth of 36 metres? How long does it take to empty the tank? Consider the following differential equation: dy dt Verify that y (12-0.5t) 0.25t- 12t + 144 is the solution to the differential equation. (Gv) Calculate dy dt Question 2 A car moves forwards and backwards along a straight line. The distance from a fixed point O is s (metres) at a time of t seconds and the velocity of the car is: Initially the distance from O is 7 metres v 3t-10t+3

Explanation / Answer

Solution

Q1

Depth of water, at time, t is given by: y = (12 – 0.5t)2 …………………………………….(1)

[Note that at t = 0, the above equation shows y = 144, which conforms to the given initial depth of 144 m.]

Part (i)

Substituting t = 6 in (1) above, y6 = 92 = 81 meter ANSWER

Part (ii)

Substituting y = 36 in (1) above and solving for t, t(36) = 12 minutes ANSWER

Part (iii)

When tank is empty, depth, y = 0.

Substituting y = 0 in (1) above solving for t, t(0) = 24 minutes ANSWER

Part (iv)

Given Differential equation is dy/dt = - y

Or, (dy/y) = - dt

Integrating both sides,

2y = - t + c, where c = constant of integration……………………………………….(2)

Given y = 144 at t = 0, (2) => 24 = c …………………………………………………(3)

(2) and (3) => y = (- t/2) + 12 = 12 – 0.5t or y = (12 – 0.5t)2 VERIFIED

Part (v)

Differentiating dy/dt = - y with respect to t,

d2y/dt2 = - (1/2y)( dy/dt) = - (1/2y)( - y) because dy/dt = - y

Thus, d2y/dt2 = ½ ANSWER

DONE

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