Fuel is flowing into a storage tank which can be filled to a depth of 6 metres.

When the fuel started flowing the tank was already filled to a depth of 2.5m. If the rate at which the depth of fuel in the tank is increasing, in metres per hour is given by d'(t) = 4t +5, find:
(i) the rate at which the height is increasing after 20 minutes
(ii) the height of water in the tank after 20 minutes
(iii) the time it takes to fill the tank

To find the answers to these questions, we need to integrate the rate equation and apply the appropriate initial conditions.

(i) To find the rate at which the height is increasing after 20 minutes, we need to evaluate d'(t) at t = 20 minutes. Since the rate equation is given by d'(t) = 4t + 5, we can integrate it to get the height function h(t). Integrating 4t + 5 with respect to t gives us 2t^2 + 5t + C, where C is the constant of integration.

To determine the constant of integration, we use the initial condition that the tank was already filled to a depth of 2.5 meters when the fuel started flowing. This means that at t = 0, h(0) = 2.5. Substituting this into the height function, we get 2(0)^2 + 5(0) + C = 2.5, which simplifies to C = 2.5.

Therefore, the height function is h(t) = 2t^2 + 5t + 2.5. Now, we can find the rate at which the height is increasing after 20 minutes by evaluating the derivative of the height function at t = 20 minutes. Taking the derivative of h(t) with respect to t gives us h'(t) = 4t + 5. Evaluating this at t = 20 minutes (which is 1/3 of an hour), we have h'(1/3) = 4(1/3) + 5 = 1.33 meters per hour.

Therefore, the rate at which the height is increasing after 20 minutes is 1.33 meters per hour.

(ii) To find the height of water in the tank after 20 minutes, we simply need to evaluate the height function h(t) at t = 20 minutes. Substituting t = 20 minutes (which is 1/3 of an hour) into the height function, we have h(1/3) = 2(1/3)^2 + 5(1/3) + 2.5 = 3.17 meters.

Therefore, the height of water in the tank after 20 minutes is 3.17 meters.

(iii) To find the time it takes to fill the tank, we need to determine the time when the height of water in the tank reaches the maximum capacity of 6 meters. Since the height function is h(t) = 2t^2 + 5t + 2.5, we can set it equal to 6 and solve for t:

2t^2 + 5t + 2.5 = 6

Rearranging and simplifying, we have:

2t^2 + 5t - 3.5 = 0

This is a quadratic equation, which we can solve by factoring or using the quadratic formula. Factoring yields (t - 0.5)(2t + 7) = 0, which gives us two possible solutions: t = 0.5 and t = -3.5/2. Since time cannot be negative, we discard t = -3.5/2 and conclude that t = 0.5.

Therefore, it takes 0.5 hours (or 30 minutes) to fill the tank.