Final answer:
To find when a cistern will be full with both fill pipes opened together, we calculate the rates of the pipes and determine the total time based on their combined rates. The correct answer is that the cistern will be full in 30 minutes.
the correct answer is (b).
Step-by-step explanation:
The question involves calculating the total time needed for two pipes with different fill rates to fill a cistern when they are used together. Let's denote the time it takes for fill pipe B to fill the cistern as t minutes. According to the given information, fill pipe A is four times faster than fill pipe B and takes 15 minutes less to fill the cistern. This means that fill pipe A will take t - 15 minutes to fill the cistern.
Now we establish the filling rates for A and B in terms of the cistern's capacity. Fill pipe B fills the cistern in t minutes, so its rate is 1/t cistern per minute. Fill pipe A, being four times faster, would fill the cistern at 4/t cistern per minute.
When both pipes are opened together, their rates are additive:
Rate of A + Rate of B = 1/t - 15 + 1/t = Capacity of cistern. Simplifying, we get 4/t + 1/t = 5/t cistern per minute. Setting up the equation 5/t = 1/t + 1/(t-15), we can solve for t.
Multiplying through by t(t-15) to clear the fractions gives us 5(t-15) = t + (t-15). This simplifies to 5t - 75 = 2t - 15, and solving for t yields t = 30.
So, if both fill pipes are opened together, the cistern will be full in 30 minutes. Thus, the correct answer is (b).