Explanation:
Let's start by using the formula for the volume of a cylinder:
V = πr^2h
where V is the volume of the cylinder, r is the radius of the cylinder, h is the height of the cylinder, and π is a mathematical constant (approximately equal to 3.14).
Since we are dealing with a drain pipe, we can assume that the height of the cylinder is fixed and does not change. Therefore, we can rewrite the formula as:
V = πr^2h = Ah
where A is the cross-sectional area of the cylinder.
Now, let's use the given information that the drain pipe can carry 36 litres per second. We know that the volume of water that passes through the pipe in one second is equal to 36 litres. We can therefore write:
36 = Ahv
where v is the velocity of the water flowing through the pipe. Since we are assuming that the height of the cylinder is fixed, we can simplify this equation to:
36 = Av
Now we need to determine the percentage increase in the diameter of the drain pipe necessary to enable it to carry 60 litres per second. Let's call the new diameter d2 and the old diameter d1. We can set up a proportion to solve for d2:
A1/A2 = d1^2/d2^2
We know that A1 and A2 are proportional to the volumes of water the pipe can carry, so we can write:
A1/A2 = 36/60
Simplifying this equation, we get:
A1/A2 = 3/5
Substituting in the formula for the cross-sectional area of a cylinder, we get:
πd1^2/4 / πd2^2/4 = 3/5
Simplifying further, we get:
d1^2/d2^2 = 3/5
Taking the square root of both sides, we get:
d1/d2 = sqrt(3/5)
Now we can solve for d2:
d2 = d1 / sqrt(3/5)
We want to know the percentage increase in the diameter, which we can find using the formula:
% Increase = (New Value - Old Value) / Old Value x 100%
Substituting in our values, we get:
% Increase = (d1 / sqrt(3/5) - d1) / d1 x 100%
Simplifying, we get:
% Increase = (1 / sqrt(3/5) - 1) x 100%
Using a calculator, we get:
% Increase ≈ 34.64%
Therefore, the percentage increase in the diameter of the drain pipe necessary to enable it to carry 60 litres per second is approximately 34.64%.