Math Problem Help?

Question

The number of houses that can be served by a water main is directly proportional to the square of the diameter of the main. This is because the cross-sectional area of the pipe, and thus the number of liters per minute that can flow, varies directly with the square of the diameter. Suppose that the City Waterworks has a 10 cm diameter water main that can supply 50 houses.

I know that the particular equation is h=0.5d^2
I just don't know how you get that. Thank you!

Answer 1

houses varies with diameter^2.

h = x * d^2

50 = x * (10^2)
50 = x * 100
x = .5

h = .5 * d^2

Answer 2

You know that the number of houses, h, is directly proportional to the square of the diameter of the main. This lets you set up the equation h=kd^2. The k is a constant. Since h and d^2 are directly proportional, when one increases, the other will and vice versa, but they are not equal. The difference between them is proportional and constant. This proportion is the constant k. To solve for k, you use the second part of the problem and plug in the values for your equation, h=kd^2, to solve for k.
50=k(10)^2
50=k100
.5=k
Now plug k back into the equation h=kd^2.
h=.5d^2
That's how you get there.