Direct Variation: Water Pipe and Number of Houses
How many houses can be served by a water pipe that has a 30-centimeter diameter?
The number of houses served by a water pipe varies directly as the square of the diameter of the pipe. If a water pipe with a 10-centimeter diameter can serve 40 houses, how many houses can a 30-centimeter diameter water pipe serve?
What size of water pipe is needed for a new subdivision of 2560 houses?
If a water pipe with a 30-centimeter diameter can serve a certain number of houses, what diameter of water pipe is needed to serve a new subdivision of 2560 houses?
Answer:
a. A water pipe with a 30-centimeter diameter can serve 360 houses.
b. To serve a new subdivision of 2560 houses, a water pipe with a 80-centimeter diameter is needed.
In the given scenario, the relationship between the number of houses served and the square of the diameter of the water pipe follows a direct variation. This can be expressed as:
Number of houses = k x Diameter^2
Where k is the constant of variation.
Given that a water pipe with a 10-centimeter diameter can serve 40 houses, we can calculate the value of k as follows:
k = Number of houses / Diameter^2
k = 40 / 100 = 0.4
Now, using the value of k, we can determine the number of houses that a 30-centimeter diameter water pipe can serve:
Number of houses = 0.4 x 30^2 = 0.4 x 900 = 360 houses
For serving a new subdivision of 2560 houses, we need to find the appropriate diameter of the water pipe:
Number of houses = 0.4 x Diameter^2 = 2560 houses
Diameter^2 = 2560 / 0.4 = 6400
Diameter = √6400 = 80 centimeters
Therefore, to serve a new subdivision of 2560 houses, a water pipe with an 80-centimeter diameter is needed.