Problem: When a certain amount of paint is spread over a sphere of radius 8.8 meters, there is a uniform coat with area density 4.7 gallons/m ^ 2. Use the technique of proportionality to determine the area density if the same amount of paint is spread over a sphere of radius 15.25 meters.
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Solution: We know that the area over which the paint is spread is 4 `pi r ^ 2. The area is therefore proportional to the square of the radius. We know also that the area density is inversely proportional to the area of the sphere. So the area density is inversely proportional to the square of the radius.
We therefore write
y = k/x ^ 2,
with y representing area density and x representing area.
Since area density is 4.7 gal/m ^ 2 when radius is 8.8 meters, it follows that 4.7 gal/m ^ 2 = k/( 8.8 meters) ^ 2.
Solving for k, we obtain k = 363.9 gallons. The relation y = k / x^2 therefore becomes
y = k/x ^ 2 = 363.9gallons / x ^ 2.
The constant k = 363.9 gallons is called the proportionality constant.
To find the area density for any radius, we simply substitute that radius for x, and y will give us the area density.
In this case, we obtain for radius 15.25 meters,
y = 363.9 gallons / ( 15.25 meters) ^ 2 = 1.564 gallons/meter ^ 2.
Generalized Response: If we know that a quantity Q, when spread uniformly over a sphere of radius r1, has density `sigma1, we can find the density of the same amount spread uniformly over a sphere of any radius r.
By the inverse square proportionality
`sigma / `sigma1 = (r1 / r)^2,
we have
`sigma = `sigma1 * (r1 / r) ^ 2.