If it takes one hundred three gallons of paint to paint a center stripe around a circular track that has a radius of one-fourth of a mile, then how much paint would it take to paint a center stripe around a circular track that had a radius of one-fifth of a mile?
The amount of paint needed is directly proportional to the circumference of the track. The formula for the circumference of a circle is:
C = 2πr
where C is the circumference and r is the radius. We can use this formula to find the ratio of the circumference of the two tracks:
C1/C2 = (2πr1)/(2πr2) = r1/r2
where C1 is the circumference of the first track (with radius r1) and C2 is the circumference of the second track (with radius r2).
We know that it takes 103 gallons of paint to paint the first track, so let's call the amount of paint needed for the second track P2. We can set up a ratio of the paint amounts:
103/P2 = C1/C2 = r1/r2
Substituting r1 = 1/4 mile and r2 = 1/5 mile:
103/P2 = (1/4)/(1/5) = 5/4
Multiplying both sides by P2:
P2 = 103 × 4/5 = 82.4
Therefore, it would take approximately 82.4 gallons of paint to paint a center stripe around a circular track that has a radius of one-fifth of a mile.
To find the amount of paint needed to paint a center stripe around a circular track with a radius of one-fifth of a mile, we can set up a ratio based on the given information.
Given:
Radius of the first track = 1/4 mile
Amount of paint needed for the first track = 103 gallons
Let's set up the ratio using the radii of the two tracks:
(1/4 mile / 1/5 mile) = (103 gallons / x gallons)
To solve for x, we can cross-multiply:
1/4 * x = 1/5 * 103
Simplifying, we have:
x/4 = 103/5
Multiply both sides by 4:
x = (103/5) * 4
x = 412/5
x = 82.4
Therefore, it would take approximately 82.4 gallons of paint to paint a center stripe around a circular track with a radius of one-fifth of a mile.