The area of two circles are in a ratio of 4:9. If both radii are integers, and r(1) - r(2) = 2, what is the radius of the larger circle?
The areas of two circles are proportional to the square of their raddi
so r1^2 / r2^2 = 9/4
r1/r2 = 3/2
r1 = 3r2/2
also r1 = r2 + 2
Thus:
3r2/2 = r2 + 2
3r2 = 2r2 + 4
r2 = 4
then r1 = 6
the larger radius is 6 units
check:
area of larger = 36π
area of smaller is 16π
ratio of larger : smaller = 36π : 16π = 36 : 16
= 9:4
To find the radius of the larger circle, we first need to set up a system of equations based on the given information.
Let's say the radius of the smaller circle is r1 and the radius of the larger circle is r2.
The ratio of the areas of the two circles is given as 4:9. Since the area of a circle is proportional to the square of its radius, we can write the equation:
π(r1^2) / π(r2^2) = 4/9
Simplifying this equation, we get:
(r1^2) / (r2^2) = 4/9
Cross-multiplying, we have:
9(r1^2) = 4(r2^2)
Now, we are also given that r1 - r2 = 2. Rearranging this equation, we have:
r1 = r2 + 2
From these two equations, we can solve for the radius of the larger circle (r2).
Substituting r1 = r2 + 2 into the area ratio equation, we get:
9(r2 + 2)^2 = 4(r2^2)
Expanding and simplifying this equation, we have:
9(r2^2 + 4r2 + 4) = 4(r2^2)
9r2^2 + 36r2 + 36 = 4r2^2
Subtracting 4r2^2 from both sides, we get:
5r2^2 + 36r2 + 36 = 0
Now we have a quadratic equation. Solving it will give us the value of r2, the radius of the larger circle. We can use the quadratic formula to find the values of r2:
r2 = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values from our equation:
a = 5, b = 36, c = 36
r2 = (-36 ± √(36^2 - 4(5)(36))) / (2(5))
Simplifying further:
r2 = (-36 ± √(1296 - 720)) / 10
r2 = (-36 ± √576) / 10
r2 = (-36 ± 24) / 10
Now we have two possible solutions for r2:
1. r2 = (-36 + 24) / 10 = -12/10 = -1.2
2. r2 = (-36 - 24) / 10 = -60/10 = -6
Since the radius cannot be negative, we discard the solution r2 = -1.2. Therefore, the radius of the larger circle (r2) is 6.