A circular pond has radius r. The area of a pond whose radius is 6m more than r is 4 times the area of the first pond. The radius r of the first pond equals
A1 = pi*r^2
A2 = pi*(r+6)^2 = 4A1 = 4*pi*r^2
pi*(r+6)^2 = 4*pi*r^2
Divide both sides by pi:
(r+6)^2 = 4r^2
r^2 + 12r + 36 = 4r^2
r^2 - 4r^2 + 12r +36 = 0
-3r^2 + 12r + 36 = 0
Divide both sides by -3:
r^2 - 4r - 12 = 0
(r-6)(r+2) = 0
r-6 = 0
r = 6.
r+2 = 0
r = -2
Select positive value of r:
r = 6 m
Let's break down the problem step by step to find the value of r.
1. Let's say the radius of the first pond is r. So, the area of the first pond is given by the formula A1 = πr^2, where A1 represents the area of the first pond.
2. Now, the problem states that the second pond has a radius which is 6 meters more than r. Therefore, the radius of the second pond is r + 6. Hence, the area of the second pond is A2 = π(r + 6)^2.
3. According to the problem, the area of the second pond is four times the area of the first pond. So, we can write the equation A2 = 4 * A1.
4. Substituting the values of A1 and A2 into the equation, we get π(r + 6)^2 = 4 * πr^2.
5. Simplifying the equation, we have (r + 6)^2 = 4r^2.
6. Expanding (r + 6)^2 gives r^2 + 12r + 36 = 4r^2.
7. Simplifying further, we have 3r^2 - 12r - 36 = 0.
8. Dividing the equation by 3, we get r^2 - 4r - 12 = 0.
9. Now, we can solve this quadratic equation to find the value of r. It can be factored as (r - 6)(r + 2) = 0.
10. Therefore, the possible values for r are 6 and -2. However, radius cannot be negative, so the only valid solution is r = 6.
Hence, the radius of the first pond is 6 meters.