the circular pond has radius r. the area of a pond whose radius is 6m more than r is 4 times the area of the the first pond. the radius r of the first pond equals
The radius of the first 1 would be 24
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Let's break down the information given in the question step by step:
1. The first circular pond has a radius, denoted as r.
2. The second circular pond has a radius that is 6 meters more than r, denoted as (r + 6).
3. The area of the second pond is 4 times the area of the first pond.
To find the radius r of the first pond, we need to use the formula for the area of a circle:
Area = π * r^2
Let's set up an equation using this information:
4 * (π * r^2) = π * (r + 6)^2
Now we can solve for r. Let's simplify the equation step by step:
1. Distribute the π on the right side of the equation:
4 * π * r^2 = π * (r^2 + 12r + 36)
2. Remove π from both sides of the equation by dividing both sides by π:
4 * r^2 = r^2 + 12r + 36
3. Subtract r^2 and 12r from both sides of the equation:
3 * r^2 - 12r - 36 = 0
4. Divide the entire equation by 3 to simplify it:
r^2 - 4r - 12 = 0
Now we have a quadratic equation. To solve it, we can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -4, and c = -12. Plugging these values into the quadratic formula, we get:
r = (-(-4) ± √((-4)^2 - 4(1)(-12))) / (2(1))
Simplifying further:
r = (4 ± √(16 + 48)) / 2
r = (4 ± √64) / 2
r = (4 ± 8) / 2
Now we have two possibilities for r:
1. r = (4 + 8) / 2 = 12 / 2 = 6
2. r = (4 - 8) / 2 = -4 / 2 = -2
Since the radius cannot be negative, the radius r of the first pond is 6 meters.
To find the radius of the first pond, let's set up an equation based on the given information.
Let r be the radius of the first pond.
According to the problem, the area of the second pond (with radius r + 6) is 4 times the area of the first pond (with radius r).
The formula for the area of a circle is A = π * r^2, where A is the area and r is the radius.
So, the equation can be written as:
π * (r + 6)^2 = 4 * π * r^2
Now, let's solve for r:
Expand the equation:
π * (r^2 + 12r + 36) = 4 * π * r^2
Distribute π:
r^2 + 12r + 36 = 4r^2
Move all terms to one side:
4r^2 - r^2 - 12r - 36 = 0
Combine like terms:
3r^2 - 12r - 36 = 0
Now, we have a quadratic equation that we can solve. To find the value of r, we can either factor the equation, complete the square, or use the quadratic formula.
Factoring:
3(r^2 - 4r - 12) = 0
Now we have:
(r - 6)(r + 2) = 0
Setting each factor equal to zero:
r - 6 = 0 or r + 2 = 0
Solving for r, we get:
r = 6 or r = -2
Since the radius cannot be negative, we discard the value of r = -2.
Therefore, the radius r of the first pond equals 6 meters.