The area of a circle is given by the formula A = ðr2, where r is the radius. If increasing the radius of a circle by 1 inch gives the resulting circle an area of 100ð square inches, what is the radius of the original circle?
well, if the original radius is r, we have
π(r+1)^1 = 100π
Or, just note that it's obvious that a radius of 10 gives an area of 100π, so the original radius was 9.
Well, in this case, we can use the formula A = πr² to figure out the radius of the original circle. If increasing the radius by 1 inch turns the area into 100π square inches, then we can say:
100π = π(r + 1)²
Now, let's do some math! We can start by dividing both sides of the equation by π:
100 = (r + 1)²
Next, we can take the square root of both sides:
√100 = √(r + 1)²
Which simplifies to:
10 = r + 1
Subtracting 1 from both sides gives us:
r = 9
So, the radius of the original circle is 9 inches. Ta-da!
To find the radius of the original circle, we need to solve the equation:
A = πr^2
Given that increasing the radius by 1 inch gives the resulting circle an area of 100π square inches, we have:
A + π(r+1)^2 = 100π
Expanding the equation:
A + π(r^2 + 2r + 1) = 100π
Substituting the formula for the area:
πr^2 + π(r^2 + 2r + 1) = 100π
Simplifying the equation:
2πr^2 + 2πr + π = 100π
Subtracting π from both sides:
2πr^2 + 2πr = 100π - π
Simplifying:
2πr^2 + 2πr = 99π
Dividing both sides by π:
2r^2 + 2r = 99
Rearranging the equation:
2r^2 + 2r - 99 = 0
Now we can solve this quadratic equation to find the value of r.
To solve this problem, we need to work backwards from the given information.
We know that increasing the radius by 1 inch gives the resulting circle an area of 100ð square inches. This means that the new radius (r + 1) is unknown, but the new area (A) is known. We can set up the equation as follows:
A = ð(r + 1)²
Substituting the given area of 100ð square inches:
100ð = ð(r + 1)²
Now we can simplify the equation:
100ð = ð(r² + 2r + 1)
Next, we can cancel out ð from both sides of the equation:
100 = r² + 2r + 1
Rearranging the equation:
r² + 2r + 1 - 100 = 0
Simplifying further:
r² + 2r - 99 = 0
Now we can solve this quadratic equation to find the values of r. We can either factor the equation, complete the square, or use the quadratic formula. Since factoring may not give us integer solutions, let's use the quadratic formula:
r = (-b ± √(b² - 4ac)) / (2a)
For our equation, a = 1, b = 2, and c = -99. Plugging these values into the quadratic formula:
r = (-2 ± √(2² - 4(1)(-99))) / (2(1))
Simplifying further:
r = (-2 ± √(4 + 396)) / 2
r = (-2 ± √400) / 2
Now we can calculate the values of r:
r = (-2 + 20) / 2 or r = (-2 - 20) / 2
Simplifying further:
r = 18 / 2 or r = -22 / 2
r = 9 or r = -11
Since a radius cannot be negative, the radius of the original circle is 9 inches.