If the radius of a circle is increased by 6 cm, the new area is 121 (pi) cm^2. Find the radius of the original border using factoring.
the answer is 5 by common sense, i would reevaluate your schools "honours" program
the area of a circle is pi(r^2)
therefore pi(x+6)^2=121pi
now just solve
(x+6)^2=121
x+6=11
x=5
Let's solve the problem step-by-step using factorization.
Step 1: Define the variables
Let's denote the original radius of the circle as "r" cm.
Step 2: Find the equation relating the original and new area
The original area of a circle with radius "r" is given by A = πr^2.
When the radius is increased by 6 cm, the new radius becomes "r + 6" cm, and the new area is given as 121π cm^2.
Step 3: Write the equation
The equation relating the original and new area can be written as:
πr^2 + 121π = π(r + 6)^2
Step 4: Expand and simplify the equation
Expanding the equation, we get:
πr^2 + 121π = π(r^2 + 12r + 36)
Simplifying, we have:
πr^2 + 121π = πr^2 + 12πr + 36π
Step 5: Cancel out π from both sides
πr^2 + 121π - πr^2 - 12πr - 36π = 0
Step 6: Factor out the common factor
π(r^2 - 12r - 36) = 0
Step 7: Solve for r^2 - 12r - 36 = 0
To solve the quadratic equation, we can either factor it or use the quadratic formula. Let's use factoring in this case.
r^2 - 12r - 36 = 0
The equation can be factored as:
(r - 6)(r - 6) = 0
Step 8: Solve for r
By setting each factor equal to zero, we can solve for r:
r - 6 = 0
r = 6
Therefore, the radius of the original circle or border is 6 cm.
To solve this problem using factoring, we need to use the formula for the area of a circle, which is given by A = πr^2, where A is the area of the circle and r is its radius.
Let's represent the original radius of the circle as x.
We are given that when the radius is increased by 6 cm, the new area is 121π cm^2. So the new radius would be (x + 6) cm.
Using the formula for the area of a circle, we can write the equation as:
π(x + 6)^2 = 121π
Now, let's simplify this equation by canceling out the π on both sides:
(x + 6)^2 = 121
To solve for x, we need to find the square root of both sides of the equation:
√[(x + 6)^2] = √121
Simplifying further, we get:
x + 6 = 11
Now, subtracting 6 from both sides of the equation, we find:
x = 11 - 6
Therefore, the original radius of the circle is x = 5 cm.