The radius of a circular mirror is increased by 2 inches due to its decorative border. This increases the area by 113 square inches. Find the radius of the mirror without the decorative border. Round to the nearest whole number.
To solve this problem, we need to use the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius.
Let's denote the original radius of the mirror as x. According to the problem statement, the radius of the mirror with the decorative border is increased by 2 inches. Therefore, the new radius with the border is x + 2.
The area of the new mirror with the border is given as the original area plus the increase in area, which is 113 square inches. Mathematically, we can write this as:
π(x + 2)^2 = πx^2 + 113
To solve this equation, we can first expand the terms on the left side:
π(x^2 + 4x + 4) = πx^2 + 113
Next, we can simplify the equation by canceling out the same terms on both sides:
πx^2 + 4πx + 4π = πx^2 + 113
Now, we can eliminate πx^2 from both sides of the equation:
4πx + 4π = 113
To isolate x, we can subtract 4π from both sides:
4πx = 113 - 4π
Finally, we divide both sides by 4π to solve for x:
x = (113 - 4π) / (4π)
Now, we can substitute the value of π as approximately 3.14 and calculate the value of x:
x = (113 - 4(3.14)) / (4(3.14))
x ≈ 3.92
So, rounding x to the nearest whole number, the radius of the mirror without the decorative border is 4 inches.
Let's assume the original radius of the mirror without the decorative border is "r" inches.
The radius of the mirror with the decorative border is "r + 2" inches.
The area of a circle is given by the formula: A = πr^2.
The original area of the mirror without the decorative border is: A = πr^2.
The increased area of the mirror with the decorative border is: A + 113.
Setting up the equation:
π(r + 2)^2 = πr^2 + 113
Expanding the equation:
π(r^2 + 4r + 4) = πr^2 + 113
Canceling out π from both sides:
r^2 + 4r + 4 = r^2 + 113
Simplifying the equation:
4r + 4 = 113
Subtracting 4 from both sides:
4r = 109
Dividing both sides by 4:
r = 27.25
Rounding to the nearest whole number:
r ≈ 27
Therefore, the radius of the mirror without the decorative border is approximately 27 inches.
original radius --- r inches
original area = πr^2 inches^2
new radius = x+2
new area = π(r+2)^2
π(r+2)^2 - πr^2 = 113
π(r^2 + 4r + 4) - πr^2 = 113
4πr = 113-4π
r = (113 - 4π)/(4π) = ...
let me know what you get