If the radius of a circle is increased by 1 unit, the area of the circle is increased by 16 pi square units. What is the radius of the original circle?
original radius --- r
new radius ------ r+1
original area = πr^2
new area = π(r+1)^2
π(r+1)^2 - πr^2 = 16π
divide by π and expand ...
r^2 + 2r + 1 - r^2 = 16
2r = 15
r = 7/5 units
check:
old area = π(7.5)^2 = 56.25π
new area = π(8.5)^2 = 72.25π
difference = 72.25P - 56.25π = 16π
all looks good!
AMAZING. Thanks Much.
Let's assume the radius of the original circle is 'r' units.
The area of the original circle is given by A = πr².
According to the problem, if the radius is increased by 1 unit, the new radius is 'r + 1' units.
So, the area of the new circle is A + 16π square units.
Now, we can write the equation: A + 16π = π(r + 1)².
Simplifying the equation:
πr² + 16π = π(r² + 2r + 1).
πr² + 16π = πr² + 2πr + π.
Canceling out the common terms:
16π = 2πr + π.
Subtracting π from both sides:
16π - π = 2πr.
15π = 2πr.
Dividing both sides by 2π:
r = (15π) / (2π).
Simplifying:
r = 7.5.
Therefore, the radius of the original circle is 7.5 units.
To solve this problem, we can first set up an equation using the given information and then solve for the radius of the original circle.
Let's consider the original radius of the circle as 'r' units.
According to the problem, if the radius is increased by 1 unit, the new radius becomes 'r + 1' units. Similarly, the respective areas will be πr² and π(r + 1)².
Now, we are given that the increase in area is 16π square units. Mathematically, this can be represented as:
π(r + 1)² - πr² = 16π
To solve for 'r', we can simplify this equation:
π(r² + 2r + 1) - πr² = 16π
πr² + 2πr + π - πr² = 16π
2πr = 15π
We can cancel out the π on both sides of the equation:
2r = 15
Finally, divide both sides of the equation by 2 to isolate 'r':
r = 15/2
Therefore, the radius of the original circle is 7.5 units.