ACT (Enhanced)MathematicsHard
An architect is designing a circular garden with a radius of 10 feet. If a path of uniform width is to be built around the garden, increasing the total area by 44π square feet, what is the width of the path?
- A2 feet
- B3 feet
- C1 foot
- D4 feet
Show answer & explanationAnswer & explanation
Correct answer: A. 2 feet
The area of the original garden is π * (10^2) = 100π sq ft. Let 'w' be the width of the path. The new radius is (10+w). The new total area is π * (10+w)^2. The increase in area is π * (10+w)^2 - 100π = 44π. Divide by π: (10+w)^2 - 100 = 44. (10+w)^2 = 144. 10+w = 12 (since width must be positive). So, w = 2 feet.
Why the other options are wrong
- B. If w=3, new area is π(13^2) = 169π. Increase = 69π, not 44π.
- C. If w=1, new area is π(11^2) = 121π. Increase = 21π, not 44π.
- D. If w=4, new area is π(14^2) = 196π. Increase = 96π, not 44π.
Area of a Circle
The measure of the two-dimensional space enclosed by a circle, calculated using the formula A = πr², where 'r' is the radius.
- Units are squared (e.g., ft²).
- Radius is half the diameter.
- Changes in radius have a squared effect on area.
Memory trick: Areas are like puzzle pieces, fitting together to make a whole.