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Difficulty: Very hardCircles

A circular crop field has a diameter of 3030 yards. A dirt road that is 55 yards wide is built along the entire outer edge of the field. What is the area of the dirt road in square yards?

  1. 175π175\piAnswer
  2. B
    325π325\pi
  3. C
    25π25\pi
  4. D
    40π40\pi

Answer

The area of the dirt road is 175π175\pi square yards.
The dirt road forms a ring (an annulus) around the circular field. To find this area, calculate the area of the total circular space (field plus road) and subtract the area of the field itself. The field's radius is 1515 yards (30÷230 \div 2), making the total outer radius 2020 yards (15+515 + 5). Subtracting the inner area (225π225\pi) from the outer area (400π400\pi) gives the road's area: 175π175\pi square yards.

Step-by-Step Solution

1
Find the radius of the inner circular field.
rinner=302=15r_{inner} = \frac{30}{2} = 15 yards
The area formula for a circle requires the radius, but the diameter of the field is given.
2
Determine the total radius of the entire area, including the road.
router=15+5=20r_{outer} = 15 + 5 = 20 yards
The dirt road adds 55 yards to the outer edge of the field's radius.
3
Calculate the total area enclosed by the outer edge of the road.
Areaouter=π(20)2=400πArea_{outer} = \pi (20)^2 = 400\pi square yards
This represents the total area of the field and the road combined.
4
Calculate the area of just the inner crop field.
Areainner=π(15)2=225πArea_{inner} = \pi (15)^2 = 225\pi square yards
This area must be subtracted to isolate the area of the road.
5
Subtract the inner area from the total outer area.
400π−225π=175π400\pi - 225\pi = 175\pi square yards
The area of the ring (annulus) is the difference between the two concentric circular areas.

Key Concept

Area of an annulus (a ring formed by two concentric circles)
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Topics

Circles

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