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Difficulty: MediumCircular Area and Circumference Applications

A city park features a circular fountain with a diameter of 2020 feet. The city plans to pour a new concrete walkway that is exactly 22 feet wide completely surrounding the fountain. If the concrete costs $5.00\$5.00 per square foot, what is the total cost of the concrete required for the new walkway? (Use π=3.14\pi = 3.14)

  1. A
    \$31.40
  2. B
    \$62.80
  3. \$690.80Answer
  4. D
    \$1,318.80

Answer

The total cost of the concrete required for the new walkway is \$690.80.
The correct answer is found by calculating the area of the walkway and multiplying it by the cost per square foot. First, find the radius of the fountain by halving the diameter (20÷2=1020 \div 2 = 10 feet). The outer radius includes both the fountain and the walkway (10+2=1210 + 2 = 12 feet). The area of the walkway is the outer area minus the inner area: (3.14×122)−(3.14×102)=452.16−314=138.16(3.14 \times 12^2) - (3.14 \times 10^2) = 452.16 - 314 = 138.16 square feet. Finally, multiplying this area by the cost of $5.00\$5.00 per square foot gives $690.80\$690.80.

Step-by-Step Solution

1
Determine the inner and outer radii of the circular areas.
Inner radius = 1010 feet; Outer radius = 1212 feet.
The area formula requires the radius, which is half of the 2020-foot diameter. The outer radius includes the 1010-foot fountain radius plus the 22-foot walkway.
2
Calculate the area of the concrete walkway.
(3.14×122)−(3.14×102)=452.16−314=138.16(3.14 \times 12^2) - (3.14 \times 10^2) = 452.16 - 314 = 138.16 square feet.
The walkway forms a ring (annulus) around the fountain. Its area is the total outer area minus the inner fountain area.
3
Calculate the total cost of the concrete.
138.16 sq ft×$5.00/sq ft=$690.80138.16 \text{ sq ft} \times \$5.00/\text{sq ft} = \$690.80.
The total material cost is the square footage of the walkway multiplied by the price per square foot.

Key Concept

Calculating the area of a circular ring (annulus) and applying unit costs.
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Topics

Circular Area and Circumference Applications

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