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Difficulty: HardCircular Area and Circumference Applications

A city is designing a circular park with a diameter of 6060 meters. In the center of the park, there will be a circular concrete plaza with a diameter of 2020 meters. The rest of the park will be covered in grass. If the cost to plant grass is $0.50\$0.50 per square meter, what will be the total cost to plant the grass, expressed in terms of π\pi?

  1. A
    $20π\$20\pi
  2. $400π\$400\piAnswer
  3. C
    $800π\$800\pi
  4. D
    $1,600π\$1,600\pi

Answer

The correct total cost is $400π\$400\pi.
The correct answer is found by first calculating the area of the entire circular park and subtracting the area of the concrete plaza. Using the formula Area=πr2Area = \pi r^2, the park's area is π(30)2=900π\pi(30)^2 = 900\pi and the plaza's area is π(10)2=100π\pi(10)^2 = 100\pi. The difference is 800π800\pi square meters. Multiplying this by the cost of $0.50\$0.50 per square meter results in a total cost of $400π\$400\pi.

Step-by-Step Solution

1
Find the radii of the entire park and the central plaza.
Park radius = 3030 m. Plaza radius = 1010 m.
The area formula for a circle requires the radius, which is half of the diameter.
2
Calculate the total area of the park and the area of the plaza in terms of π\pi.
Total Area = π(30)2=900π\pi(30)^2 = 900\pi. Plaza Area = π(10)2=100π\pi(10)^2 = 100\pi.
We must determine the individual areas to find the space remaining for the grass.
3
Subtract the plaza area from the total park area to find the area to be covered in grass.
Grass Area = 900π−100π=800π900\pi - 100\pi = 800\pi square meters.
The grass will cover the portion of the park that is not occupied by the concrete plaza.
4
Multiply the grass area by the unit cost to plant the grass.
Total Cost = 800π×$0.50=$400π800\pi \times \$0.50 = \$400\pi.
The final question asks for the total cost, which is the area multiplied by the price per square meter.

Key Concept

Calculating the area of an annulus (a ring shape) by subtracting the area of an inner circle from an outer circle, and applying unit rates.
Estimated Time:2m 0s
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Topics

Circular Area and Circumference Applications

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