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Zorluk: OrtaCircular Area and Circumference Applications

A local radio station broadcasts its signal in a circular area with a radius of 4040 miles. The station upgrades its transmitter, which increases the broadcasting radius to 5050 miles. How many additional square miles of coverage does the upgraded transmitter provide? (Use π=3.14\pi = 3.14)

Cevap: 2826 square miles

Cevap

2826 square miles
The additional coverage is represented geometrically by the area of the ring (annulus) formed between the two circular boundaries. This is found by calculating both areas and subtracting the original area (3.14×402=50243.14 \times 40^2 = 5024) from the new area (3.14×502=78503.14 \times 50^2 = 7850), resulting in 28262826 square miles of extra coverage.

Adım Adım Çözüm

1
Calculate the original coverage area.
Aoriginal=3.14×402=3.14×1600=5024A_{original} = 3.14 \times 40^2 = 3.14 \times 1600 = 5024 square miles
We must determine the initial broadcast area before the upgrade.
2
Calculate the new coverage area.
Anew=3.14×502=3.14×2500=7850A_{new} = 3.14 \times 50^2 = 3.14 \times 2500 = 7850 square miles
We need the total broadcast area provided by the new transmitter.
3
Calculate the additional coverage area by finding the difference.
7850−5024=28267850 - 5024 = 2826 square miles
The additional area is the exact difference between the new area and the original area.

Anahtar Kavram

Calculating the difference between two circular areas (area of an annulus).

Alternatif Yöntem

Instead of calculating both areas separately, you can factor the expression for the difference in area: ΔA=π(R2−r2)=3.14×(502−402)=3.14×(2500−1600)=3.14×900=2826\Delta A = \pi(R^2 - r^2) = 3.14 \times (50^2 - 40^2) = 3.14 \times (2500 - 1600) = 3.14 \times 900 = 2826.
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Circular Area and Circumference Applications

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