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Difficulty: MediumCircles, Arc Lengths, and Sector Areas

In a circle, an arc of length 4π4\pi corresponds to a central angle of 40∘40^\circ. If the area of the sector formed by this central angle is kπk\pi, what is the value of kk?

Answer: 36

Answer

The value of kk is 36.
Using the arc length equation 4π=40360⋅2πr4\pi = \frac{40}{360} \cdot 2\pi r, we solve for the radius r=18r = 18. Substituting r=18r = 18 into the sector area formula A=40360⋅π(18)2A = \frac{40}{360} \cdot \pi (18)^2 results in A=36πA = 36\pi. Thus, k=36k = 36.

Step-by-Step Solution

1
Calculate the radius of the circle using the arc length formula
Radius r=18r = 18
Arc length is related to central angle and radius by L=θ360∘⋅2πrL = \frac{\theta}{360^\circ} \cdot 2\pi r. Substituting L=4πL = 4\pi and θ=40∘\theta = 40^\circ gives 4π=19⋅2πr  ⟹  r=184\pi = \frac{1}{9} \cdot 2\pi r \implies r = 18.
2
Calculate the area of the sector using the radius and central angle
Sector Area A=36πA = 36\pi
Sector area is calculated using A=θ360∘⋅πr2A = \frac{\theta}{360^\circ} \cdot \pi r^2. Substituting θ=40∘\theta = 40^\circ and r=18r = 18 gives A=19⋅π(182)=36πA = \frac{1}{9} \cdot \pi (18^2) = 36\pi.
3
Extract the coefficient kk from kπk\pi
k=36k = 36
Comparing 36π36\pi to kπk\pi directly yields k=36k = 36.

Key Concept

Relationship between central angle, arc length, radius, and sector area
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Topics

Circles, Arc Lengths, and Sector Areas

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