Question

1. what is the area of a sector with a radius of 10 meters and a central angle of 45°?

a. \\(\frac{10\pi}{2}\\) square meters
b. 50\\(\pi\\) square meters
c. 25\\(\pi\\) square meters
d. \\(\frac{25\pi}{2}\\) square meters

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector with radius \( r \) and central angle \( \theta \) (in degrees) is \( A=\frac{\theta}{360^\circ}\times\pi r^{2} \).

Step2: Identify the given values

Here, \( r = 10 \) meters and \( \theta=45^\circ \).

Step3: Substitute the values into the formula

Substitute \( r = 10 \) and \( \theta = 45^\circ \) into the formula:

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Wait, no, wait. Wait, \( \frac{45}{360}=\frac{1}{8} \), \( 10^2 = 100 \), \( \frac{1}{8}\times100\pi=\frac{25\pi}{2} \)? Wait, no, wait, maybe I made a mistake. Wait, no, let's recalculate. Wait, \( 45\div360=\frac{1}{8} \), \( 10^2 = 100 \), \( \frac{1}{8}\times100\pi=\frac{25\pi}{2} \)? Wait, no, \( 100\div8 = 12.5 \), which is \( \frac{25}{2} \). Wait, but let's check the options. Option d is \( \frac{25\pi}{2} \) square meters. Wait, but wait, maybe I messed up the formula. Wait, the formula for the area of a sector is also \( A=\frac{1}{2}r^{2}\theta \) when \( \theta \) is in radians. Let's convert \( 45^\circ \) to radians. \( 45^\circ=\frac{\pi}{4} \) radians. Then \( A=\frac{1}{2}\times(10)^{2}\times\frac{\pi}{4}=\frac{1}{2}\times100\times\frac{\pi}{4}=\frac{100\pi}{8}=\frac{25\pi}{2} \). Yes, that's correct. So the area is \( \frac{25\pi}{2} \) square meters, which is option d? Wait, no, wait the options: a. \( \frac{10\pi}{2} \), b. \( 50\pi \), c. \( 25\pi \), d. \( \frac{25\pi}{2} \). Wait, let's recalculate with the degree formula. \( \frac{45}{360}\times\pi\times10^2=\frac{1}{8}\times100\pi=\frac{25\pi}{2} \). Yes, so the area is \( \frac{25\pi}{2} \) square meters, which is option d. Wait, but wait, maybe I made a mistake in the first calculation. Wait, no, \( 45/360 = 1/8 \), \( 10^2 = 100 \), \( 100\times1/8 = 12.5 \), \( 12.5\pi=\frac{25\pi}{2} \). Yes, that's correct. So the correct answer is option d.

Answer:

d. \(\frac{25\pi}{2}\) square meters