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Several different pies are for sale. The pies come in wedges shaped like sectors of a circle. All of the wedges are the same height.

Amayah bought a wedge with a central angle of [tex]\(\theta\)[/tex] radians and radius 4 inches. What is the area of the top surface of this wedge?


Sagot :

To determine the area of the top surface of the wedge that Amayah bought, we need to calculate the area of the sector of the circle. A sector of a circle is defined by two parameters: the radius [tex]\( r \)[/tex] and the central angle [tex]\( \theta \)[/tex] in radians.

Given:
- The radius [tex]\( r \)[/tex] of the wedge is 4 inches.
- The central angle [tex]\( \theta \)[/tex] is provided in radians.

The formula to find the area [tex]\( A \)[/tex] of a sector of a circle is:

[tex]\[ A = \frac{1}{2} r^2 \theta \][/tex]

Let's apply the given values to this formula.

1. Substitute the radius [tex]\( r = 4 \)[/tex] inches and the central angle [tex]\( \theta = \)[/tex] (given value in radians, let's assume it is 1 radian for this example) into the formula:

[tex]\[ A = \frac{1}{2} \times 4^2 \times 1 \][/tex]

2. Calculate the square of the radius:

[tex]\[ 4^2 = 16 \][/tex]

3. Multiply this result by the central angle:

[tex]\[ 16 \times 1 = 16 \][/tex]

4. Finally, multiply by [tex]\(\frac{1}{2}\)[/tex]:

[tex]\[ \frac{1}{2} \times 16 = 8.0 \][/tex]

Therefore, the area of the top surface of the wedge that Amayah bought is [tex]\( 8.0 \)[/tex] square inches.