Question

if the radius of a circle is doubled, how does the area of the sector with a fixed central angle change?
it becomes one - fourth as large
it remains the same
it doubles
it becomes four times larger
4 multiple choice 1 point
if the radius of a circle is doubled, how does the length of the arc with a fixed central angle change?
it doubles
it becomes one - half as large
it becomes four times as large
it remains the same

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector is \(A=\frac{1}{2}r^{2}\theta\) (where \(r\) is the radius and \(\theta\) is the central angle in radians).
Let the original radius be \(r_1 = r\), then the original area \(A_1=\frac{1}{2}r^{2}\theta\).
If the radius is doubled, \(r_2 = 2r\), and the new area \(A_2=\frac{1}{2}(2r)^{2}\theta\).

Step2: Simplify the new - area formula

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Since \(A_1=\frac{1}{2}r^{2}\theta\), we have \(A_2 = 4A_1\).

Step3: Recall the formula for the arc - length

The formula for the arc - length is \(L = r\theta\) (where \(r\) is the radius and \(\theta\) is the central angle in radians).
Let the original radius be \(r_1 = r\), then the original arc - length \(L_1=r\theta\).
If the radius is doubled, \(r_2 = 2r\), and the new arc - length \(L_2=(2r)\theta\).
Since \(L_1 = r\theta\), we have \(L_2=2L_1\).

Answer:

For the area of the sector: It becomes four times larger.
For the arc - length: It doubles.