Question

company logo a company has two options for logos as shown. the radius of each logo is 1 foot. jason knows the radius of each circle is 1 foot, so the area of each circle is π square feet. he claims that for any sector in the logos, the angle sector in radians, α, is the same as the numerical value for the arc length in feet and the area of the sector in square feet. which two statements best support or disprove the claim? logo a logo b the area of each sector is θ/360 π where θ is the measure of the angle in degrees. by substitution, θ/2π π = θ/2. this means the numerical value for the area of the sector is one - half the numerical value for the measure of the angle. a the area of each sector is θ/360 π where θ is the measure of the angle in degrees. by substitution, θ/π π = θ. this means the numerical value for the area of the sector is equal to the numerical value for the measure of the angle. b the arc length is rθ where θ is the measure of the angle in degrees. the angle measure α and the radius is 1 foot. by substitution, rθ = 1(1/a) = 1/a. this means the numerical value for the arc length is the quotient of 1 and the angle measure. c the arc length is rθ where θ is the measure of the angle in degrees. the angle measure α and the radius is 1 foot. by substitution, rθ = 1(α) = α. this means the numerical value for the arc length is the same as the numerical value of the angle measure. d the area of each sector is θ/360 π where θ is the measure of the angle in degrees. by substitution, θ/360 π = α. this means the numerical value for the area of the sector is equal to the numerical value for the measure of the angle. by substitution, 2α/360 π = α. this means the numerical value for the area of the sector is equal to the numerical value for the measure of the angle. e

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle is \(A=\frac{1}{2}r^{2}\theta\) (where \(r\) is the radius and \(\theta\) is the angle in radians). Given \(r = 1\) foot, so \(A=\frac{1}{2}(1)^{2}\theta=\frac{\theta}{2}\). Also, if \(\theta\) is in degrees, we know that \(\theta\) (in radians) \(=\frac{\pi}{180}\times\theta\) (in degrees). So \(A=\frac{\theta}{360}\pi r^{2}\), with \(r = 1\), \(A=\frac{\theta}{360}\pi\).

Step2: Recall the formula for the arc - length

The formula for the arc - length of a circle is \(s=r\theta\) (where \(r\) is the radius and \(\theta\) is the angle in radians). Given \(r = 1\) foot, so \(s=\theta\). If \(\theta\) is in degrees, \(\theta\) (in radians) \(=\frac{\pi}{180}\times\theta\) (in degrees), and \(s = r\theta\), with \(r = 1\), \(s=\frac{\theta}{180}\pi\). But if we consider the relationship when \(\theta=\alpha\) (in radians) and \(r = 1\), \(s=\alpha\).

Step3: Analyze each option

• Option A:

The area of a sector \(A=\frac{1}{2}r^{2}\theta\). With \(r = 1\), \(A=\frac{\theta}{2}\). If \(\theta\) is in degrees, \(\theta\) (in radians) \(=\frac{\pi}{180}\theta_{d}\) (where \(\theta_{d}\) is the degree measure). So \(A=\frac{\theta_{d}}{360}\pi\). But \(\frac{\theta}{2}\pi=\frac{\theta}{2}\) is wrong.

• Option B:

The area of a sector \(A=\frac{1}{2}r^{2}\theta\). With \(r = 1\), \(A=\frac{\theta}{2}\). If \(\theta=\alpha\) (in radians), and if we use the degree formula \(A=\frac{\theta_{d}}{360}\pi\), when \(\alpha=\frac{\theta_{d}}{180}\pi\) (since \(\theta\) (radians) \(=\frac{\pi}{180}\theta_{d}\)), then \(A=\frac{\alpha}{2\pi}\pi=\frac{\alpha}{2}\). So \(\frac{\alpha}{\pi}\pi=\alpha\) is wrong.

• Option C:

The arc - length formula is \(s = r\theta\). With \(r = 1\), \(s=\theta\). If \(\theta=\alpha\) (in radians) and \(s=\frac{\theta_{d}}{180}\pi\) (where \(\theta_{d}\) is the degree measure), and \(\alpha=\frac{\theta_{d}}{180}\pi\), then \(\frac{1}{\alpha}= \frac{180}{\theta_{d}\pi}\) is not relevant to the claim.

• Option D:

The arc - length formula \(s=r\theta\). With \(r = 1\), \(s = \theta\). If \(\theta=\alpha\) (in radians) and \(s=\frac{\theta_{d}}{180}\pi\) (where \(\theta_{d}\) is the degree measure), when \(\alpha=\frac{\theta_{d}}{180}\pi\), then \(r\theta=1\times\alpha=\alpha\). And \(\frac{\theta_{d}}{180}\pi=\alpha\). So the arc - length (numerical value) is the same as the numerical value of the angle measure (since \(r = 1\) and \(s=\theta\) (when \(\theta\) is in radians))

• Option E:

The area of a sector \(A=\frac{1}{2}r^{2}\theta\). With \(r = 1\), \(A=\frac{\theta}{2}\). If \(\theta\) is in degrees, \(A=\frac{\theta_{d}}{360}\pi\). If \(2\alpha=\theta_{d}\) (where \(\alpha\) is in radians and \(\theta_{d}\) is in degrees) and \(\alpha=\frac{\theta_{d}}{180}\pi\), then \(\frac{\theta_{d}}{360}\pi=\frac{\alpha}{2}\). So the area of the sector (numerical value) is equal to \(\frac{\alpha}{2}\) and if \(2\alpha=\theta_{d}\), \(\frac{\theta_{d}}{360}\pi=\frac{\alpha}{2}\)

Answer:

D. The arc length is \(r\theta\) where \(\theta\) is the measure of the angle in degrees. The angle measure \(\alpha\) and the radius is \(1\) foot. By substitution, \(r\theta=1(\alpha)\). This means the numerical value for the arc length is the same as the numerical value of the angle measure.

E. The area of each sector is \(\frac{\theta}{360}\pi\) where \(\theta\) is the measure of the angle in degrees. The angle measure \(\alpha\) (in radians). By substitution, \(\frac{\theta}{360}\pi=\frac{\alpha}{2}\). This means the numerical value for the area of the sector is equal to the numerical value for \(\frac{\alpha}{2}\). By substitution, \(\frac{\theta}{360}\pi=\frac{\alpha}{2}\) (since \(\alpha=\frac{\theta}{180}\pi\))