Question

NO LINKS!!! URGENT HELP PLEASE!!!

Consider the following
θ = 1.4, d= 160 cm
a. Find the length of the arc that subtends the given central angle θ on a circle of diameter d. _________ cm

b. Find the area of the sector determined by θ. _________ cm^2

Answer 1

a) 112 cm

b) 4480 cm²

Explanation:

Given:

• θ = 1.4 radians
• d = 160 cm

The diameter of a circle is twice its radius. Therefore, the radius of the circle is:

`\implies r=(d)/(2)=(160)/(2)=80\; \sf cm`

Part (a)

To find the length of the arc that subtends the given central angle θ on a circle of diameter d, use the arc length formula.

`\boxed{\begin{minipage}{6.4 cm}\underline{Arc length}\\\\Arc length $=r \theta$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle measured in radians.\\\end{minipage}}`

Substitute the values into the formula:

`\begin{aligned}\implies \sf Arc\; length&=r \theta\\&=80 \cdot 1.4\\&=112\; \sf cm\end{aligned}`

Therefore, the length of the arc that subtends the given central angle θ on a circle of diameter d is 112 cm.

Part (b)

To calculate the area of the sector determined by θ, we can use the area of a sector formula.

`\boxed{\begin{minipage}{6.4 cm}\underline{Area of a sector}\\\\$A=\frac12 r^2 \theta$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle measured in radians.\\\end{minipage}}`

Substitute the values into the formula:

`\begin{aligned}\implies \sf Area\;of\;a\;sector&=(1)/(2)r^2\theta\\\\&=(1)/(2) \cdot 80^2 \cdot 1.4\\\\&=(1)/(2) \cdot 6400 \cdot 1.4\\\\&=3200 \cdot 1.4\\\\&=4480\; \sf cm^2\end{aligned}`

Therefore, the area of the sector determined by θ is 4480 cm².