Question

1. determine the arc length of $widehat{tw}$. give your answer in terms of $pi$

arc length = ________

1. determine the sector area of the shaded region $qop$. give your answer in terms of $pi$

sector area = ________

Explanation:

Step1: Recall arc length formula

Arc length: $s = \frac{\theta}{360^\circ} \times 2\pi r$

Step2: Substitute values for $\overparen{TW}$

$\theta=135^\circ$, $r=24$ ft
$s = \frac{135^\circ}{360^\circ} \times 2\pi \times 24$

Step3: Simplify the expression

$\frac{135}{360} = \frac{3}{8}$, so $s = \frac{3}{8} \times 48\pi = 18\pi$

Step4: Recall sector area formula

Sector area: $A = \frac{\theta}{360^\circ} \times \pi r^2$

Step5: Find shaded sector angle

Full circle is $360^\circ$, right angle is $90^\circ$, so $\theta=360^\circ-90^\circ=270^\circ$

Step6: Substitute values for sector $QOP$

$r=4$ in, $\theta=270^\circ$
$A = \frac{270^\circ}{360^\circ} \times \pi \times 4^2$

Step7: Simplify the expression

$\frac{270}{360} = \frac{3}{4}$, so $A = \frac{3}{4} \times 16\pi = 12\pi$

Answer:

Arc Length = $18\pi$ ft
Sector Area = $12\pi$ in$^2$