Question

arc length and area of a sector hw name: daniel tello mejlerfind the area of the shaded sector in terms of $pi$.$a=\frac{45}{360}pi(4^2)$find the length of the gray arc in terms of $pi$.find the area of the shaded sector in terms of $pi$.find the length of the gray arc in terms of $pi$.find the area of the shaded sector in terms of $pi$.find the length of the gray arc in terms of $pi$.find the area of the shaded sector in terms of $pi$.find the length of the gray arc in terms of $pi$.challenge: three friends are splitting a 16-inch pizza evenly, how much pizza does each friend receive?

Explanation:

---

Problem 1: Shaded Sector Area (45°, r=4)

Step1: Apply sector area formula

$A = \frac{45}{360} \pi (4^2)$

Step2: Simplify the expression

$A = \frac{1}{8} \pi (16) = 2\pi$

---

Problem 2: Gray Arc Length (135°, r=12)

Step1: Apply arc length formula

$L = \frac{135}{360} \cdot 2\pi(12)$

Step2: Simplify the expression

$L = \frac{3}{8} \cdot 24\pi = 9\pi$

---

Problem 3: Shaded Sector Area (216°, r=10)

Step1: Apply sector area formula

$A = \frac{216}{360} \pi (10^2)$

Step2: Simplify the expression

$A = \frac{3}{5} \pi (100) = 60\pi$

---

Problem 4: Gray Arc Length (90°, r=8)

Step1: Identify arc central angle

Right angle = $90^\circ$

Step2: Apply arc length formula

$L = \frac{90}{360} \cdot 2\pi(8)$

Step3: Simplify the expression

$L = \frac{1}{4} \cdot 16\pi = 4\pi$

---

Problem 5: Shaded Sector Area (90°, r=16)

Step1: Identify sector central angle

Right angle = $90^\circ$

Step2: Apply sector area formula

$A = \frac{90}{360} \pi (16^2)$

Step3: Simplify the expression

$A = \frac{1}{4} \pi (256) = 64\pi$

---

Problem 6: Gray Arc Length (72°, r=15)

Step1: Apply arc length formula

$L = \frac{72}{360} \cdot 2\pi(15)$

Step2: Simplify the expression

$L = \frac{1}{5} \cdot 30\pi = 6\pi$

---

Problem 7: Shaded Sector Area (135°, r=8)

Step1: Apply sector area formula

$A = \frac{135}{360} \pi (8^2)$

Step2: Simplify the expression

$A = \frac{3}{8} \pi (64) = 24\pi$

---

Problem 8: Gray Arc Length (252°, r=5)

Step1: Apply arc length formula

$L = \frac{252}{360} \cdot 2\pi(5)$

Step2: Simplify the expression

$L = \frac{7}{10} \cdot 10\pi = 7\pi$

---

Challenge: Pizza Portion (16-inch diameter)

Step1: Calculate pizza radius

$r = \frac{16}{2} = 8$ inches

Step2: Calculate total pizza area

$A_{total} = \pi(8^2) = 64\pi$

Step3: Divide by 3 friends

$A_{each} = \frac{64\pi}{3}$

---

Answer:

1. $2\pi$
2. $9\pi$
3. $60\pi$
4. $4\pi$
5. $64\pi$
6. $6\pi$
7. $24\pi$
8. $7\pi$

Challenge: $\frac{64\pi}{3}$ square inches