Math / Geometry

QuestionFor Exercises $95-98$ , find the exact area of the sector. Then round the result to the nearest tenth of a unit. (See Example 10) 95. 96. 97. 98.

Studdy Solution

STEP 1

1. The formula for the area of a sector is $A = \frac{1}{2} r^2 \theta$ , where $\theta$ is in radians.
2. If the angle is given in degrees, convert it to radians using $\theta = \frac{\pi}{180} \times \text{angle in degrees}$ .
3. We need to find both the exact area and the rounded area to the nearest tenth.

STEP 2

1. Convert angles from degrees to radians (if necessary).
2. Calculate the exact area of the sector.
3. Round the area to the nearest tenth.
Exercise 95:

STEP 3

The angle is given in radians, $\theta = \frac{5\pi}{3}$ . No conversion needed.

STEP 4

Use the formula for the area of a sector:
$A = \frac{1}{2} r^2 \theta$
Substitute $r = 6$ and $\theta = \frac{5\pi}{3}$ :
$A = \frac{1}{2} \times 6^2 \times \frac{5\pi}{3}$
$A = \frac{1}{2} \times 36 \times \frac{5\pi}{3}$
$A = 18 \times \frac{5\pi}{3}$
$A = 30\pi$

STEP 5

Round $30\pi$ to the nearest tenth. Using $\pi \approx 3.14159$ :
$30\pi \approx 94.2478$
Rounded to the nearest tenth:
$94.2$
Exercise 96:
STEP_1: The angle is given in radians, $\theta = \frac{\pi}{6}$ . No conversion needed.
STEP_2: Use the formula for the area of a sector:
$A = \frac{1}{2} r^2 \theta$
Substitute $r = 1.2$ and $\theta = \frac{\pi}{6}$ :
$A = \frac{1}{2} \times (1.2)^2 \times \frac{\pi}{6}$
$A = \frac{1}{2} \times 1.44 \times \frac{\pi}{6}$
$A = 0.72 \times \frac{\pi}{6}$
$A = 0.12\pi$
STEP_3: Round $0.12\pi$ to the nearest tenth. Using $\pi \approx 3.14159$ :
$0.12\pi \approx 0.37699$
Rounded to the nearest tenth:
$0.4$
Exercise 97:
STEP_1: Convert the angle from degrees to radians: $\theta = 120^\circ$ .
$\theta = \frac{\pi}{180} \times 120$
$\theta = \frac{2\pi}{3}$
STEP_2: Use the formula for the area of a sector:
$A = \frac{1}{2} r^2 \theta$
Substitute $r = 3$ and $\theta = \frac{2\pi}{3}$ :
$A = \frac{1}{2} \times 3^2 \times \frac{2\pi}{3}$
$A = \frac{1}{2} \times 9 \times \frac{2\pi}{3}$
$A = 9 \times \frac{\pi}{3}$
$A = 3\pi$
STEP_3: Round $3\pi$ to the nearest tenth. Using $\pi \approx 3.14159$ :
$3\pi \approx 9.42477$
Rounded to the nearest tenth:
$9.4$
Exercise 98:
STEP_1: Convert the angle from degrees to radians: $\theta = 225^\circ$ .
$\theta = \frac{\pi}{180} \times 225$
$\theta = \frac{5\pi}{4}$
STEP_2: Use the formula for the area of a sector:
$A = \frac{1}{2} r^2 \theta$
Substitute $r = 4$ and $\theta = \frac{5\pi}{4}$ :
$A = \frac{1}{2} \times 4^2 \times \frac{5\pi}{4}$
$A = \frac{1}{2} \times 16 \times \frac{5\pi}{4}$
$A = 8 \times \frac{5\pi}{4}$
$A = 10\pi$
STEP_3: Round $10\pi$ to the nearest tenth. Using $\pi \approx 3.14159$ :
$10\pi \approx 31.4159$
Rounded to the nearest tenth:
$31.4$
The exact and rounded areas for each exercise are:
- 95: Exact: $30\pi$ , Rounded: $94.2$ - 96: Exact: $0.12\pi$ , Rounded: $0.4$ - 97: Exact: $3\pi$ , Rounded: $9.4$ - 98: Exact: $10\pi$ , Rounded: $31.4$

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