Question

calculate the perimeter of the following object to the nearest tenth.

Explanation:

Step1: Find the length of arc \( CD \)

The formula for the length of an arc is \( L = \frac{\theta}{360^\circ} \times 2\pi r \), where \( \theta \) is the central angle and \( r \) is the radius. Here, \( \theta = 93^\circ \) and \( r \) is the distance from \( B(0,6) \) to \( D(-8,13.5) \) (or \( C(8,13.5) \)). Using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), for \( B(0,6) \) and \( D(-8,13.5) \):
\[
r = \sqrt{(-8 - 0)^2 + (13.5 - 6)^2} = \sqrt{64 + 56.25} = \sqrt{120.25} = 11
\]
Now, calculate the arc length:
\[
L_{CD} = \frac{93^\circ}{360^\circ} \times 2\pi \times 11 = \frac{93}{360} \times 22\pi \approx \frac{93 \times 22\pi}{360} \approx 17.9
\]

Step2: Find the length of \( AD \) and \( AC \)

Using the distance formula for \( A(0,-8) \) and \( D(-8,13.5) \):
\[
AD = \sqrt{(-8 - 0)^2 + (13.5 - (-8))^2} = \sqrt{64 + (21.5)^2} = \sqrt{64 + 462.25} = \sqrt{526.25} = 22.94
\]
Since \( AC = AD \) (symmetry), \( AC \approx 22.94 \)

Step3: Calculate the perimeter

The perimeter is \( AD + AC + L_{CD} \):
\[
\text{Perimeter} = 22.94 + 22.94 + 17.9 \approx 63.8
\]

Answer:

\( 63.8 \)