Math / Geometry

QuestionFind the area of $\triangle RPQ$ given that $p = 18 \, \text{mi}$ , $m \angle R = 17^\circ$ , and $q = 28 \, \text{mi}$ .

Studdy Solution

STEP 1

1. We are given a triangle $\triangle RPQ$ .
2. Side $p = 18$ miles, side $q = 28$ miles.
3. Angle $\angle R = 17^\circ$ .
4. We need to find the area of the triangle.
5. We will use the formula for the area of a triangle using two sides and the included angle: $\text{Area} = \frac{1}{2} \times p \times q \times \sin(\angle R)$ .

STEP 2

1. Identify the known values and the formula to use.
2. Calculate the sine of the given angle.
3. Substitute the known values into the area formula and calculate the area.

STEP 3

Identify the known values: - Side $p = 18$ miles. - Side $q = 28$ miles. - Angle $\angle R = 17^\circ$ .
We will use the formula for the area of a triangle using two sides and the included angle: $\text{Area} = \frac{1}{2} \times p \times q \times \sin(\angle R)$

STEP 4

Calculate the sine of the given angle $\angle R = 17^\circ$ .
Use a calculator to find: $\sin(17^\circ)$

STEP 5

Substitute the known values into the area formula: $\text{Area} = \frac{1}{2} \times 18 \times 28 \times \sin(17^\circ)$
Calculate the area using the sine value obtained from the calculator.

STEP 6

Perform the calculation: - Assume $\sin(17^\circ) \approx 0.2924$ (using a calculator). - Substitute into the formula:
$\text{Area} = \frac{1}{2} \times 18 \times 28 \times 0.2924$
$= 9 \times 28 \times 0.2924$
$= 252 \times 0.2924$
$\approx 73.7088$
The area of $\triangle RPQ$ is approximately $73.71 \, \text{square miles}$ .
The number of square miles in the area of the triangle is approximately:
$\boxed{73.71}$

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