19. You inherit a triangular plot of land. There is a stake at each corner of the lot. From one stake, the angle between the other two stakes is 30°. The distance from the first stake to the others is 400 feet and 500 feet, respectively. What is the area of the land you have inherited?
1. Understanding the given information: - You have a triangular plot of land. - From one corner (let's call it vertex [tex]\(A\)[/tex]), the angle between the other two corners (vertices [tex]\(B\)[/tex] and [tex]\(C\)[/tex]) is 30°. - The distance from vertex [tex]\(A\)[/tex] to vertex [tex]\(B\)[/tex] is 400 feet. - The distance from vertex [tex]\(A\)[/tex] to vertex [tex]\(C\)[/tex] is 500 feet.
2. Labeling the triangle: Let's denote the vertices of the triangle as follows: - [tex]\(A\)[/tex] is the vertex where the 30° angle is. - [tex]\(B\)[/tex] and [tex]\(C\)[/tex] are the other two vertices. - [tex]\(AB = 400\)[/tex] feet. - [tex]\(AC = 500\)[/tex] feet. - [tex]\(\angle BAC = 30^\circ\)[/tex].
3. Converting the angle to radians: We need to convert the angle [tex]\(\angle BAC\)[/tex] from degrees to radians for further calculations. The conversion formula is: [tex]\[
\text{Angle in radians} = \text{Angle in degrees} \times \frac{\pi}{180}
\][/tex] Thus, [tex]\[
\angle BAC = 30^\circ \times \frac{\pi}{180} = \frac{\pi}{6} \approx 0.5235987755982988 \text{ radians}
\][/tex]
4. Using the formula for the area of a triangle with two sides and the included angle: The formula to find the area ([tex]\( \text{Area} \)[/tex]) of a triangle when you know two sides and the included angle is: [tex]\[
\text{Area} = 0.5 \times AB \times AC \times \sin(\angle BAC)
\][/tex] Here, - [tex]\(AB = 400 \text{ feet}\)[/tex] - [tex]\(AC = 500 \text{ feet}\)[/tex] - [tex]\(\angle BAC = 0.5235987755982988 \text{ radians} \)[/tex]
5. Calculating [tex]\(\sin(\angle BAC)\)[/tex]: Using the known value: [tex]\[
\sin(30^\circ) = 0.5
\][/tex]
