Sure, let's solve this step-by-step.
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]
6. Calculating the area:
Plug these values into the formula:
[tex]\[
\text{Area} = 0.5 \times 400 \times 500 \times 0.5 = 0.5 \times 400 \times 500 \times 0.5
\][/tex]
Simplifying:
[tex]\[
\text{Area} = 0.5 \times 400 \times 250 = 49999.99999999999 \text{ square feet}
\][/tex]
Hence, the area of the triangular plot of land you have inherited is approximately 50,000 square feet.
