Question
86. A construction company owns a triangular lot in at a " Y " intersection. The length of the property line AB is 140 feet. The length of the property line BC is 220 feet. The angle between the property lines AB and BC is \( 125^{\circ} \). What is the area of the lot? Round to the nearest hundredth.
Ask by Bolton Bowman. in the United States
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The area of the lot is approximately 12,614.94 square feet.
Solution
To determine the area of the triangular lot, we'll use the **Law of Sines** combined with the formula for the area of a triangle when two sides and the included angle are known.
**Given:**
- Side \( AB = 140 \) feet
- Side \( BC = 220 \) feet
- Angle \( \angle ABC = 125^\circ \)
**Formula for Area:**
\[
\text{Area} = \frac{1}{2} \times AB \times BC \times \sin(\angle ABC)
\]
**Step-by-Step Calculation:**
1. **Plugging in the values:**
\[
\text{Area} = \frac{1}{2} \times 140 \times 220 \times \sin(125^\circ)
\]
2. **Calculate the product of the sides and the constant:**
\[
\frac{1}{2} \times 140 \times 220 = 70 \times 220 = 15,400
\]
3. **Determine \( \sin(125^\circ) \):**
\[
\sin(125^\circ) \approx 0.819152
\]
4. **Multiply the results:**
\[
\text{Area} = 15,400 \times 0.819152 \approx 12,614.94 \text{ square feet}
\]
**Final Answer:**
The area of the lot is **12,614.94 square feet**.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To find the area of the triangular lot, we can use the formula: \[ \text{Area} = \frac{1}{2} \times AB \times BC \times \sin(\theta) \] In this case: - \( AB = 140 \) feet - \( BC = 220 \) feet - \( \theta = 125^\circ \) First, we calculate the sine of \( 125^\circ \): \[ \sin(125^\circ) \approx 0.8192 \] Now, we plug the values into the area formula: \[ \text{Area} = \frac{1}{2} \times 140 \times 220 \times 0.8192 \] Calculating this gives: \[ \text{Area} \approx \frac{1}{2} \times 140 \times 220 \times 0.8192 \approx 12,211.84 \text{ square feet} \] So, the area of the lot is approximately **12,211.84 square feet** when rounded to the nearest hundredth.
