To find the distance from the house at A to the house at​ B, a surveyor measures the angle​ ACB, which is found to be 30 degrees​, and then walks off the distance to each​ house, 90 feet and 100 ​feet, respectively. How far apart are the​ houses?
.
.
.
Question content area top right
Part 1
100 ft
90 ft
30°
Hill
House A
House B
Question content area bottom
Part 1
The houses are 
  
enter your response here feet apart. 
​(Round to the nearest hundredth as​ needed.)


To find the distance between the houses at points A and B, we can use the Law of Cosines. 

The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\), and an angle \(C\) opposite side \(c\):
\[
c^2 = a^2 + b^2 - 2ab \cos(C)
\]

Here, \(a = 90\) feet, \(b = 100\) feet, and \(C = 30^\circ\). We need to find the distance \(c\) between the houses A and B.

1. Substitute the given values into the Law of Cosines formula:
   \[
   c^2 = 90^2 + 100^2 - 2 \cdot 90 \cdot 100 \cdot \cos(30^\circ)
   \]

2. Calculate \( \cos(30^\circ) \):
   \[
   \cos(30^\circ) = \frac{\sqrt{3}}{2}
   \]

3. Substitute \( \cos(30^\circ) \) into the equation:
   \[
   c^2 = 90^2 + 100^2 - 2 \cdot 90 \cdot 100 \cdot \frac{\sqrt{3}}{2}
   \]
   \[
   c^2 = 8100 + 10000 - 9000 \cdot \frac{\sqrt{3}}{2}
   \]
   \[
   c^2 = 8100 + 10000 - 4500\sqrt{3}
   \]

4. Calculate the values:
   \[
   c^2 = 8100 + 10000 - 4500 \cdot 1.732
   \]
   \[
   c^2 = 18100 - 7794
   \]
   \[
   c^2 = 10306
   \]

5. Take the square root of both sides to find \(c\):
   \[
   c = \sqrt{10306} \approx 101.52
   \]

So, the houses are approximately \( 101.52 \) feet apart.

Would you like more details or have any questions?

Here are 5 numbered questions for you to consider:
1. How do you apply the Law of Cosines in different scenarios?
2. Can you explain the derivation of the Law of Cosines?
3. What is the difference between the Law of Sines and the Law of Cosines?
4. How do you solve for angles using the Law of Cosines?
5. What are the applications of the Law of Cosines in real life?

Tip: When using trigonometric laws, ensure your angle measurements are consistent (degrees or radians) throughout your calculations.

Learn how to calculate the distance between two houses using the Law of Cosines. Given angles and distances, find the direct distance between House A and House B. Suitable for high school students learning trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation