two towns A and B are 96km apart a is due north of B. town C is 80km from B and is on a bearing of 56 degree from B find distance of town A and C
using the law of cosines,
AC^2 = 96^2 + 80^2 - 2*96*80 cos56°
AC^2=96^2+80^2-2(96*80)cos56
AC=\/15616-8477.63
AC=9.2km
To find the distance between town A and town C, we can use trigonometry and the given information. Here's how to solve it step by step:
Step 1: Draw a diagram
Draw a diagram with town B at the center. Place town A directly above town B and town C on a bearing of 56 degrees from B. The distance from B to C is 80 km.
Step 2: Split the triangle
Split the triangle into two right-angled triangles by drawing a vertical line from town C to town A. Let's call the point where this line intersects the line connecting A and B, point D.
Step 3: Calculate the distance from A to D
Since town A is directly north of B, the line connecting A and B is vertical. Thus, the distance from A to D is the same as the distance from B to D, which is 96 km.
Step 4: Calculate the distance from B to C
Using the given information, we know that the distance from B to C is 80 km.
Step 5: Use trigonometry to solve for AC
Now, we have a right-angled triangle BCD, where we know the lengths of two sides: BD (96 km) and BC (80 km). We want to find the length of the third side, AC.
Using the trigonometric function tangent (tan), we can set up the following equation:
tan(56 degrees) = AC / BC
Step 6: Solve for AC
Rearrange the equation to solve for AC:
AC = tan(56 degrees) * BC
Using a calculator, calculate the value of tan(56 degrees) and then multiply it by BC (80 km) to get the distance of AC.
Step 7: Calculate AC
AC ≈ tan(56 degrees) * 80 km ≈ 111.82 km
Hence, the distance between town A and town C is approximately 111.82 km.