A surveyor wants to find the distance from a rock on east side of river to a tree across on the opposite bank. On east side he locates a rock 135 ft from first rock. From each rock he measures the angle between the line connecting the two rocks and the tree. The angle from first rock is 87 degrees and from the second rock is 82 degrees. Find the distance from the first rock to the tree.

Question

A surveyor wants to find the distance from a rock on east side of river to a tree across on the opposite bank. On east side he locates a rock 135 ft from first rock. From each rock he measures the angle between the line connecting the two rocks and the tree. The angle from first rock is 87 degrees and from the second rock is 82 degrees. Find the distance from the first rock to the tree.

Answer 1

draw a diagram. Label the two rocks R and S, and the tree T. And label the sides r,s,t in the usual way. We want to find RT (side s). Let h be the distance straight across the river.

h cotR + h cotS = 135
h/s = sinR
Now plug in your numbers to find s.

Answer 2

To find the distance from the first rock to the tree, we can use the Law of Sines. The Law of Sines states that for any triangle, the side lengths are proportional to the sines of their opposite angles.

Let's denote the distance from the first rock to the tree as x (in feet).

According to the given information:
The distance from the first rock to the second rock is 135 ft.
The angle from the first rock to the tree is 87 degrees.
The angle from the second rock to the tree is 82 degrees.

Using the Law of Sines, we have:

sin(87°) / 135 = sin(82°) / x

To find x, we can rearrange the equation:

x = (135 * sin(82°)) / sin(87°)

Now let's calculate the value of x.

Answer 3

To find the distance from the first rock to the tree, we can use the Law of Sines. This law states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all sides and angles in the triangle.

Let's label the distance from the first rock to the tree as x. We can set up the equation using the Law of Sines:

sin(87°) / x = sin(82°) / 135ft

To find x, we can rearrange the equation by cross-multiplying:

x * sin(82°) = 135ft * sin(87°)

Now, we can divide both sides by sin(82°) to solve for x:

x = (135ft * sin(87°)) / sin(82°)

Using a calculator, we can find the values of sin(87°) and sin(82°), and then perform the calculation to find x.

Upon calculating the values, we find that x is approximately 144.87 feet. Therefore, the distance from the first rock to the tree is approximately 144.87 feet.