Two dogs are 180 ft apart on opposite sides of a tree. The angles of elevation from the dogs to the top of the tree are 35 degrees and 23 degrees.
What is the height of the tree?
Let X be the distance of the dog with the 35 degree elevation angle from the tree. The other dog is 180-X away (in feet). Let H be the height of the tree. You have these two equations in two unknowns:
H/X = tan 35 = 0.7002075
H/(180-X) = tan 23 = 0.4244748
Solve for X first by eliminating H.
(180-X)/X = tan35/tan23 = 1.64959
180 - X = 1.64959 X
X = 180/2.64959 = 67.935 feet
180-X = 112.065 ft
H = X tan 35 = 47.569 ft
Thanks!
To find the height of the tree, we can use trigonometric ratios and the concept of similar triangles. Let's break down the problem step by step:
Step 1: Draw a diagram
Draw a diagram representing the situation described. Label the two dogs as Dog A and Dog B, and denote the height of the tree as h.
Tree (h)
/\
/ \
/ \
/ \
/ \
/__________\
Dog A Dog B
Step 2: Identify the relevant angles
From the information given, we know that the angle of elevation from Dog A to the top of the tree is 35 degrees, and the angle of elevation from Dog B to the top of the tree is 23 degrees.
Step 3: Define the trigonometric ratios
For any right triangle, we have three trigonometric ratios: sine (sin), cosine (cos), and tangent (tan). In this case, we will be using the tangent ratio, as it relates the opposite side of an angle to the adjacent side.
Step 4: Solve for the height of the tree
Consider Dog A. The tangent of the angle of elevation (θ) is defined as the ratio of the opposite side (h) to the adjacent side (d), where d is the distance between Dog A and the tree. In this case, d is the distance between Dog A and the tree, which is 180 ft.
Using tangent, we can write the equation as:
tan(35°) = h / 180
Rearranging the equation, we have:
h = 180 × tan(35°)
Calculating this expression gives us the height of the tree, which is the distance h.
Step 5: Calculate the height of the tree
Now, we can plug in the values and calculate the height of the tree:
h = 180 × tan(35°)
Using a scientific calculator, we find:
h ≈ 146.33 ft
Therefore, the height of the tree is approximately 146.33 feet.