Each of two observers 475 feet apart measures the angle of elevation to the top of a tree that sits on the straight line between them. These angles are 55° and 59°, for observers A and B, respectively. How tall is the tree? How far is the base of its trunk from each observer?
To solve this problem, we can use the properties of triangles and trigonometry.
Let's denote the height of the tree as h and the distance from observer A to the base of the tree as x. Similarly, let's denote the distance from observer B to the base of the tree as y.
We can start by visualizing the triangle formed by the tree, observer A, and the top of the tree. This triangle will have a right angle at the top of the tree.
Now, let's look at observer A's perspective. From observer A's point of view, the angle of elevation to the top of the tree is 55°. This means that the opposite side (height of the tree, h) is the side opposite to the 55° angle, and the adjacent side (distance from observer A to the base of the tree, x) is the side adjacent to the 55° angle.
Using trigonometry, we can set up the following equation:
tan(55°) = h / x
Similarly, let's set up a similar equation for observer B's perspective:
tan(59°) = h / y
Now, we have two equations with two unknowns (h and x). To solve for both h and x, we need to eliminate one of the unknowns. Let's solve for h in terms of x from the first equation:
h = x * tan(55°)
Now, substitute this expression for h in the second equation:
tan(59°) = (x * tan(55°)) / y
To solve for y, rearrange the equation:
y = (x * tan(55°)) / tan(59°)
We have the relationship between y and x, which means we can find the value of y once we know the value of x.
To find the value of x, we can use the fact that the distance between observer A and observer B is given as 475 feet. So, the sum of the distances x and y should be equal to 475:
x + y = 475
Substitute the expression for y in terms of x in the above equation:
x + (x * tan(55°)) / tan(59°) = 475
Now, we can solve this equation to find the value of x. Once we have the value of x, we can find the value of y, and then substitute the values of x and h into any of the previous equations to find the height of the tree, h.
Note: It's important to use a scientific calculator or trigonometric table to calculate the values of tan(55°) and tan(59°) accurately.
To solve this problem, we can use the concept of trigonometry and set up a system of equations. Let's denote the height of the tree as "h" and the distance from observer A to the base of the tree as "x". The distance from observer B to the base of the tree will then be "475 - x" (since the observers are 475 feet apart).
Using the tangent function, we can write the following equations:
For observer A:
tan(55°) = h / x
For observer B:
tan(59°) = h / (475 - x)
Now let's solve this system of equations to find the height of the tree and the distance from each observer to the base.
Step 1: Solve the equation for observer A:
tan(55°) = h / x
h = x * tan(55°)
Step 2: Solve the equation for observer B:
tan(59°) = h / (475 - x)
h = (475 - x) * tan(59°)
Step 3: Set the two expressions for h equal to each other:
x * tan(55°) = (475 - x) * tan(59°)
Step 4: Solve for x:
x * tan(55°) = 475 * tan(59°) - x * tan(59°)
x * tan(55°) + x * tan(59°) = 475 * tan(59°)
x * (tan(55°) + tan(59°)) = 475 * tan(59°)
x = (475 * tan(59°)) / (tan(55°) + tan(59°))
Step 5: Calculate the height of the tree using either observer A or B:
h = x * tan(55°)
Now we can plug in the values and calculate the height of the tree and the distance to the base for each observer.
Using a calculator, we find:
x ≈ 267.3 feet
h ≈ 267.3 feet * tan(55°) ≈ 398.3 feet
Therefore, the height of the tree is approximately 398.3 feet, and the distance from each observer to the base of the tree is approximately 267.3 feet.