Please can someone show me the steps to do this problem?? Also, please don't tell me to just google the solution because I've already tried and have tried to do this problem by myself for hours now...Thank you so much.
You are measuring trees in a forest. Standing on the ground exactly half -waybetweentwo trees, you measure the angle the top of each tree makes with the horizon: one angle is 67◦, and the other is 82◦. If one tree is 80 feet taller than the other, how far apart (horizontal distance along the ground) are the two trees?
First we draw a picture that represents
the problem:
1. Draw a hor. line which represents the distance between the 2 trees.
2. Draw a vertical line from the Lt end
of the hor line upward and label it X.
This is the shorter tree.
3. Draw a hyp. from the top of ver line
to center of hor line. Label each half
of hor line "a."
4. Draw a ver. line at the rt end of hor line and label it X+80. This is the taller tree.
5. Draw a hyp from the top of taller tree to center of hor line.
6. The angle between hyp and hor of smaller tree = 67 deg. It is 82 deg for larger tree.
tan67 = X/a,
a = X/tan67.
tan82 = (X+80)/a,
a = (X+80)/tan82.
X/tan67 = (x+80)/tan82,
Cross multiply:
(X+80)tan67 = Xtan82,
2.36X + 188.47 = 7.12X,
7.12x - 2.36x = 188.47,
4.76x = 188.47,
X = 39.6 Ft.
X + 80 = 119.6 Ft.
a = X/tan67 = 39.6/tan67 = 16,80 Ft.
d = 2a = 2 * 16.8 = 33.6 Ft. = Distance
between trees.
google the probelm
To solve this problem, we can use some trigonometry principles and set up a system of equations. Let's go through the steps together:
Step 1: Draw a diagram
Start by drawing a simple diagram to help visualize the problem. Label the two trees as Tree 1 and Tree 2, and draw a horizontal line representing the ground.
Step 2: Define the variables
Let's denote the horizontal distance between the trees as "x". Since we are given that one tree is 80 feet taller than the other, we can define the height of Tree 1 as "h", and the height of Tree 2 as "h + 80".
Step 3: Identify the relevant trigonometric relationships
From the information given, we can identify two right-angled triangles. The angle between the line connecting the observer to each tree (the line pointing upwards) and the horizontal ground is given as 67◦ and 82◦, respectively.
Step 4: Apply the tangent function
Since we know the angle and the height of each tree, we can use the tangent function to find the distance between the trees. The tangent of an angle is equal to the opposite side divided by the adjacent side in a right triangle.
For Tree 1: tan(67◦) = h / x
For Tree 2: tan(82◦) = (h + 80) / x
Step 5: Solve the system of equations
Now, we have two equations with two variables (h and x). We can solve this system of equations to find the values.
Using the first equation, we can isolate h:
h = x * tan(67◦)
Substitute this value of h into the second equation:
tan(82◦) = (x * tan(67◦) + 80) / x
Simplify the equation:
tan(82◦) = tan(67◦) + 80/x
Rearrange the equation to isolate x:
80/x = tan(82◦) - tan(67◦)
x = 80 / (tan(82◦) - tan(67◦))
Step 6: Calculate the distance between the trees
Using a scientific calculator, calculate the value of x using the above equation. The result will give you the horizontal distance between the two trees.
That's it! By following these steps, you should be able to find the horizontal distance between the two trees using the given angles and the height difference.