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?
make a diagram, showing the two trees at A and B, labeled h for the height of the shorter, and h+80 for the height of the taller.
let P be the midpoint of AB , where AP = BP
Then the angle at P for the taller tree is 82° , and the angle at P for the smaller tree is 67°
then h/AP = tan67° and (h+80)/BP = tan82°
AP = h/tan67 and BP = (h+80)/tan82
but AP and BP are equal, so
h/tan67 = (h+80)/tan82
htan82 = htan67 + 80tan67
htan82 - htan67 = 80tan67
h(tan82-tan67) = 80tan67
h = 80tan67/(tan82-tan67) = .....
You do the button - pushing
once you have h, sub it back into the little equation for AP
(I got AP= 16.8)
What does tan67 equal?
To solve this problem, we can use trigonometry and create a right triangle. Here are the steps:
Step 1: Draw a diagram: Visualize two trees in a forest with the observer standing on the ground, equidistant from the trees.
Step 2: Label the diagram: Label the angle between the observer's line of sight and the horizontal line as angle A (67 degrees) and the angle between the observer's line of sight and the horizontal line for the other tree as angle B (82 degrees). Also, label the vertical distance between the observer's line of sight and the top of the shorter tree as X.
Step 3: Set up the trigonometric equations: We know that the tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height difference between the two trees (80 feet), and the adjacent side is the horizontal distance between the observer and the trees (which we want to find). For angle A, we can write the equation:
tan(67) = 80 / (horizontal distance)
Similarly, for angle B, the equation becomes:
tan(82) = X / (horizontal distance)
Step 4: Solve the equations: Rearrange the equations to solve for the horizontal distance:
horizontal distance = 80 / tan(67)
horizontal distance = X / tan(82)
Step 5: Substitute the value of X: Since X is the vertical distance, we can assume it's the height difference between the two trees (80 feet). Substitute this value into the second equation:
horizontal distance = 80 / tan(67)
horizontal distance = 80 / tan(82)
Step 6: Calculate the values: Use a calculator to determine the values of tan(67) and tan(82) and perform the calculations:
horizontal distance = 80 / 2.4656
horizontal distance = 32.4 feet
Therefore, the two trees are approximately 32.4 feet apart horizontally along the ground.