A cat is sitting on top of a tree, and spots a dog 10m south of the tree with an angle of depression of 65 degrees. Then it spots a firefighter a certain distance east of the tree with an angle of depression at 50 degrees. How far is the dog from the firefighter?
i know the answer but have a hard time forming a mental image of what it looks like. drawing it does not help either.
we know the height h of the tree, since
tan65° = h/10
We know the distance d from the firefighter to the base of the tree, since
h/d = tan50°
Knowing d, the distance z from the dog to the firefighter can be found, since
z^2 = 10^2 + d^2
because the two lines of sight are perpendicular at the tree.
To visualize this scenario, it can be helpful to imagine a right-angled triangle. Let's denote the tree as T, the cat as C, the dog as D, and the firefighter as F. Here's a step-by-step breakdown to help you create a mental image:
1. Draw a horizontal line, representing the ground.
2. Draw a vertical line from point T upward, representing the tree trunk.
3. Place a dot at the top of the tree trunk as point T.
4. Draw a downward slanting line from T to the right, representing the cat's line of sight to the dog.
5. Mark a point at the intersection of the line (from step 4) and the ground as point D.
6. Draw a horizontal line from D to the right, representing the ground.
7. Draw a vertical line upward from D, at an angle of 65 degrees to the horizontal line. This represents the dog's line of sight to the tree.
8. Mark a point at the intersection of the line (from step 7) and the tree trunk as point C.
9. Draw a line from T to the right, representing the ground.
10. Place a dot on this line, to the right of T, as point F.
11. Draw a line from F to the right, at an angle of 50 degrees to the horizontal line. This represents the firefighter's line of sight to the tree.
Now, the distance between the dog and the firefighter (DF) can be found by using trigonometric ratios. Since we know the angle of depression and the distance of the tree from each observer, we can use the tangent function. Recall that for a right-angled triangle:
tan(angle) = opposite/adjacent
In this case, for the dog's line of sight to the tree:
tan(65 degrees) = TC/DC (opposite/adjacent)
tan(65 degrees) = TC/10m (distance given in the question)
Now, solve for TC:
TC = tan(65 degrees) * 10m
Similarly, for the firefighter's line of sight to the tree:
tan(50 degrees) = TF/DF
Solve for DF:
DF = TF / tan(50 degrees)
Note: To perform these calculations, convert the angles to radians if necessary when using a calculator or mathematical software.
Once you have calculated TC and DF separately, subtract the two to find the distance between the dog and the firefighter:
Distance between dog and firefighter (DFD) = DF - TC
That's it! You have successfully visualized the scenario and determined the distance between the dog and the firefighter using trigonometric ratios.