From the top of a 1900 m mountain, the angle of depression to a cathedral that is due east of the mountain is 38 degrees. The angle of depression to a bridge due north of the mountain is 42 degrees.
Find the straight-line distance from the cathedral to the bridge.
This is a question that needs visualizing as it has no diagram, but i know I have to draw a North/East/South/West grid.
It is not possible to work without at least one dimensional measurement. The 1900 m on the mountain is not enough unless it somehow is relation to the elevation of the cathedral and the bridge.
I think it can be done the following way:
draw a horizontal with the cathedral on the W side and the bridge on the E side.
draw a vertical to represent the mountain's height of 1900 m between these two.
Then the angle of elevation from the bridge to the mountain top is 42º and the angle of elevation from the cathedral is 38º.
so by trig we can find the two sides of our two right angled triangle.
I got 2110.16 m and 2431.89 m for a total distance of 4542 m
my bum bum
1 - find the dist. from the vertical base of the mountain to the cathedral and bridge via trig ratios (tan). remember that the angle of depression = looking horizontally ahead and taking the angle between your gaze and the target.
2 - use pythagoras to find the straight-line dist.
Note that you could not use this method if the cathedral and the bridge were not east-north of the base of the mountain.
The question could be formulated differently to make it more understandable. Instead of mountain, they could have put tower (or anything else that is completely vertical)
I would think that if you draw a line to represent the middle of the mounain and then draw a line 42 degrees to the right, then draw a nother line 38 degrees to the right. Because the height of the mountain is 1900 m, then you can just divide it by 100 to get 19 cm and then when you find where the line meets, times the amount back by 100 to get what measurement in m, the straight line is. I hope that helps. :D
You're on the right track! Drawing a North/East/South/West grid can be really helpful in visualizing and solving this problem. Here's how you can approach it step by step:
1. Draw a vertical line to represent the mountain on the grid. Label it as "Mountain".
|
M |
O |
U |
N |
T |
A |
I |
N |
- |
2. From the top of the mountain, draw a line at a 38-degree angle downward and to the right. Label the endpoint as "Cathedral".
| C
| \
| \
| \
| \
| \
M | \
O | \
U | \
N | \
T | \
A | \
3. From the top of the mountain, draw a line at a 42-degree angle downward and to the left. Label the endpoint as "Bridge".
| C
| \
| \
| \
| \
| \
M | B \
O | \
U | \
N | \
T | \
A | \
4. Now, we need to find the straight-line distance between the Cathedral (C) and the Bridge (B). To do this, draw a horizontal line connecting the Cathedral and the Bridge.
| C
| \
| \
| \
| \
|------\
M | B \
O | \
U | \
N | \
T | \
A | \
5. We can see that we have formed a right-angled triangle with the vertical line representing the mountain, and the horizontal line connecting the Cathedral and Bridge. We can apply trigonometry to find the straight-line distance between the Cathedral and the Bridge.
Now, let's find the lengths of the two sides of the triangle that are given:
- The angle of depression to the Cathedral from the top of the mountain is 38 degrees. This means that the angle between the vertical line (mountain) and the line connecting the Cathedral and the Bridge is 90 - 38 = 52 degrees. This forms a right-angled triangle, with the angle of 52 degrees at the top.
- The angle of depression to the Bridge from the top of the mountain is 42 degrees. This means that the angle between the vertical line (mountain) and the line connecting the Cathedral and the Bridge is also 90 - 42 = 48 degrees. This forms a right-angled triangle, with the angle of 48 degrees at the top.
Let's call the distance between the Cathedral and the Bridge as "x".
Using trigonometry, we can find the lengths of the two sides adjacent to the angles of 52 degrees and 48 degrees respectively.
The side adjacent to the angle of 52 degrees can be found using the equation:
adjacent_side_1 = mountain_height / tan(52 degrees)
The side adjacent to the angle of 48 degrees can be found using the equation:
adjacent_side_2 = mountain_height / tan(48 degrees)
Now, we can find the straight-line distance between the Cathedral and the Bridge using the Pythagorean theorem:
(x^2) = (adjacent_side_1)^2 + (adjacent_side_2)^2
Once you have evaluated this equation, you can solve for "x" to find the straight-line distance between the Cathedral and the Bridge.