a smokestack is 205 m high. from 2 points C and D on the same side of the smoke stacks base b, the angles of elevation to the top of the smokestack are 40 degrees and 36 degrees respectively. find the distance between C and D
I hope you made a diagram
I let the top of the smoke-stack be P and its bottom be Q
then in triangle PCD, angle C = 36º
angle CDP = 140º, thus angle CPD = 4º
triangle PDQ is right-angles, so
sin40 = 205/DP
DP = 40/sin40 = ....
then in the other triangle by the Sine Law
CD/sin4 = DP/sin36
CD = DPsin4/sin36
= ....
let me know what you got
To find the distance between points C and D, we can use the trigonometric relationship between angles of elevation and distances.
Let's label the distance between C and the smokestack as x, and the distance between D and the smokestack as y.
First, let's draw a diagram to visualize the problem:
```
C
\
\ 40°
\ \
\ \
\ \
\ \
\ 205m \
\ \
\ \
\_______B
/ /
/ /
/ /
/ /
/ /
/ /
D / 36°
```
Now, we have two right triangles, with angles of 40 degrees and 36 degrees, and a common side with length 205m (the height of the smokestack).
Using trigonometry, we can write the following equations for each triangle:
For triangle ABC:
tan(40°) = 205 / x
For triangle ABD:
tan(36°) = 205 / y
To find the distance between C and D, we need to solve for x and y and subtract them:
x - y = distance between C and D
Now, let's solve for x and y using the given information:
From the first equation, we can isolate x:
x = 205 / tan(40°)
From the second equation, we can isolate y:
y = 205 / tan(36°)
Now, substitute these values into the equation for the distance between C and D:
x - y = 205 / tan(40°) - 205 / tan(36°)
Evaluate this expression using a calculator, and you will find the distance between C and D.