From a tower 57 ft high two objects in a straight line from it are sighted at angles of depression of 27degrees and 40degrees5' respectively. Find the distance between the two objects.
40° 5' = (40 + 5/60) = 481/12 °
You will have two right-angled triangle
For the distance of the closer object
tan (481/12)° = 57/d1
d1 = 57/tan(481/12)°
For the distance of the farther away object
tan 27 = 57/d2
d2 = 57/tan27°
distance between objects = d2 - d1 = .....
Ah, the old "downward angle of depression" problem. It takes a lot to make me feel down, but I'll do my best to help you out!
Let's call the distance between the first object and the tower "x" and the distance between the second object and the tower "y". We can now start applying some trigonometry!
Now, we have two right triangles formed by the tower and the objects. In the first triangle, the angle of depression is 27 degrees, and in the second triangle, it's 40 degrees 5 minutes (which we can convert to decimals for convenience, giving us 40.083 degrees).
In the first triangle, we have the opposite side (57 ft) and the adjacent side (x). So, we can use the tangent function to get the value of x: tan(27°) = 57 / x.
Similarly, in the second triangle, we have the opposite side (57 ft) and the adjacent side (y). Using the tangent function again, we get: tan(40.083°) = 57 / y.
Now we have two equations with two variables:
1) x = 57 / tan(27°)
2) y = 57 / tan(40.083°)
We can now plug these values into our calculator and find the individual distances.
Once we have the two distances, we can find the distance between the two objects by simply subtracting one distance from the other.
Keep in mind that my calculations might not be 100% accurate, but hopefully, it'll give you a good starting point. Just remember, no matter how down you feel, just look up and keep moving forward!
To find the distance between the two objects, we can use trigonometry. Let's refer to the tower as point A and the two objects as points B and C.
Step 1: Set up a diagram. Draw a vertical line to represent the tower (point A) with a length of 57 feet. From point A, draw two lines at the given angles of depression: 27° and 40°5'. Label the points where these lines meet the ground as B and C, respectively.
Step 2: Convert the angle of depression in minutes to decimal form. The angle of depression 40°5' can be converted to decimal form as follows:
40° + (5'/60') = 40.0833°
Step 3: Use trigonometry to calculate the distances AB and AC.
For triangle ABD:
BD/AB = tangent(angle of depression 27°)
Let's assume BD = x.
tan(27°) = x / AB
For triangle ACD:
CD/AC = tangent(angle of depression 40.0833°)
Let's assume CD = y.
tan(40.0833°) = y / AC
Step 4: Solve the equations to find AB and AC.
From triangle ABD:
tan(27°) = x / AB
AB = x / tan(27°)
From triangle ACD:
tan(40.0833°) = y / AC
AC = y / tan(40.0833°)
Step 5: Find the distance between the two objects.
To find the distance between points B and C, we need to calculate the difference between AB and AC:
Distance between B and C = AB - AC
Now, using the given information, plug in the values and calculate the distances AB and AC. Finally, subtract AC from AB to find the distance between the two objects.
To find the distance between the two objects, we can use trigonometry and apply the concept of angles of depression.
First, let's draw a diagram to visualize the situation. Let T represent the top of the tower, and A and B represent the two objects.
```
A
/
/
/ θ1
/ /
T _______/________
θ2
B
```
Now, let's label the given information.
- The height of the tower, T to the ground, is 57 ft.
- The angle of depression from the tower to object A is 27 degrees.
- The angle of depression from the tower to object B is 40 degrees 5 minutes.
To find the distance between the two objects, we can consider the right triangles formed by the tower, the objects, and the horizontal ground.
For object A, we have a right triangle with the tower as the vertical side and the distance to object A as the horizontal side. The angle between the vertical side and the horizontal side is the angle of depression (θ1 = 27 degrees).
Similarly, for object B, we have a right triangle with the same vertical side (the tower) and the distance to object B as the horizontal side. The angle between the vertical side and the horizontal side is the angle of depression (θ2 = 40 degrees 5 minutes).
Let's use the trigonometric tan function to find the distances to objects A and B.
For object A:
tan(θ1) = height of tower / distance to A
tan(27 degrees) = 57 ft / distance to A
For object B:
tan(θ2) = height of tower / distance to B
tan(40 degrees 5 minutes) = 57 ft / distance to B
To find the distance to object A:
distance to A = 57 ft / tan(27 degrees)
To find the distance to object B:
distance to B = 57 ft / tan(40 degrees 5 minutes)
Now, we can calculate these distances using a calculator or trigonometric tables. Let's substitute the values and compute the distances.
distance to A = 57 ft / tan(27 degrees) ≈ 103.23 ft
distance to B = 57 ft / tan(40 degrees 5 minutes) ≈ 71.82 ft
Therefore, the distance between the two objects (A and B) is approximately 103.23 ft - 71.82 ft = 31.41 ft.
Hence, the distance between the two objects is approximately 31.41 ft.