From a point 50 feet in front of a church, the angles of elevation to the base of the steeple and the top of the steeple are 35° and 47° 20', respectively. Find the height of the steeple.
I missed class this day and have no idea how to go about this problem. I'd really appreciate any help with this. Thanks in advance!
Make a sketch showing the steeple on top of the church.
label the height of the church c and the height of the steeple as s
I see two right-angled triangles:
in the first:
tan 35° = c/50
c = 5tan35 = appr 35.01 m
in the second:
(c+s)/50 - tan 47.33333..
c+s= 54.248
so the church part is 35.01 m, and to the top of the steeple is 54.248
so the steeple is 54.248 - 35.01 = 19.24 m high
To solve this problem, we can use trigonometry and the concept of similar triangles.
Let's use the following notation:
- Let A be the observer's position in front of the church.
- Let B be the base of the steeple.
- Let C be the top of the steeple.
- Let x be the distance from A to B (the distance from the observer to the base of the steeple).
- Let h be the height of the steeple.
We need to find the value of h.
First, let's find the distance from A to C (the distance from the observer to the top of the steeple). We can do this by using the concept of similar triangles.
In triangle ABC, we have the following proportions:
(Tangent Ratio)
tan(35°) = h / x [1]
tan(47° 20') = h / (x + 50) [2]
Now, let's solve equations [1] and [2] simultaneously to find the values of x and h.
From equation [1], we have:
tan(35°) = h / x
Rearranging the equation, we get:
x = h / tan(35°)
Substituting this value of x in equation [2], we have:
tan(47° 20') = h / (h / tan(35°) + 50)
Simplifying this equation, we get:
tan(47° 20') = tan(35°) / (1 + (50 / h))
Using a calculator, find the value of tan(47° 20') and tan(35°), and substitute them into the equation.
After obtaining the value of h, you will have the height of the steeple.
To find the height of the steeple, we can use trigonometry. Here's how you can approach this problem step by step:
Step 1: Draw a diagram
Draw a diagram representing the situation described in the problem. Label the relevant points: the church, the point 50 feet in front of the church, the base of the steeple, and the top of the steeple.
Step 2: Identify the given information
From the problem statement, we know that the angle of elevation to the base of the steeple is 35° and the angle of elevation to the top of the steeple is 47° 20'. We also know that the distance between the point 50 feet in front of the church and the base of the steeple is 50 feet.
Step 3: Define variables
Let's define some variables to represent the unknown quantities. Let 'h' represent the height of the steeple and 'd' represent the distance between the base of the steeple and the point 50 feet in front of the church.
Step 4: Apply trigonometry to find the height
Using the information from the diagram and the given angles, we can set up two right triangles. The first triangle includes the point 50 feet in front of the church, the base of the steeple, and the top of the steeple. The second triangle includes the base of the steeple, the top of the steeple, and the height of the steeple.
In the first triangle:
tan(35°) = h/d
In the second triangle:
tan(47° 20') = h/(d + 50)
Step 5: Solve the equations
Now you have two equations with two variables. You can solve this system of equations using algebra or a graphing calculator.
From the first equation, we can get:
h = d * tan(35°)
Substituting this expression for h in the second equation, we get:
tan(47° 20') = (d * tan(35°))/(d + 50)
Now you can solve this equation for the value of d.
Step 6: Calculate the height of the steeple
Once you find the value of d, you can substitute it back into the equation h = d * tan(35°) to get the height of the steeple.
That's it! Follow these steps, and you should be able to find the height of the steeple.