A billboard rests on top of a base. From a point level with the bottom of the base, the angle of elevation of the bottom of the billboard is 26 degrees and the top of the billboard is 31 degrees. From a point 21 meters further away, the angle of elevation to the top of the billboard is 17 degrees. Find the height of the billboard.
Let's make a diagram.
Call the top of the billboard A and the bottom of the billboard E, and the point below AE we will call C, where C is at ground level.
Label D as the first point of observation, and B as the second point of observation, so that BD = 21
Let's look at triangle ABD.
Angle BDA = 180-31 = 149º
then angle BAD = 14º
and angle B = 17º
So by the sine law,
AD/sin17 = 21/sin14
We can find AD.
Now let's look at triangle ADC which is right-angled.
cos 31 = DC/AD, thus we can find DC
also sin 31 = AC/AD, giving us AC
and finally in triangle DEC
tan 26 = EC/DC giving us EC
so the billboard's height is AC - EC
Do not round off preliminary answers, good luck.
Let a be the original horizontal distance from which the billboard is viewed.
Let h be the height of the billboard.
Let b be the height of the base.
b/a = tan 26 = 0.4877
(b+h)/a = tan 31 = 0.6009
(b+h)/(a + 21) = tan 17 = 0.3057
You have to solve three equations in three unknowns (a, b, h) to get h.
To find the height of the billboard, we can consider the two triangles formed by the top and bottom of the billboard.
Let's label the following:
Let "h" be the height of the billboard.
Let "d" be the distance from the original point to the base of the billboard.
Let "d + 21" be the distance from the second point to the base of the billboard.
First, consider the triangle formed by the bottom of the billboard:
In this triangle,
The angle of elevation is 26 degrees.
The side opposite the angle of elevation is "h".
The side adjacent to the angle of elevation is "d".
Using trigonometry, we can use the formula for tangent:
tangent(angle) = opposite/adjacent
So, for the bottom triangle, we have:
tan(26 degrees) = h/d
Next, consider the triangle formed by the top of the billboard:
In this triangle,
The angle of elevation is 31 degrees.
The side opposite the angle of elevation is "h".
The side adjacent to the angle of elevation is "d".
Again, using the formula for tangent, we have:
tan(31 degrees) = h/d
Now, let's consider the second point 21 meters further away:
In this case,
The angle of elevation is 17 degrees.
The side opposite the angle of elevation is "h".
The side adjacent to the angle of elevation is "d + 21".
Using the formula for tangent once again, we have:
tan(17 degrees) = h/(d + 21)
Now we have a system of three equations:
1) tan(26 degrees) = h/d
2) tan(31 degrees) = h/d
3) tan(17 degrees) = h/(d + 21)
To solve this system, we can use substitution or elimination.
By rearranging equation 1) for h, we have:
h = d * tan(26 degrees)
By rearranging equation 2) for h, we have:
h = d * tan(31 degrees)
Since both equations have h on one side, we can equate them:
d * tan(26 degrees) = d * tan(31 degrees)
Now, divide both sides by "d":
tan(26 degrees) = tan(31 degrees)
We can now solve this equation using a scientific calculator or trigonometric tables to find the value of "d".
Once we find the value of "d", we can substitute it back into one of the equations (either 1) or 2)) to find the value of "h".
After finding the values of "d" and "h", we can calculate the height of the billboard by using the equation:
height = h + base length
Since the base length is not given, we cannot provide a final numerical answer.