a building 60 feet high. from a distance at point A on the ground, the angle of elevation to the top of the building is 40 degree. from a little nearer at point B, the angle of elevation to the top of the building is 70 degree. What's the distance between point A and B
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49.67
49.7
To find the distance between point A and B, we can use the concept of trigonometry and the relationships between the angles of elevation, height of the building, and distance.
Let's denote the distance between point A and B as 'x'.
From point A, we have an angle of elevation of 40 degrees to the top of the building, and from point B, we have an angle of elevation of 70 degrees.
We can create two right-angled triangles to represent the situations at points A and B. In both triangles, the height of the building is the opposite side, and the distance between the points and the building is the adjacent side.
For the triangle at point A:
The opposite side (height of the building) = 60 feet
The adjacent side (distance between point A and the building) = x feet
The angle of elevation = 40 degrees
Using the trigonometric function for tangent (tan), we have the equation:
tan(40 degrees) = opposite side / adjacent side
tan(40 degrees) = 60 / x
For the triangle at point B:
The opposite side (height of the building) = 60 feet
The adjacent side (distance between point B and the building) = (x - d) feet (where 'd' is the horizontal distance between point A and B)
The angle of elevation = 70 degrees
Using the same equation, we have:
tan(70 degrees) = opposite side / adjacent side
tan(70 degrees) = 60 / (x - d)
To find the value of 'd', we need to subtract the adjacent sides at points A and B:
x - d = x - (((60 feet) * tan(40 degrees)) / tan(70 degrees))
Now, we have an equation in terms of 'x' and 'd'.
Since 'd' is the horizontal distance between point A and B, it will not affect the value of 'x'. Therefore, we can ignore 'd' for now and solve for 'x' using the equation:
tan(40 degrees) = 60 / x
Rearranging the equation to solve for 'x', we have:
x = 60 / tan(40 degrees)
Applying the values to the formula, we get:
x = 60 / tan(40 degrees)
x ≈ 60 / 0.839
x ≈ 71.50 feet
Thus, the approximate distance between point A and B is 71.50 feet.
Make your sketch, label the base of the building C
tan40 = 60/AC
AC = 60/tan40
tan 70 = 60/BC
BC = 70/tan70
AB = AC - BC
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