two office towers are 31 m apart. From the top of the shorter tower, the angle of elevation to the top of the tall tower is 25 degrees . The angle of depression to the base of the taller tower is 31 degrees. Calculate the height of each tower
Tan(90-31) = Y1/31
Tan(59) = Y1/31 = 1.66428
Y1=31 * 1.66428 = 51.6 = Ht. of shorter
tower.
Y1+Y2 = 51.6 + 31*sin25 = 64.7 m. = Ht.
of taller tower.
To calculate the height of each tower, we can use trigonometric ratios.
Let's consider the shorter tower first. We need to find its height. We can create a right triangle with the shorter tower's height as the opposite side, the distance between the towers as the adjacent side, and the angle of elevation as the angle opposite to the height.
Using the tangent function:
tan(angle) = opposite/adjacent
In this case, the angle is 25 degrees, the adjacent side is the distance between the towers (31 m), and we want to find the opposite side (height of the shorter tower).
Thus, we have:
tan(25 degrees) = height_shorter/31
Now, we can solve for the height_shorter:
height_shorter = tan(25 degrees) * 31
Next, let's calculate the height of the taller tower. We can create another right triangle using the taller tower's height as the opposite side, the distance between the towers as the adjacent side, and the angle of depression as the angle between the adjacent side and the horizontal line.
Using the tangent function again:
tan(angle) = opposite/adjacent
Now, the angle is 31 degrees, the adjacent side is still the distance between the towers (31 m), and we want to find the opposite side (height of the taller tower).
So, we have:
tan(31 degrees) = height_taller/31
Solving for the height_taller:
height_taller = tan(31 degrees) * 31
Now that we have the formulas, let's calculate the heights:
Height of the shorter tower:
height_shorter = tan(25 degrees) * 31
Height of the taller tower:
height_taller = tan(31 degrees) * 31
Using a calculator, we can plug in the values and solve for the heights.