To estimate the height of a building, two students find the angle of elevation from a point (at ground level) down the street from the building to the top of the building is 33 degrees. From a point that is 300 feet closer to the building, the angle of elevation (at ground level) to the top of the building is 49 degrees. If we assume that the street is level, use this information to estimate the height of the building.
The height of the building is_____
To estimate the height of the building, we can use the concept of trigonometry. Let's denote the distance from the first point to the building as "x" and the height of the building as "h".
From the first point, the angle of elevation is 33 degrees. This means that the tangent of the angle (tan(33)) is equal to the height of the building divided by the distance "x". So we have the equation:
tan(33) = h / x
Similarly, from the second point, the angle of elevation is 49 degrees, and the distance is "x - 300" (as it is 300 feet closer to the building). So we have:
tan(49) = h / (x - 300)
Now we can solve these two equations simultaneously to find the values of "h" and "x". Let's start by isolating "h" in the first equation:
h = tan(33) * x
Now we substitute this value of "h" into the second equation:
tan(49) = (tan(33) * x) / (x - 300)
Now we can solve for "x" using algebraic manipulation. Multiply both sides of the equation by (x - 300) to get:
(x - 300) * tan(49) = tan(33) * x
Expanding and rearranging the equation, we get:
xtan(49) - 300tan(49) = xtan(33)
Next, isolate "x" terms on one side and constants on the other side:
xtan(49) - xtan(33) = 300tan(49)
Now factor out "x" from both terms:
x(tan(49) - tan(33)) = 300tan(49)
Finally, divide both sides by (tan(49) - tan(33)) to solve for "x":
x = 300tan(49) / (tan(49) - tan(33))
Now we can substitute the value of "x" back into the first equation to find the height "h":
h = tan(33) * x
By plugging in the values and evaluating the expressions, we can estimate the height of the building.
To estimate the height of the building, we can use trigonometry and set up a proportion based on the angles of elevation and the given distances.
Let's denote the height of the building as 'h' and the initial distance from the building as 'x'. From the first observation point, we have an angle of elevation of 33 degrees.
Using the tangent function, we can write the following equation:
tan(33) = h / x
Now, let's consider the second observation point, which is 300 feet closer to the building. The new distance from the building can be expressed as (x - 300). The angle of elevation from this point is 49 degrees.
Using the same equation, we can write:
tan(49) = h / (x - 300)
Now we have two equations with two unknowns (h and x), which we can solve simultaneously.
Rearranging the first equation, we get:
h = x * tan(33)
Similarly, rearranging the second equation, we get:
h = (x - 300) * tan(49)
Since both expressions equal 'h', we can set them equal to each other:
x * tan(33) = (x - 300) * tan(49)
Simplifying this equation, we can solve for 'x':
x = 300 * tan(49) / (tan(49) - tan(33))
Now that we have the value of 'x', we can substitute it back into one of the previous equations to calculate the height 'h'.
Therefore, the estimated height of the building can be found by evaluating the equation:
h = x * tan(33)
Note: Make sure to use the appropriate units when performing the calculation.
h/tan 49° = h/tan 33° - 300
solve for h