from a window 30 feet above the street, the angle of depression of the curb on the near side of the street is 50 degree that of the curb on the far side is 13 degree. how wide is the street from curb to curb? Oblique triangles by using right triangles
Just in case that wasn't enough of a hint, draw your diagram and notice that
streetwidth = 30 cot15° - 30cot50°
use the cot function.
To find the width of the street from curb to curb, we can use trigonometry and the concept of oblique triangles.
Let's start by drawing a diagram to visualize the situation. Let's say we have a window 30 feet above the street. From this window, we can see two curbs, one on the near side and one on the far side. The angle of depression of the curb on the near side is 50 degrees, and the angle of depression of the curb on the far side is 13 degrees. Let's label the width of the street as 'x'.
Now, let's consider the right triangles formed with the window as the top vertex, the curbs as the two bottom vertices, and the vertical line from the window to the street as the hypotenuse.
For the triangle on the near side curb, we can consider it as a right triangle with the angle of depression as 50 degrees. The opposite side of this right triangle is the height of the window, which is 30 feet. The adjacent side is the distance from the curb to the window, which is half the width of the street, that is x/2. Using the trigonometric ratio tan(50°), we have:
tan(50°) = height of window / distance from near curb to window
tan(50°) = 30 / (x/2)
tan(50°) = 60 / x
Next, let's consider the right triangle on the far side curb. We can consider it as a right triangle with the angle of depression as 13 degrees. The opposite side of this right triangle is also the height of the window, which is 30 feet. The adjacent side is the distance from the curb to the window, which is half the width of the street, that is x/2. Using the trigonometric ratio tan(13°), we have:
tan(13°) = height of window / distance from far curb to window
tan(13°) = 30 / (x/2)
tan(13°) = 60 / x
Now, we have two equations:
tan(50°) = 60 / x
tan(13°) = 60 / x
We can solve these two equations simultaneously to find the value of x, which represents the width of the street.
By rearranging the first equation, we have:
x = 60 / tan(50°)
And by rearranging the second equation, we have:
x = 60 / tan(13°)
Therefore, the width of the street from curb to curb is the average of these two values, which can be calculated as:
x = (60 / tan(50°) + 60 / tan(13°)) / 2
To find the width of the street from curb to curb, we can use the concept of oblique triangles by utilizing right triangles. Here's how you can approach the problem:
1. Start by drawing a diagram to visualize the situation. Draw a vertical line to represent the window, and label it with 30 feet to indicate the height from the street. Draw two lines from the window down to the curbs on both sides of the street, forming two right angles with the horizontal street.
Curb on Near Side
/
/
/ 50°
Window
\
\
\
Curb on Far Side
2. Identify the given angles of depression:
- Angle of depression for the curb on the near side = 50 degrees
- Angle of depression for the curb on the far side = 13 degrees
3. You can use the property of complementary angles to find the angle of elevation for the near side curb:
- Angle of elevation for the near side curb = 90 degrees - 50 degrees = 40 degrees
4. Now, we can create two right triangles:
a) Triangle 1: Window-Near Side Curb-Street
b) Triangle 2: Window-Far Side Curb-Street
5. Use trigonometric ratios (specifically the tangent function) to relate the angles and sides of the triangles:
a) In Triangle 1, the ratio of the opposite (30 feet, height from the window to the near side curb) to the adjacent (width of the street) is equal to the tangent of the angle of elevation (40 degrees):
tan(40 degrees) = opposite/adjacent
b) In Triangle 2, the ratio of the opposite (30 feet, height from the window to the far side curb) to the adjacent (width of the street) is equal to the tangent of the angle of depression (13 degrees):
tan(13 degrees) = opposite/adjacent
6. Rearrange the equations from step 5 to solve for the width of the street in both cases:
a) Width = 30 feet / tan(40 degrees)
b) Width = 30 feet / tan(13 degrees)
7. Calculate the width of the street using a scientific calculator or an online trigonometric calculator:
a) Width (near side) ≈ 30 feet / 0.8391 ≈ 35.76 feet
b) Width (far side) ≈ 30 feet / 0.2240 ≈ 133.93 feet
8. The actual width of the street from curb to curb can be estimated by taking the average of the two calculated values:
Width (average) ≈ (35.76 feet + 133.93 feet) / 2 ≈ 84.85 feet
So, the estimated width of the street from curb to curb is approximately 84.85 feet.