Lindsay is out one night and notices that her body casts a shadow that is 1.8m long when she is standing 5mfrom the pole supporting a street light Lindsay is 1.5m tall.
1-what is the angle of elevation to the street light ?
2- how high is the street light
The angle is given by her height, and shadow length. tanTheta= 1.5/1.8
You can to the second by similar triangles
h/1.5=1.8/(5+1.8)
check that.
To solve this problem, we can use similar triangles. Here's how you can find the answers:
1. Angle of Elevation to the Street Light:
First, we need to determine the height of the pole supporting the street light. To do this, we can use the concept of similar triangles.
We have two right-angled triangles: one formed by Lindsay's shadow and her height, and the other formed by the shadow of the pole and the height of the pole.
Let's denote:
- Lindsay's height = H
- Lindsay's shadow length = S
- Pole's height = P
- Pole's shadow length = X
Using the similar triangles, we can set up a proportion:
H / S = P / X
Substituting the given values:
1.5 / 1.8 = P / X
Cross-multiplying, we get: 1.5X = 1.8P
Now, we can calculate the ratio of the pole's shadow length to Lindsay's shadow length:
X / S = P / H
Plug in the known values:
X / 1.8 = P / 1.5
Now, we can substitute the value of X from the earlier equation:
1.8P / 1.8 = P / 1.5
Simplifying, we find: P = 1.5P / 1.8
Divide both sides by P and simplify again:
1.8 = 1.5 / X
Cross-multiplying, we find: 1.8X = 1.5
Finally, solve for X:
X = 1.5 / 1.8
X ≈ 0.83m
Now, we have the shadow length of the pole, which is X. The angle of elevation can be found using the tangent function:
Tangent(angle) = Opposite / Adjacent
Tangent(angle) = P / X
Substituting the known values:
Tangent(angle) = 1.5 / 0.83
To find the angle, take the inverse tangent (arctan) of both sides:
Angle = arctan(1.5 / 0.83)
Using a calculator, we find that the angle of elevation to the street light is approximately 58.18 degrees.
2. Height of the Street Light:
Now that we know the angle of elevation (58.18 degrees) and the distance from Lindsay to the pole (5m), we can use trigonometry to find the height of the street light.
Using the tangent function:
Tangent(angle) = Opposite / Adjacent
Tangent(58.18 degrees) = P / 5
Rearranging the equation to solve for P:
P = Tangent(58.18 degrees) * 5
Calculating using a calculator:
P ≈ 8.61m
Therefore, the height of the street light is approximately 8.61 meters.