A woman that is 5'4" stands 15 ft from a streetlight and casts a four-foot long shadow. Determine the height of the streetlight and the degree measure of the angle of elevation form the tip of her shadow to the top of the streetlight, both accurate to two decimal places.
To determine the height of the streetlight and the degree measure of the angle of elevation, we can use similar triangles. Here's how you can find the answer:
1. First, let's label the known measurements and the unknowns. The height of the woman is given as 5'4", which can be converted to 5.33 feet. The distance between the woman and the streetlight is 15 feet. The length of the shadow cast by the woman is 4 feet. We need to determine the height of the streetlight and the angle of elevation.
2. Draw a diagram to visualize the problem. Sketch a triangle where the streetlight is at the top, the base is the distance between the woman and the streetlight, and the shadow cast by the woman forms the other side. Label the height of the streetlight as "h," the distance between the woman and the streetlight as "d," and the length of the woman's shadow as "s."
/\|
/ |\
/ | \
/ | \
/ | \
/ | \
/ | \
/θ | \
/________|_______\
d s
3. Based on the information given, we know that we have two similar triangles: the woman's shadow triangle and the streetlight triangle.
4. Since the triangles are similar, we can set up the following proportion based on the corresponding sides:
height of woman / length of shadow = height of streetlight / distance to streetlight
5.33 / 4 = h / 15
5. Solve the proportion for h:
5.33 * 15 = h * 4
79.95 = 4h
h ≈ 19.99 feet (rounded to two decimal places)
Therefore, the height of the streetlight is approximately 19.99 feet.
6. To find the angle of elevation (θ), we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the streetlight (h) and the adjacent side is the distance to the streetlight (d).
tangent(θ) = h / d
tangent(θ) = 19.99 / 15
7. Solve for θ by taking the inverse tangent (arctan) of both sides. Use a scientific calculator or online tool to calculate the arctan:
θ = arctan(19.99 / 15)
θ ≈ 51.34 degrees (rounded to two decimal places)
Therefore, the degree measure of the angle of elevation from the tip of her shadow to the top of the streetlight is approximately 51.34 degrees.