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Math  /  Geometry

QuestionA person is standing 40 ft away from a street light that is 30 ft tall and casts a shadow that is 50 feet long. How tall is the person if his shadow is 10.ft10 . \mathrm{ft} long?

Studdy Solution

STEP 1

1. The street light and the person are both perpendicular to the ground.
2. The light from the street light creates similar triangles between the street light and its shadow, and the person and his shadow.
3. The street light is 30 feet tall.
4. The shadow of the street light is 50 feet long.
5. The person is standing 40 feet away from the street light.
6. The person's shadow is 10 feet long.

STEP 2

1. Use the concept of similar triangles.
2. Set up a proportion based on the similar triangles.
3. Solve for the height of the person.

STEP 3

Use the concept of similar triangles:
The triangles formed by the street light and its shadow, and the person and his shadow, are similar.

STEP 4

Set up a proportion based on the similar triangles:
Let h h be the height of the person. The proportion is:
Height of Street LightLength of Street Light’s Shadow=Height of PersonLength of Person’s Shadow\frac{\text{Height of Street Light}}{\text{Length of Street Light's Shadow}} = \frac{\text{Height of Person}}{\text{Length of Person's Shadow}}
Substitute the known values:
30 ft50 ft=h10 ft\frac{30 \text{ ft}}{50 \text{ ft}} = \frac{h}{10 \text{ ft}}

STEP 5

Solve for the height of the person:
Cross-multiply to solve for h h :
30×10=50×h30 \times 10 = 50 \times h
300=50h300 = 50h
Divide both sides by 50:
h=30050h = \frac{300}{50}
h=6 fth = 6 \text{ ft}
The height of the person is:
6 ft \boxed{6 \text{ ft}}

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