Math  /  Geometry

QuestionGabrielle knows that a fir tree that is 8 feet tall casts a shadow that is 15 feet long. She wants to calculate the height of a nearby plum tree. If its shadow is 30 feet long, how tall is the plum tree?
Write your answer as a whole number or a decimal. Do not round. 新 \square feet Submit

Studdy Solution

STEP 1

1. The trees and their shadows form similar triangles.
2. The height of the fir tree is 8 8 feet.
3. The shadow of the fir tree is 15 15 feet long.
4. The shadow of the plum tree is 30 30 feet long.

STEP 2

1. Set up a proportion using the similar triangles.
2. Solve the proportion for the height of the plum tree.

STEP 3

Set up a proportion using the similar triangles. The ratio of the height of the fir tree to its shadow is equal to the ratio of the height of the plum tree to its shadow:
Height of Fir TreeShadow of Fir Tree=Height of Plum TreeShadow of Plum Tree\frac{\text{Height of Fir Tree}}{\text{Shadow of Fir Tree}} = \frac{\text{Height of Plum Tree}}{\text{Shadow of Plum Tree}}
Substitute the known values:
815=h30\frac{8}{15} = \frac{h}{30}
where h h is the height of the plum tree.

STEP 4

Solve the proportion for h h :
First, cross-multiply:
8×30=15×h8 \times 30 = 15 \times h
Simplify:
240=15h240 = 15h
Divide both sides by 15 15 to solve for h h :
h=24015h = \frac{240}{15}
Calculate the division:
h=16h = 16
The height of the plum tree is:
16 \boxed{16} feet

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