A tree is 12 feet tall and casts a shadow 9 feet long. A nearby building casts a shadow that is
24 feet long. What is the height of the building? (Draw a picture, set up a proportion)

Question

A tree is 12 feet tall and casts a shadow 9 feet long. A nearby building casts a shadow that is 24 feet long. What is the height of the building? (Draw a picture, set up a proportion)

UpStudy AI · Accepted Answer

To solve the problem, we can use the concept of similar triangles. The height of the tree and the length of its shadow form one triangle, while the height of the building and the length of its shadow form another triangle. Since the sun's rays create similar angles, the triangles are similar.

### Known Conditions:
- Height of the tree ($$h_t$$) = 12 feet
- Length of the tree's shadow ($$s_t$$) = 9 feet
- Length of the building's shadow ($$s_b$$) = 24 feet
- Height of the building ($$h_b$$) = ?

### Step 1: Set up the proportion
Since the triangles are similar, we can set up the following proportion:

$$
\frac{h_t}{s_t} = \frac{h_b}{s_b}
$$

### Step 2: Substitute the known values
Substituting the known values into the proportion gives us:

$$
\frac{12}{9} = \frac{h_b}{24}
$$

### Step 3: Solve for $$h_b$$
To find $$h_b$$, we can cross-multiply:

$$
12 \cdot 24 = 9 \cdot h_b
$$

Calculating the left side:

$$
288 = 9 \cdot h_b
$$

Now, divide both sides by 9 to isolate $$h_b$$:

$$
h_b = \frac{288}{9}
$$

### Step 4: Calculate $$h_b$$
Now, let's perform the calculation:

$$
h_b = 32
$$

### Final Answer
The height of the building is $$32$$ feet.
The building is 32 feet tall.

A tree is 12 feet tall and casts a shadow 9 feet long. A nearby building casts a shadow that is
24 feet long. What is the height of the building? (Draw a picture, set up a proportion)

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