If a tree casts a 18 feet shadow and at the same time, a child of height 3 feet casts a 2 feet shadow, then the height of the tree is

Options

A. 27 feet
B. 32 feet
C. 45 feet
D. 36 feet

Answer

To find the height of the tree based on the shadow lengths of the tree and the child, we can use the concept of similar triangles. Here's the step-by-step solution:

### Step 1: Understand the relationship between the heights and shadows
The height of the tree (let's call it \( h \)) and the length of its shadow (18 feet) form one triangle. The height of the child (3 feet) and the length of the child's shadow (2 feet) form another triangle. Since both triangles are formed under the same angle of elevation from the sun, they are similar.

**Hint:** Remember that in similar triangles, the ratios of corresponding sides are equal.

### Step 2: Set up the proportion
From the similarity of the triangles, we can set up the following proportion:

\[ \frac{h}{18} = \frac{3}{2} \]

**Hint:** This proportion states that the height of the tree over its shadow length is equal to the height of the child over its shadow length.

### Step 3: Cross-multiply to solve for \( h \)
Cross-multiplying gives us:

\[ h \cdot 2 = 3 \cdot 18 \]

This simplifies to:

\[ 2h = 54 \]

**Hint:** Cross-multiplying helps eliminate the fraction and makes it easier to isolate the variable.

### Step 4: Solve for \( h \)
Now, divide both sides by 2 to find \( h \):

\[ h = \frac{54}{2} = 27 \]

**Hint:** Dividing both sides by the same number helps to isolate the variable.

### Conclusion
The height of the tree is \( 27 \) feet.

**Final Answer:** The height of the tree is 27 feet.