If the heights of two poles are 22 m and 37 m and the distance between their tops is 39 m, then find the distance between the feet of the poles.

Answer

To find the distance between the feet of the two poles, we can use the Pythagorean theorem. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have two poles with heights of 22 m and 37 m. The distance between the tops of the poles is given as 39 m. We need to find the horizontal distance between the feet of the poles.

### Step 2: Set Up the Diagram
Let’s label the poles:
- Let Pole A be the one with a height of 22 m.
- Let Pole B be the one with a height of 37 m.

### Step 3: Determine the Vertical Distance Between the Tops of the Poles
The vertical distance between the tops of the poles can be calculated as:
\[ \text{Vertical Distance} = \text{Height of Pole B} - \text{Height of Pole A} = 37 \, \text{m} - 22 \, \text{m} = 15 \, \text{m} \]

### Step 4: Use the Pythagorean Theorem
We can visualize a right triangle where:
- One leg (vertical distance) is 15 m (the difference in height).
- The hypotenuse (distance between the tops of the poles) is 39 m.
- The other leg (horizontal distance between the feet of the poles) is what we need to find.

According to the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
Where:
- \( c \) is the hypotenuse (39 m),
- \( a \) is the vertical distance (15 m),
- \( b \) is the horizontal distance (distance between the feet of the poles, which we will denote as \( d \)).

### Step 5: Set Up the Equation
Using the Pythagorean theorem:
\[ 39^2 = 15^2 + d^2 \]

### Step 6: Calculate the Squares
Calculating the squares:
\[ 39^2 = 1521 \]
\[ 15^2 = 225 \]

### Step 7: Substitute and Solve for \( d^2 \)
Substituting the values into the equation:
\[ 1521 = 225 + d^2 \]
Now, isolate \( d^2 \):
\[ d^2 = 1521 - 225 \]
\[ d^2 = 1296 \]

### Step 8: Find \( d \)
Taking the square root of both sides to find \( d \):
\[ d = \sqrt{1296} = 36 \, \text{m} \]

### Conclusion
The distance between the feet of the poles is **36 m**.

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