Illustration of the Pythagorean Theorem using a ladder, house, and firefighter.

This image presents a classic application of the Pythagorean Theorem. Let's break down the problem and solve it step-by-step.

Step 1: Understand the Problem

The image shows a fireman placing a ladder against a house to rescue someone from a window. We are given:
- The height of the window from the ground is 40 ft.
- The distance from the base of the ladder to the house is 60 ft.
- The length of the ladder, labeled as x, is what we need to find.

This forms a right triangle where:
- The height of the window (40 ft) is one leg of the triangle.
- The distance from the base of the ladder to the house (60 ft) is the other leg.
- The ladder (x) is the hypotenuse.

Step 2: Apply the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle:

$$
a^2 + b^2 = c^2
$$

Where:
- $a$ and $b$ are the lengths of the legs.
- $c$ is the length of the hypotenuse.

In this case:
- $a = 40$ ft
- $b = 60$ ft
- $c = x$

So, we can write:

$$
40^2 + 60^2 = x^2
$$

Step 3: Calculate the Values

First, calculate the squares:

$$
40^2 = 1600
$$
$$
60^2 = 3600
$$

Now, add them together:

$$
1600 + 3600 = 5200
$$

So,

$$
x^2 = 5200
$$

Step 4: Solve for $x$

To find $x$, take the square root of both sides:

$$
x = \sqrt{5200}
$$

Simplify the square root:

$$
\sqrt{5200} = \sqrt{100 \times 52} = 10 \sqrt{52}
$$

Further simplify $\sqrt{52}$:

$$
\sqrt{52} = \sqrt{4 \times 13} = 2 \sqrt{13}
$$

So,

$$
x = 10 \times 2 \sqrt{13} = 20 \sqrt{13}
$$

Step 5: Approximate the Value (if needed)

If you need a decimal approximation:

$$
\sqrt{13} \approx 3.6056
$$

So,

$$
x \approx 20 \times 3.6056 = 72.112
$$

Final Answer

The length of the ladder, $x$, is approximately 72.11 feet.

This means the fireman needs a ladder that is at least 72.11 feet long to reach the window safely.