Entering a darkened room, Gustav strikes a match in an attempt to see his surroundings. At once he sees what looks like another match about $4 \mathrm{m}$ away from him. As it turns out, a mirror hangs on one wall of the room. How far is Gustav from the wall with the mirror?

First, let's draw a diagram to represent the situation. Label the points as follows: - G: Gustav's position - M: The position of the match in Gustav's hand - R: The position of the reflection of the match in the mirror - W: The wall with the mirror Gustav, the match, and its reflection form a right triangle.

Since we have a right triangle, we can use the Pythagorean theorem to find the distance between Gustav and the mirror. The theorem states that the sum of the squares on the legs of a right triangle is equal to the square of the length of the hypotenuse: \(legs^2 + base^2 = hypotenuse^2\) In our case, the hypotenuse is the distance between Gustav and the reflection of the match (4 meters). The base is the distance between Gustav and the wall with the mirror, which we are trying to find (let's call this distance x meters). The legs represent the distance between the mirror and the reflection, which is equal to the distance from Gustav to the mirror (x meters). So, we can write the equation as: \(x^2 + x^2 = 4^2\)

Now we need to solve the equation for x, the distance between Gustav and the wall with the mirror. \(x^2 + x^2 = 16\) Combining the x terms gives: \(2x^2 = 16\) Divide both sides by 2: \(x^2 = 8\) Finally, find the square root of both sides: \(x = \sqrt{8}\) To find the distance in meters, we can simplify the square root of 8: \(x = 2\sqrt{2} \approx 2.83\) So, Gustav is about 2.83 meters away from the wall with the mirror.