A light source at a height of 8 metres is shining a laser beam onto the mirror on the ground. The light beam needs to be observed by a sensor at a height of 4 metres. If the total distance the laser beam travels is 15 metres, what is the distance between the bases of the light source and the sensor? Your answer should be a numerical value.

To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Here, the total distance the laser beam travels (15 metres) is the hypotenuse, the height difference between the light source and the sensor (8 - 4 = 4 metres) is one side, and the distance between the bases of the light source and the sensor is the other side.

Let's denote the distance between the bases of the light source and the sensor as x. According to the Pythagorean theorem, we have:

15^2 = 4^2 + x^2

225 = 16 + x^2

x^2 = 225 - 16

x^2 = 209

Taking the square root of both sides, we get:

x = sqrt(209)

So, the distance between the bases of the light source and the sensor is approximately 14.46 metres.

To solve this problem, we can use the Pythagorean theorem. The total distance the laser beam travels is the hypotenuse of a right triangle, and the difference in height between the light source and the sensor forms one of the sides of the triangle.

Step 1: Find the difference in height between the light source and the sensor. The light source is at a height of 8 metres and the sensor is at a height of 4 metres. So, the difference in height is 8 - 4 = 4 metres.

Step 2: Use the Pythagorean theorem to find the distance between the bases of the light source and the sensor. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. We can write this as: c² = a² + b²

Here, c is the hypotenuse (the total distance the laser beam travels), a is one of the sides (the difference in height between the light source and the sensor), and b is the other side (the distance between the bases of the light source and the sensor, which we are trying to find).

We know that c = 15 metres and a = 4 metres. We can substitute these values into the Pythagorean theorem and solve for b:

15² = 4² + b² 225 = 16 + b² b² = 225 - 16 b² = 209 b = √209

So, the distance between the bases of the light source and the sensor is approximately 14.45 metres.