Using the Pythagorean theorem, we can find the distance between the bases of the light source and the sensor. Let's call this distance "x".
We know that the light source is at a height of 8 metres and the sensor is at a height of 4 metres, so the vertical distance between them is 8 - 4 = 4 metres.
We also know that the total distance the laser beam travels is 15 metres. Let's call the horizontal distance between the mirror and the sensor "y". Then, the horizontal distance between the light source and the mirror is also "y".
Using the Pythagorean theorem, we can create an equation:
x^2 = y^2 + 4^2 (equation 1)
x^2 = y^2 + (15-y)^2 (equation 2)
We can simplify equation 2 by expanding (15-y)^2:
x^2 = y^2 + (225 - 30y + y^2)
Combining like terms:
x^2 = 2y^2 - 30y + 225
Now we can substitute equation 1 into this equation:
y^2 + 4^2 = 2y^2 - 30y + 225
Simplifying:
y^2 + 16 = 2y^2 - 30y + 225
Rearranging and simplifying:
y^2 - 30y + 209 = 0
Using the quadratic formula:
y = (30 ± sqrt(30^2 - 4*1*209)) / (2*1)
y = (30 ± sqrt(16)) / 2
y = 15 ± 2
Since we're looking for a positive value for "y", we can take the solution y = 15 - 2 = 13.
Now we can substitute this value of "y" into equation 1:
x^2 = 13^2 + 4^2
x^2 = 185
x = sqrt(185)
Therefore, the distance between the bases of the light source and the sensor is approximately 13.6 metres.