Final answer:
Using principles from parabolic shapes, the lamp's width is found to be 20.8 centimeters, calculated by determining the 'a' value from the distance between the vertex and focus, and then using the parabolic equation to solve for the width at the depth of the lamp.
Step-by-step explanation:
To find out how wide the lamp is, we need to understand the properties of a parabola and the fact that the lamp is shaped as a parabola when viewed from the side. The question states that the light source, acting as the focus of the parabola, is 3 centimeters from the bottom of the lamp and that the lamp is 12 centimeters deep. In other words, the vertex of the parabola (bottom of the lamp) is 3 centimeters from the focus, and the depth of the lamp is therefore 12 - 3 = 9 centimeters from the vertex to the directrix.
The standard form of a parabolic equation is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola, and the value of 'a' determines the width (or steepness) of the parabola. Since the vertex is at the origin, h and k are 0. The distance from the vertex to the focus is 1/(4a), meaning that 1/(4a) = 3 cm, so 'a' is 1/12. To find the width at the directrix, which is 9 cm away from the vertex, we plug y = 9 into the equation and solve for x.
y = a(x^2), so 9 = 1/12(x^2), which simplifies to x^2 = 108. Therefore, x = ±10.4 centimeters, giving us a total width of 20.8 centimeters for the lamp.