Final answer:
The distance from the vertex to the focus of the parabolic spotlight is 0.8 feet. We derived this by using the standard parabolic equation and the given dimensions of the parabolic cross-section of the spotlight.
Step-by-step explanation:
The question asks for the distance of the focus from the vertex of a parabolic spotlight. To solve this, we use the standard equation of a parabola in the form (y - k)^2 = 4p(x - h), where (h,k) is the vertex of the parabola and p is the distance from the vertex to the focus. Given that the parabolic cross-section is 10 ft wide at the opening and 4 ft deep at the vertex, we can infer that the parabola opens upwards and its vertex is at the origin (0,0), making h and k both 0.
The width of the parabola is the distance from one point on the parabola to another across the vertex, which corresponds to a change in x (Δx) that is twice the distance of the focus from the vertex, or 2p. The depth of 4 ft corresponds to the change in y (Δy) from the vertex to the point on the parabola; hence if Δy is 4 ft, then Δx must be 10 ft.
Using these measurements to adapt our equation, we get (4)^2 = 4p(10/2), which simplifies to 16 = 20p. Solving for p gives us p = 0.8 ft. Therefore, the distance from the vertex to the focus of the parabolic spotlight is 0.8 ft.