Final answer:
The receiver, which is placed at the focus of a parabolic satellite dish that is 144 cm wide and 108 cm deep, is 12 centimeters from the bottom of the dish.
Step-by-step explanation:
The question involves finding the distance the receiver, placed at the focus, is from the bottom of the parabolic satellite dish. Given that the dish is 144 centimeters wide and 108 centimeters deep, we can use the properties of a parabola to determine this distance. The equation of a parabola in vertex form is y = a(x-h)^2 + k, where (h, k) is the vertex of the parabola, and the focus is at (h, k + 1/(4a)).
Since the dish is 144 cm wide, the vertex will be at the origin of our coordinate system (h = 0) assuming that the dish opens upwards, and the width of the dish will spread from -72 cm to 72 cm. We are given the depth (height) of the parabola as 108 cm, so k = -108. To find the value of a, we use the point (72, 0) which lies on the parabola. Plugging this into the equation, we get 0 = a(72-0)^2 - 108, solving for a gives us a = 1/48. The focus lies 1/(4a) units above the vertex, which is 1/(4(1/48)) = 12 cm above the vertex, or in other words, it is at the point (0, -96). Hence, the receiver (focus) is 12 centimeters from the bottom of the dish.