Final answer:
To find 'a' in the equation y = ax² for a satellite dish with a 10-ft diameter and 3-ft depth, set x = 5 and y = 3, yielding a = 0.12. The surface area S of the dish can be obtained by integrating S = 2π∫ y sqrt(1 + (dy/dx)²) dx from x = 0 to 5 and doubling the result, substituting a = 0.12 into the equation.
Step-by-step explanation:
The student asks to determine the value of the coefficient a in the equation of a parabola y = ax², which defines a parabolic satellite dish, as well as the surface area S of the dish, given that it will have a diameter of 10 feet and a maximum depth of 3 feet.
Firstly, the diameter of 10 feet corresponds to a radius of 5 feet. Hence, when x = 5, y = 3. Substituting these values into the equation gives 3 = a(5)², which simplifies to 3 = 25a. Solving for a, we find that a = 3/25 or 0.12.
The surface area of the satellite dish can be found using the formula for the surface area of a solid of revolution, the general form of which is S = 2π∫ y sqrt(1 + (dy/dx)²) dx, where y = ax² and dy/dx = 2ax. We will integrate from x = 0 to x = 5 to find the complete surface area of one half of the dish (as it's symmetric) and then double the result.
Thus, the surface area S can be calculated by the following integration: S = 2 * 2π∫(0 to 5) (ax²) sqrt(1 + (2ax)²) dx. Upon solving this integral and substituting in the value we found for a, we can calculate the overall surface area of the parabolic dish.