Satellite dish antennas have the shape of a parabola. Consider a satellite dish that is 80 cm across. Its cross-sectional shape can be described by the function d(x) = 0.0125x^2 - x, where d is the depth, in centimetres, of the dish at a horizontal distance of x centimetres from one edge of the dish.
1. What is the maximum depth of the dish in cm?
0.0125x^2 - x
= 1/80 x^2 - x
= 1/80 (x^2 - x + 1/4) - 1/80 * 1/4
= 1/80 (x - 1/2)^2 - 1/320
max depth is 1/320 cm
To find the maximum depth of the dish, we need to find the vertex of the parabolic function d(x) = 0.0125x^2 - x.
The vertex of a parabola in the form y = ax^2 + bx + c is given by the coordinates (h, k), where h = -b / (2a) and k = f(h), where f(h) represents the value of y at x = h.
In this case, a = 0.0125 and b = -1. Plugging these values into the formula, we get:
h = -(-1) / (2 * 0.0125) = 1 / 0.025 = 40
To find the depth at x = 40, we substitute x = 40 into the equation:
d(40) = 0.0125 * 40^2 - 40 = 20 - 40 = -20
Therefore, the maximum depth of the dish is -20 cm. Note that the negative sign indicates that the depth is below the surface of the dish.