Final answer:
The coordinates of the suspension cable function h(d) at its minimum value are calculated using the vertex formula. Due to an initial error made when designating the variables, a correction provides the right minimum coordinates as (50, 5), which is not listed in the provided options.
Step-by-step explanation:
To find the coordinates of the function at its minimum value, we need to look at the function given for the height of the suspension cable: h(d) = 0.01d² - d + 30. This is a quadratic function in the form of ax² + bx + c, where the vertex represents the maximum or minimum point depending on the direction of the parabola (which is concave up if a > 0 and concave down if a < 0). In our case, a = 0.01 is positive, so the parabola opens upwards, and the vertex represents the minimum point.
The vertex of a quadratic function is given by the formula h = -b/(2a). Inserting the coefficients from our function we get h = -(-1)/(2 * 0.01), which simplifies to h = 50. Plugging d = 50 back into the formula for h(d), we get h(50) = 0.01 * 50² - 50 + 30, which simplifies to h(50) = 25 - 50 + 30, finally giving us h(50) = 5. However, this method led to a calculation mistake as we did not properly find the value of d which should be d = -b/(2a), not h.
Let us correct the calculation. For the vertex, calculate d = -b/(2a), which is d = -(-1)/(2 * 0.01) and simplifies to d = 50. To find the minimum value of the function, h(d), we plug d = 50 back into the original equation: h(50) = 0.01(50)² - 50 + 30 which simplifies to h(50) = 25 - 50 + 30 and results in h(50) = 5. Therefore, the coordinates of the function at its minimum value are (50, 5), which is not one of the provided options.