Question

The cost, ( c(x) ), in dollars per hour of running a trolley at an amusement park is modelled by the function ( c(x) = 2.1x² - 12.7x + 167.4 ), where ( x ) is the speed in kilometers per hour. At what approximate speed should the trolley travel to achieve minimum cost?

(A) 6.04 km/h
(B) 12.74 km/h
(C) 9.67 km/h
(D) 8.23 km/h

Answer 1

The minimum cost for running the trolley occurs at a speed of approximately 3.02 km/h, which can be found by calculating the vertex of the quadratic cost function.

Step-by-step explanation:

The cost of running a trolley at an amusement park is given by the function c(x) = 2.1x² - 12.7x + 167.4, where x represents the speed in kilometers per hour. To find the speed that minimizes the cost, we need to calculate the vertex of the quadratic function since the coefficient of x² is positive, indicating a parabolic curve that opens upwards. The vertex formula for a parabola given by y = ax² + bx + c is x = -b/(2a).

For our function, a = 2.1 and b = -12.7. Using the vertex formula: x = -(-12.7)/(2 × 2.1) = 12.7/4.2 = 3.0238. Hence, the speed that achieves the minimum cost is approximately 3.02 km/h, which is not one of the options provided, therefore, there might be an error in the options or the function's coefficients.