Final answer:
To determine the lowest operating cost of a video store, given by the quadratic function C(x), the vertex of the function provides the minimum cost. The vertex formula yields x=4.5; checking integer values, C(4) and C(5) both result in a cost of $640. Thus, the lowest cost is $640. The correct answer is option A) $640.
Step-by-step explanation:
The question asks to find the lowest cost for operating a video store, given by the quadratic function C(x) = 2x2 - 18x + 680, where x represents the number of videos rented daily. To find the lowest cost, we must determine the vertex of this parabola, as the coefficient of the x2 term is positive, indicating that the parabola opens upwards and the vertex will give the minimum value.
The vertex of a parabola given by f(x) = ax2 + bx + c is found using the formula -b/(2a). For the given function C(x), a = 2 and b = -18. Plugging these values into the vertex formula, we get x = -(-18)/(2*2) = 4.5. Since we cannot rent half a video, we need to check the cost for renting 4 and 5 videos respectively.
Calculating C(4): 2(4)2 - 18(4) + 680 = 2(16) - 72 + 680 = 32 - 72 + 680 = 640.
Calculating C(5): 2(5)2 - 18(5) + 680 = 2(25) - 90 + 680 = 50 - 90 + 680 = 640.
Upon comparison, renting 5 videos is the better option since the cost of renting 4 and 5 videos is the same; renting more videos for the same operating cost is advantageous. Therefore, the lowest cost to the nearest dollar is $640, which corresponds to option A).