Final answer:
The maximum amount of profit the company can make is $978 (to the nearest dollar).
Step-by-step explanation:
The company's profit function is given by the equation y = -x^2 + 78x - 543. To find the maximum amount of profit, we need to determine the value of x that corresponds to the vertex of the parabolic graph represented by this equation. The x-coordinate of the vertex can be found using the formula x = -b/(2a), where a, b, and c are the coefficients of the quadratic equation. In this case, a = -1, b = 78, and c = -543. Plugging these values into the formula, we get x = -78/(2*(-1)) = 39. Substitute this value of x back into the profit function to find the maximum profit:
y = -(39^2) + 78(39) - 543 = -1521 + 3042 - 543 = 978
Hence, the maximum profit the company can make is $978 (to the nearest dollar).