Final answer:
The maximum area of the deck is obtained when the dimensions are 21 meters by 42 meters, giving an area of 882 m2. So, the x-coordinate of the vertex is 21, which represents the width of the deck. To find the length of the deck, we can substitute this value back into the function A(x). A(21) = 84(21) - 2(21^2) = 1764 - 882 = 882.
Step-by-step explanation:
The student's question asks about finding the maximum area of a rectangular deck modeled by the function A(x)=84x-2x^2. To find the dimensions that give the maximum area, we need to find the vertex of the quadratic function, which represents the maximum point for a downward-opening parabola. Since the coefficient of the x^2 term is negative, we know the graph opens downwards, and thus the vertex will give us the maximum area.
First, let's find the x-coordinate of the vertex using the formula x = -b/2a, where a is the coefficient of x^2 and b is the coefficient of x. Plugging in the values, we get x = -84/(2*(-2)) = 21. Now we substitute x = 21 back into the original equation to find the maximum area: A(21) = 84(21) - 2(21)^2 = 1764 - 882 = 882 m^2.
The dimensions that give this maximum area would then be 21 meters by 42 meters since a rectangle's area is found by multiplying its length by its width, and since we are maximizing a quadratic function representing area in terms of one dimension, we infer that the rectangle is twice as long as it is wide for the maximum area.