I've done the first question but I cannot get any farther. I'm so stuck. Please explain or just help me to figure out the next steps.
A printed poster is to have a total area of 837 square inches with top and bottom margins of 7 inches and side margins of 6 inches. What should be the dimensions of the poster so that the printed area be as large as possible?
*To solve this problem let x denote the width of the poster and let y denote the length. We need to maximize the following function of x and y:
(x-12)(y-14)
*We can reexpress this as the following function of x alone: f(x)=__________
*We find that f(x) has a critical number at x=_________
*To verify that f(x) has a maximum at this critical number we compute the second derivative f(x) and find that its value at the critical number is_________, a negative number.
*Thus the optimal dimensions of the poster are ________ inches in width and ________ inches in height giving us a maximal printed area of ________square inches.
To solve this problem, we can follow these steps:
Step 1: Express the function of x and y
The given function is (x-12)(y-14). We want to express this function in terms of x alone, so we need to eliminate the y variable.
Expanding the function, we get xy - 14x - 12y + 168.
Now, replace y with its expression in terms of x. Since the total area is given as 837 square inches and the margins are 7 inches on top and bottom, and 6 inches on each side, we have:
xy = 837 - 2(7)(x) - 2(6)(y)
xy = 837 - 14x - 12y
Substituting this into the expanded function, we get:
f(x) = 837 - 14x - 12y + 168
f(x) = 1005 - 14x - 12y
Therefore, the reexpressed function of x alone is f(x) = 1005 - 14x - 12y.
Step 2: Find the critical number
To find the critical number, we set the derivative of f(x) equal to zero.
f'(x) = -14
Setting -14 equal to zero, we find that there are no critical numbers for this function.
Step 3: Verify if f(x) has a maximum at the critical number
Since we don't have a critical number, we cannot verify if f(x) has a maximum at that point.
Step 4: Optimal dimensions and maximal printed area
As we couldn't find a critical number, we need to take a different approach to find the optimal dimensions and maximal printed area. We will use the concept of vertex of a parabola.
The function f(x) = 1005 - 14x - 12y can be visualized as a upward-opening parabola on a coordinate system.
The x-coordinate of the vertex of this parabola will give us the optimal width, and the corresponding y-coordinate will give us the optimal length.
To find the x-coordinate of the vertex, we use the formula x = -b/(2a), where a = -14 and b = 0 (since there is no term with x in f(x)).
x = -0/(2*-14) = 0/(-28) = 0
Therefore, the optimal width is 0 inches.
Substituting x = 0 into the function f(x), we get:
f(0) = 1005 - 14(0) - 12y
f(0) = 1005 - 0 - 12y
f(0) = 1005 - 12y
To maximize the printed area, we need to minimize y. Since y represents the length, the minimum possible value for y would be the sum of the top and bottom margins, which is 7 + 7 = 14 inches.
Thus, the optimal height (length) would be 14 inches.
Substituting the optimal width (0 inches) and height (14 inches) into the original function, we get:
(x-12)(y-14) = (0-12)(14-14) = (0)(0) = 0
Therefore, the maximal printed area is 0 square inches.
In summary, according to the given function, the optimal dimensions for the poster are 0 inches in width and 14 inches in height, resulting in a maximal printed area of 0 square inches.