Maximum printable area A rectangular page in a text (with width x and length y) has an area of 98 in² , top and bottom margins set at 1 in, and left and right margins set at 1/2 in. The printable area of the page is the rectangle that lies within the margins. What are the dimensions of the page that maximize the printable area?

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Define the dimensions of the page: Let the width of the page be x inches and the length be y inches. The total area of the page is given by the equation x * y = 98 square inches.

Determine the dimensions of the printable area: The printable width is (x - 1) inches, accounting for 1/2 inch margins on each side, and the printable length is (y - 2) inches, accounting for 1 inch margins at the top and bottom.

Express the printable area: The printable area A can be expressed as A = (x - 1) * (y - 2).

Substitute the constraint into the printable area equation: Since x * y = 98, express y in terms of x, y = 98/x, and substitute into the printable area equation to get A = (x - 1) * (98/x - 2).

Find the critical points: Differentiate the expression for A with respect to x, set the derivative equal to zero, and solve for x to find the critical points. Use the second derivative test to determine if the critical point is a maximum.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this context, we need to determine the dimensions of the page that maximize the printable area, which requires setting up a function that represents the area and then using techniques such as taking derivatives to find critical points.

Area of a Rectangle

The area of a rectangle is calculated by multiplying its width (x) by its length (y). In this problem, the total area is constrained to 98 in², and we must account for the margins to find the effective dimensions that contribute to the printable area, which is the area available after subtracting the margins.

Constraints

Constraints are conditions that must be satisfied in an optimization problem. Here, the constraints include the fixed total area of 98 in² and the specific margin sizes, which limit the possible values of width and length. Understanding these constraints is crucial for correctly formulating the problem and finding the optimal solution.