a page that x inches wide & y inches high contains 26 square inches of print. the margins on the top & bottom are 1.5 inches deep and the margins in the sides are 1 inches wide.
a. sketch the situation above
b. find the total area of the page
c. find the domain based on the physical constraints of the problem
as they said, the total area is xy
as for the print area, that is
(x-2)(y-3) = 26
a. To sketch the situation, draw a rectangle representing the page. Label the length of the page as y inches and the width as x inches. Divide the page into different sections to represent the margins. The top and bottom margins are 1.5 inches deep, so draw two horizontal lines parallel to the shorter sides, each 1.5 inches away from the edge. The side margins are 1 inch wide, so draw two vertical lines parallel to the longer sides, each 1 inch away from the edge.
b. To find the total area of the page, we need to subtract the area of the margins from the total area of the page. The total area of the page is the product of its length and width, which is x * y inches^2. To find the area of the margins, we need to find the area of each margin separately and then sum them up.
The area of each top and bottom margin is the product of the length of the margin (1.5 inches) and the width of the page (x inches). So the total area of the top and bottom margins is 2 * (1.5 inches * x inches).
Similarly, the area of each side margin is the product of the width of the margin (1 inch) and the length of the page (y inches). So the total area of the side margins is 2 * (1 inch * y inches).
Therefore, the total area of the margins is 2 * (1.5 inches * x inches) + 2 * (1 inch * y inches).
To find the area of the printed section on the page, subtract the area of the margins from the total page area: (x * y inches^2) - [2 * (1.5 inches * x inches) + 2 * (1 inch * y inches)].
c. The domain is the set of possible values for x and y based on the physical constraints of the problem. From the problem statement, we know that the margins on the top and bottom are 1.5 inches deep and the margins on the sides are 1 inch wide. Therefore, the width of the printed section on the page must be x - 2 inches (subtracting 2 inches for the total width of the side margins). Similarly, the height of the printed section on the page must be y - 3 inches (subtracting 3 inches for the total height of the top and bottom margins).
Since we are given that the printed section on the page contains 26 square inches of print, we can set up the equation:
(x - 2 inches) * (y - 3 inches) = 26 square inches.
The domain of the problem would be the range of valid values for x and y that satisfy this equation while also considering any other physical constraints that might exist.