A page of 150 square inches is to have margins of 1 inch on either side and 1 1/2 inches at the top and bottom, as in the diagram. Find the dimensions of the page that maximize the print area.

12X18

Well, let's analyze this like a jigsaw puzzle. We have a page with margins on the sides and top and bottom. So if we subtract the margins from the dimensions of the page, we can find the maximum print area.

Let's call the width of the page "w" and the height of the page "h." We can express the width and height of the print area by subtracting the margins:

Width of print area = w - 2 * 1 inch
Height of print area = h - 2 * 1.5 inches

The total area of the print is then:

Print area = (w - 2) * (h - 3)

To find the dimensions that maximize the print area, we need to maximize the function (w - 2) * (h - 3). We can do this by finding critical points. Taking the derivative of the function with respect to both w and h, we get:

d(Print area) / d(w) = h - 3
d(Print area) / d(h) = w - 2

Setting these derivatives equal to zero, we find that the critical point occurs when h = 3 and w = 2.

Hence, the dimensions of the page that maximize the print area are 2 inches by 3 inches. I hope that brings you plenty of laughs while you maximize your print area!

To find the dimensions of the page that maximize the print area, we need to determine the length and width of the print area on the page.

Let's denote the width of the print area as x inches and the length as y inches.

From the given diagram, we can see that the total width of the page, including the margins, is x+2 inches, and the total length is y+3 inches.

Since the area of a rectangle is calculated by multiplying the length by the width, the print area can be expressed as:

Print Area = x * y

The total area of the page, including the margins, can be expressed as:

Total Area = (x + 2) * (y + 3)

However, we are given that the total area is 150 square inches. So, we can set up the equation:

(x + 2) * (y + 3) = 150

Expanding the equation, we get:

xy + 3x + 2y + 6 = 150

Rearranging the equation, we have:

xy = 150 - 3x - 2y - 6
xy = 144 - 3x - 2y

Now, we can isolate y in terms of x by dividing both sides of the equation by x:

y = (144 - 3x - 2y) / x

Next, we can substitute this value of y into the equation for the print area:

Print Area = x * y
Print Area = x * [(144 - 3x - 2y) / x]

Simplifying further, we have:

Print Area = 144 - 3x - 2y

To maximize the print area, we need to find the values of x and y that maximize the expression 144 - 3x - 2y.

Now, we can take partial derivatives of the expression with respect to both x and y and set them equal to zero to find the critical points:

d(Print Area) / dx = -3
d(Print Area) / dy = -2

Setting both derivatives equal to zero:

-3 = 0 and -2 = 0

These equations have no solutions.

Since there are no critical points, we can conclude that the print area is not maximized within the given constraints.

Therefore, there are no dimensions for the page that maximize the print area.

To find the dimensions of the page that maximize the print area, we first need to determine the dimensions of the page itself. Let's assume the dimensions of the page are x inches (width) and y inches (height).

Given that there are 1 inch margins on either side, the actual print area width will be x - 2 inches. Similarly, considering the 1 1/2 inch margins at the top and bottom, the actual print area height will be y - 3 inches.

The area of the print on the page, known as the print area, is given by the product of these dimensions:

Print area = (x - 2) inches * (y - 3) inches

Now, we want to maximize this print area. To do so, we need to find the critical points of the function representing the print area and determine whether they correspond to a maximum or minimum.

To find the critical points, we take partial derivatives of the print area equation with respect to both x and y. Then, we set these derivatives equal to zero and solve for x and y.

Let's differentiate the print area equation with respect to x:

d(Print area) / dx = (d/dx)(x - 2)(y - 3)
= (y - 3)

Setting this equal to zero:

(y - 3) = 0

Solving for y:

y = 3

Now, let's differentiate the print area equation with respect to y:

d(Print area) / dy = (d/dy)(x - 2)(y - 3)
= (x - 2)

Setting this equal to zero:

(x - 2) = 0

Solving for x:

x = 2

Therefore, the critical point for maximizing the print area occurs at x = 2 inches and y = 3 inches.

To determine whether this critical point represents a maximum or minimum, we can use the second partial derivative test. However, in this case, the test is not necessary as we have a very simple scenario.

Since both x and y are positive dimensions, a smaller width x and a smaller height y would result in even smaller print areas. Therefore, we can conclude that the dimensions maximizing the print area are x = 2 inches and y = 3 inches.

Hence, the dimensions of the page that maximize the print area are 2 inches for the width and 3 inches for the height.

If the page has dimensions x and y, then we know

xy=150

Since it has margins all around, we want to maximize the print area

p = (x-2)(y-3) = (x-2)(150/x-3) = 156 - 3x - 300/x
dp/dx=0 when x=10
So, a 10 by 15 page has maximum print area.