A book designer has decided that the pages of a book should have margins at the top and bottom and margins on the sides.She further stipulated that each page should have an area of (see the figure). Determine the page dimensions that will result in the maximum printed area on the page.

Question

A book designer has decided that the pages of a book should have  margins at the top and bottom and  margins on the sides.She further stipulated that each page should have an area of  (see the figure).  Determine the page dimensions that will result in the maximum printed area on the page.

Quizplus · Accepted Answer

Let x be the width of the printed area and y be the height of the printed area.

From the given information: 
- The top and bottom margins are both 11eb12b0_a9dd_78a4_b394_6d0d96952e83_TB7866_11, so the total height of the page is y + 2(11eb12b0_a9dd_78a4_b394_6d0d96952e83_TB7866_11) 
- The left and right margins are both 11eb12b0_a9dd_78a5_b394_eb4e3a63880f_TB7866_11, so the total width of the page is x + 2(11eb12b0_a9dd_78a5_b394_eb4e3a63880f_TB7866_11) 
- The page area is 11eb12b0_a9dd_78a6_b394_c3763176427e_TB7866_11, so xy = 11eb12b0_a9dd_78a6_b394_c3763176427e_TB7866_11

We want to maximize the printed area xy.

Using the information above, we can write:

y + 2(11eb12b0_a9dd_78a4_b394_6d0d96952e83_TB7866_11) = y + 22eb12b0_a9dd_78a4_b394_6d0d96952e83_TB7866_11 
x + 2(11eb12b0_a9dd_78a5_b394_eb4e3a63880f_TB7866_11) = x + 22eb12b0_a9dd_78a5_b394_eb4e3a63880f_TB7866_11

Since we want to maximize xy, we can solve for y in terms of x using the equation xy = 11eb12b0_a9dd_78a6_b394_c3763176427e_TB7866_11:

y = 11eb12b0_a9dd_78a6_b394_c3763176427e_TB7866_11 / x

Substituting this into the equation for the total height of the page, we get:

y + 22eb12b0_a9dd_78a4_b394_6d0d96952e83_TB7866_11 = 11eb12b0_a9dd_78a6_b394_c3763176427e_TB7866_11 / x + 22eb12b0_a9dd_78a4_b394_6d0d96952e83_TB7866_11

Simplifying this, we get:

y + 22eb12b0_a9dd_78a4_b394_6d0d96952e83_TB7866_11 = (11eb12b0_a9dd_78a6_b394_c3763176427e_TB7866_11 + 2x) / x

To maximize xy, we want to maximize the right-hand side of this equation. We can find the maximum value of the right-hand side by finding its derivative with respect to x and setting it equal to zero:

d/dx [(11eb12b0_a9dd_78a6_b394_c3763176427e_TB7866_11 + 2x) / x] = 0

Simplifying this, we get:

11eb12b0_a9dd_78a6_b394_c3763176427e_TB7866_11 - 2x = 0

Solving for x, we get:

x = 11eb12b0_a9dd_9fba_b394_f1042e106039_TB7866_11 / 2

Substituting this back into the equation for y, we get:

y = 11eb12b0_a9dd_9fba_b394_f1042e106039_TB7866_11 / x = 11eb12b0_a9dd_9fba_b394_f1042e106039_TB7866_11 / (11eb12b0_a9dd_9fba_b394_f1042e106039_TB7866_11 / 2) = 2

Therefore, the width of the printed area is 11eb12b0_a9dd_9fba_b394_f1042e106039_TB7866_11 / 2 and the height of the printed area is 2.

The total width of the page is 11eb12b0_a9dd_9fba_b394_f1042e106039_TB7866_11, since the printed area takes up the full width of the page.

The total height of the page is y + 22eb12b0_a9dd_78a4_b394_6d0d96952e83_TB7866_11 = 2 + 22eb12b0_a9dd_78a4_b394_6d0d96952e83_TB7866_11

Therefore, the dimensions of the page are 11eb12b0_a9dd_9fba_b394_f1042e106039_TB7866_11 by 2 + 22eb12b0_a9dd_78a4_b394_6d0d96952e83_TB7866_11.

The correct answer is C.

A Book Designer Has Decided That the Pages of a Book