The top and bottom margins of a rectangular poster are 6 centimeters each, and the side margins are 4 centimeters each. If the area printed on the poster is fixed at 384 cm2,
a) Sketch the figure.
b) Find the dimensions of the poster using the least amount of paper.
if the printed area has width and height x and y, then the total area is
a = (x+8)(y+12)
and xy=384
a = (x+8)(384/x + 12) = 12x + 3072/x + 480
that is minimum when da/dx = 0, so we need
12 - 3072/x^2 = 0
x = 16
so the poster is (16+8) by (24+6) or 24x30
To find the dimensions of the poster using the least amount of paper, we can start by sketching the figure and understanding the given information.
a) Sketching the figure:
To sketch the figure, draw a rectangle representing the entire poster. Label the top and bottom margins as 6 cm each and the side margins as 4 cm each. This will leave a printed area within the margins.
Here is a rough sketch of the figure:
______6 cm______
|_________________|
| ____________ |
| | 384 cm² | |
| |___________| |
|_________________|
______6 cm______
b) Finding the dimensions of the poster:
Let's consider the length and width of the printed area as "x" and "y" respectively.
Since the entire printed area has a fixed area of 384 cm², we can write an equation using the formula for the area of a rectangle:
area = length * width
384 = x * y
Now let's consider the dimensions of the entire poster, including the margins.
Since the top and bottom margins are each 6 cm, the total length of the poster is (x + 12) cm.
Similarly, the total width of the poster with side margins is (y + 8) cm.
To find the dimensions of the entire poster, we add the margins to the dimensions of the printed area:
Length of poster = (x + 12) cm
Width of poster = (y + 8) cm
We can now set up another equation using the fact that the area of the entire poster is fixed at 384 cm²:
area = length * width
384 = (x + 12) * (y + 8)
Now we have two equations:
Equation 1: 384 = x * y
Equation 2: 384 = (x + 12) * (y + 8)
To find the dimensions that use the least amount of paper, we'll solve these equations simultaneously.
Using Equation 1, x * y = 384, we can solve for one variable in terms of the other, for example:
x = 384 / y
Substituting this expression for x in Equation 2:
384 = (384 / y + 12) * (y + 8)
Now simplify and solve this equation to find the value of y. Once you've found y, substitute it back into x = 384 / y to find the value of x. These values of x and y will give you the dimensions of the printed area that use the least amount of paper.