Question

A sheet of paper contains 18 square feet. The top and bottom margins are 9inches and the side margins are 6 inches. What are the dimensions of the pagethat has the largest printed area?

Answer 1

Answer:

The dimensions of the page are

3.46 ft by 5.20 ft

Step-by-step explanation:

Let

x---> the length of the sheet of paper in feet

y ---> the width of the sheet of paper in feet

$A=xy$

$A=18\ ft^2$

so

$18=xy$

$y=\frac{18}{x}$ -----> equation A

Remember that

$1\ ft=12\ in$

Convert the margins into feet

$9\ in=9\12=0.75\ ft$

$6\ in=6\12=0.50\ ft$

so

we know that

The area of the largest printed area is given by

$A=(y-0.75-0.75)(x-0.50-0.50)$

$A=(y-1.50)(x-1)$

$A=xy-y-1.50x+1.50$

substitute equation A in the above expression

$A=x(\frac{18}{x})-\frac{18}{x}-1.50x+1.50$

$A=18-\frac{18}{x}-1.50x+1.50$

$A=19.50-\frac{18}{x}-1.50x$

Now we have an output (A) in terms of only one variable (x),

so

we differentiate:

$\frac{dA}{dx}=\frac{18}{x^2}-1.50$

equate to zero

$\frac{18}{x^2}=1.50$

$x^2=12\\x=3.46\ ft$

Find the value of y

$18=(3.46)y\\y=5.20\ ft$

therefore

The dimensions of the page are

3.46 ft by 5.20 ft