Question

A graphic designer wants to create a rectangular graphic that has a 2-inch margin on each side and a 4-inch margin on the top and the bottom. The design, including the margins, should have an area of 392 square inches. What overall dimensions will
maximize the size of the design, excluding the margins?
(Hint: If one side of the design is x, then the other side is
392 divided by x.)

Answer 1

The area of a shape is the amount of space the shape can occupy.

The dimension that maximizes the design is 20 inches by 20 inches

From the question, we have:

$Area = 392in^2$

Let the length and width of the graphic (without the margin) be l and w.

So, the dimension of the whole graphic is:

$L = l + 2 + 2 = l +4$

$W = w + 4 + 4 = w +8$

The area is then calculated as:

$Area = L \times W$

$Area = (l + 4) \times (w + 8)$

Substitute $Area = 392in^2$

$392 = (l + 4) \times (w + 8)$

Make l + 4, the subject

$l + 4 =\frac{392}{w + 8}$

Subtract 4 from both sides

$l =\frac{392}{w + 8} - 4$

The area of the design is:

$A= lw$

So, we have:

$A =(\frac{392}{w + 8} - 4)w$

$A =\frac{392w}{w + 8} - 4w$

Plot the graph of $A =\frac{392w}{w + 8} - 4w$

From the graph (see attachment), the maximum is at:

$(w,A) = (20,200)$

Substitute $(w,A) = (20,200)$ in $A= lw$

$200 = l \times 20$

Divide both sides by 20

$l = 20$

Hence, the dimension that maximizes the design (excluding the margin) is 20 inches by 20 inches

Read more about maximizing areas at:

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