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A billboard designer has decided that a sign should have 5-ft margins at the top and bottom and 2-ft margins on the left and right sides. Furthermore, the billboard should have a total area of 700ft2
(including the margins).
A. O
Printed Region
If x denotes the width (in feet) of the billboard, find a function in the variable x giving the area of the printed region of the billboard,
Area, as a function of x
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Determine the domain of the function for area. Enter your answer using interval notation
Domain of the function for area
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The area as a function of x is: [tex]A(x) = \frac{4(x - 10)(175-x)}x[/tex].
The domain of x in interval notation is [tex][10, \infty)[/tex]

Given that:

[tex]x \to length[/tex]

[tex]y \to width[/tex]

So, the area is:

[tex]Area = Length \times Width[/tex]

[tex]A = x \times y[/tex]

[tex]x \times y =700[/tex]

Make y the subject:

[tex]y = \frac{700}x[/tex]

When the margins are removed, the dimension of the billboard is:

[tex]Length = x - 2\times 5[/tex]

[tex]Length = x - 10[/tex]

[tex]Width = y - 2 \times 2[/tex]

[tex]Width = y - 4[/tex]

The print area is calculated as:

[tex]A_p = (x - 10) \times (y -4)[/tex]

So, we have:

[tex]A_p = (x - 10) \times (\frac{700}x -4)[/tex]

Take LCM

[tex]A_p = (x - 10) \times (\frac{700-4x}x)[/tex]

[tex]A_p = \frac{(x - 10)(700-4x)}x[/tex]

So, as a function of x;

The print area is:

[tex]A(x) = \frac{(x - 10)(700-4x)}x[/tex]

[tex]A(x) = \frac{4(x - 10)(175-x)}x[/tex]

So, the area as a function of x is: [tex]A(x) = \frac{4(x - 10)(175-x)}x[/tex]

To determine the domain of x, I will plot the graph of A(x) (See attachment).

From the attached graph, we can see that the values of x starts from 10.

Hence, the domain of x in interval notation is [tex](10, \infty)[/tex]