Mr. Jones is opening a restaurant. A town ordinance states that his
restaurant sign must have an area no greater than 96 square feet.
1. Mr. Jones wants his restaurant's sign to have the greatest area
allowable. The dimensions of the sign will be whole numbers
greater than 1. What are the possible dimensions of the sign?
Explain.

(l = 2 ft, b = 48 ft ), (l = 4 ft, b = 24 ft ), (l = 8 ft, b = 12 ft ), (l = 16 ft, b = 6) (l = 32 ft, b = 3 ft ) are the set of possible length and breadth dimension for sign board.

Step-by-step explanation:

Let us assume the possible length of the sign board = a ft

And the possible width of the sign board = b ft

The maximum possible area covered by board = 96 sq ft.

Now,as we know, AREA OF A RECTANGLE = LENGTH x WIDTH

So, here:

Area of sign board = Possible ( length x Width) of board

⇒ 96 sq ft = a x b .......... (1)

Now, by prime factorization, 96 = 2 x 2 x 2 x 2 x 2 x 3

Now, also, 96 = 2 x 48 ⇒ a = 2 ft, b = 48 ft

= 4 x 24 ⇒ a = 4 ft, b = 24 ft

= 8 x 12 ⇒ a = 8 ft, b = 12 ft

= 16 x 6 ⇒ a = 16 ft, b = 6 ft

= 32 x 3 ⇒ a = 32 ft, b = 3 ft

As, all above mentioned values of a and b gives the area = 96 sq ft

Hence, these are the given possible dimensions of the sign board