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                        

x= -3 and

step-by-step explanation:

to find the roots of a cubic, use division or factoring to find the factors then set equal to 0. as this is a difference of cubes, it factors into

where a is the cube root of .

here   this is 27 and the cube root of 27 is 3. so we write using the form:

.

set each factor to 0 and solve for x.

x-3=0

x=3

the quadratic expression does not factor and must be solved using the quadratic formula.

where a = 1, b=3, and c=9

this cubic has two complex roots.

for the friend request.

when a figure, such as triangle (xyz) is reflected over the line y = x to create x'y'z' the x-coordinates and y-coordinates of the original image (pre-image) will be reversed for the image. (e.g., (x, y) > (y, x)) this means that the pre-image position of the x- and y-coordinates switch places which in turn provides a mirror like image of the original xyz.  

i notice that when i draw a line segment from x to x', that the points are parallel from each other -- they are exactly across from each other and do not intersect with any other points.  

yes, think i would see the same characteristic if you drew the line segment connecting y with the reflecting line and then y' with the reflecting line because of the essence of reflections. because it is a mirror image, each point was effected the same way which leads me to conclude that they same would apply to y and y'.