A square rug has area 49x^2 – 56x +16 . A second square rug has area 16x^2 + 24x + 9. A) What expression can be written

Question

for the difference in rug areas? B) What significance does the variable x have with respect to the areas of the rugs, and C) what significance do the roots of the expression for the difference of areas have (values of x for which the expression is = 0)?

Answer 1

These are square rugs.

Rug1: area 49x^2 – 56x +16 This rug is (7x-4) units by (7x-4) units .

Rug2. area 16x^2 + 24x + 9. This rug is (4x+3) units by (4x+3) units

A) What expression can be written for the difference in rug areas?

(49x^2 – 56x +16) - (16x^2 + 24x + 9)

33x^2 - 32x + 7 units²

B) What significance does the variable x have with respect to the areas of the rugs

For Rug1, each dimension is increased by 7x units. For Rug2, each dimension is increased by 3x units.

C) what significance do the roots of the expression for the difference of areas have (values of x for which the expression is = 0)?

33x² -32x + 7 = ( 3x - 1 )*( 11x - 7 ) = 0

By the multiplicative property of 0, either x=1/3 or x=7/11 or both. Thus if x=1/3 or x=7/11, then the rugs are the same size (that is, the size difference is 0).

Answer 2

Let A be the area of the 1st rug and B the area of the 2nd rug.

a) The difference in areas is A - B =

(49x² - 56x + 16) - (16x² + 24x +9) =

(49-16)x² - (56-24)x + (16-9) =

33x² -32x + 7 units²

b) x determines the size of the difference between areas. If x=0, the A-B = 7 units²; if x=1 then A-B = 33-32+7 = 8 units²; if x=5 then A-B = 672 units²; etc.

c) x = [-b +/- sqrt(b² - 4ac)] / 2a, where a=33, b=-32 and c=7

Plugging in values for a, b and c we get x = [-(-32) +/- sqrt((-32)² - 4*33*7) / 2(33)

= [32 +/- sqrt(1024 - 924)] / 66

= [32 +/- sqrt(100)] / 66

= (32 +/- 10) / 66

= 42/66 or 22/66

Reduce to find roots are 7/11 and 1/3

Answer 3

Strategy: (A) is straightforward. (B) and (C) require a little thought!

Tactics: In passing, recognize that the rug areas are respectively (7x – 4)^2 and (4x + 3)^2, i.e. each is neatly parceled out as (expression)^2 .

(A) Obtain the difference in areas by subtraction: #1 - #2 = 33x^2 – 80x + 7 .

This factors into (11x – 1) * (3x – 7) , by the way.

(B) The variable “x” is just a dummy variable – it has no particular significance, except that it cannot be +(4/7) nor -(3/4) , so as to avoid “0” sized rugs (which would not be square, having no finite size). By inspection, the lengths of sides of the rugs variously correspond to +(expression), where (expression) > 0, or to -(expression), where (expression) < 0 , which comes to the same thing!

The difference expression, with zeros at x = (1/11) and at (7/3) , indicates x values for which the two square rugs are equal in size (this occurs at two different sizes for the pair of rugs!).

{{If you only indicated that a single value of x yielded identical sized rugs, could you be called on the carpet for it?}}