SAT Algebra question (Heart of Algebra)

Question

A landscaper is designing a rectangular fountain with a 4-foot-wide path around it. The equation A = 4p + 64 will relate the area A, in square feet, of the path to the perimeter p, in feet, of the fountain. In the design, how many feet will the perimeter of the fountain increase for each additional square foot of the path’s area?

The answer is 1/4, but I do not know how. Anyone please show me with every vital steps? Thank you!

Answer 1

Area of initial design:  A = 4p + 64

 

Let x = increase in path's area

     y = increase in fountain's perimeter

 

Find y/x  ["how many feet will the perimeter of the fountain increase for each additional square foot of the path’s area?"]  This is a unit value ["how many of these for one of those"].

 

A+x = 4(p+y) + 64                       [increased values]

(4p + 64) + x = 4p + 4y + 64      [substitute for A]

   x = 4y                                        

   1/4 = y/x

 

The fountain's Perimeter increases 1/4 foot for each 1 square foot increase of path's Area.

 

 

Answer 2

A = 4p + 64

A + 1 = 4p + 64 + 1        area increases by 1

A + 1 = 4p + 1 + 64        commutative property

A + 1 = 4(p + 1/4) + 64   factor out 4

 

When the area increase by 1 the perimeter increases by 1/4.

Answer 3

Another way to look at the problem is:

 

             A = 4p + 64

 

      A - 64 = 4p

 

Divide both sides by 4

 

  (A/4) - 16 = p

      

 

If 1 is added to A, then you are increasing the fraction A/4 by 1/4;

 

Therefor the perimeter has been increased by 1/4