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

The problem asks how much the fountain's perimeter increases for every 1-square-foot increase in the area of the surrounding path, based on the equation A = 4p + 64.

Here are three ways to find the answer, which is 1/4.

Method 1: The Slope Approach

In algebra, when an equation is written to show how one thing changes in relation to another, the number attached to the variable is the rate of change.

1. Move the numbers around to solve for p:
2. A - 64 = 4p
3. Divide everything by 4:
4. A/4 - 16 = p
5. Looking at A/4, which is the same as 1/4 times A, we can see that for every 1 unit A goes up, p only goes up by 1/4.

Method 2: Using Example Numbers

If you plug in real numbers, the relationship becomes clear:

1. Start: If the perimeter p is 10, then the area A is 4 times 10 plus 64, which equals 104.
2. Change: If we increase the area A by 1 square foot so it becomes 105, we solve for the new perimeter: 105 = 4p + 64.
3. Result: Subtracting 64 gives us 41 = 4p. Dividing 41 by 4 gives us a new perimeter of 10.25.
4. Difference: The perimeter grew from 10 to 10.25, an increase of 0.25 or 1/4.

Method 3: Grouping the Change

You can see the relationship by looking at how the math balances when you add 1 to the area.

As shown in the expert solutions on Wyzant, adding 1 to the area looks like this:

1. A + 1 = 4p + 64 + 1

To keep the 4p structure of the original design, we rewrite the extra 1 as 4 times 1/4:

1. A + 1 = 4 * p + 1/4 + 64

This confirms that the new perimeter is the old perimeter plus 1/4.

Answer 4

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