Question is in the description.

Question

Involves The quadratic Formula, I know how to use this formula i just need to know how to get to that point.

Question: A rectangular garden has dimensions of 20 feet by 15 feet. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 234 square feet?

Answer 1

Hi Tyler,

 

Since you are given that the area of the path is 234 sq. ft., subtract the area of the garden from the area of the garden including the path. The difference is 234 square feet. Your equation would be

          (20+2x)(15+2x) - 20(15) = 234   where x is the width of the path. Remember that the path is on both sides of the length and the width. That is why the length is (20+2x) and not (20+x) and similarly for the width.

Multiplying the two binomials, we get

           300+40x+30x+4x²- 300 = 234   Combine like terms to get

                4x² +70x = 234

               4x² + 70x - 234 = 0  Now use the quadratic formula.

Answer 2

Draw a rectangle.  Label it 20 feet on the long side, and 15 feet on the short side.  (I have mine with 15 feet on top and bottom, while the 20 feet are the left and right sides.  This will help as I explain below if you set yours up the same.)

 

Now, draw a bigger rectangle around the 20 x 15 rectangle,  the 20 x 15 rectangle should be completely inside of the new rectangle.  Make sure that there is the same amount of space between the two rectangles on all four sides.  Label this width (the space between the rectangles) "x".

 

The amount of gravel you need is the area of the space between these two rectangles.  This space between the smaller inside rectangle (20 x 15) and the outer rectangle [ (20 + 2x) x (15 + 2x)] is the walkway.

 

If you break the space between the two larger rectangles into 4 skinny rectangles:  (I wish I could draw a picture for you)  But do it carefully.  If you extend the vertical sides of the inner rectangle to the top and bottom of the outer rectangle.  You should get 4 skinny rectangles.  Two vertical skinny rectangles that look like "|" and two narrow  horizontal rectangles that look like "-".

 

The dimensions of the small horizontal rectangles are 15 * x.  There are 2 of them, one on top and one on bottom.

The dimensions of the tall vertical rectangles are (20+2x)(x)  There are 2 of these, one on each side.

 

Add all of this area and it must = 234.

 

15x   +   15x    + (20+2x)(x) + (20+2x)(x) = 234

(top) +    (bot)  +    (left)       +   (right)      = total area of gravel walkway

 

Distribute and combine like terms:

 

30x + 20x + 2x² + 20x + 2x² = 234

 

Combine like terms again

 

4x² + 70x = 234

 

Put everything on the left side

 

4x² + 70x - 234 = 0

 

Now use the Quadratic formula.