SOLUTION: a rectangular garden has a perimeter of 44m. A walkway 2m wide surrounds the garden. The combined area of the garden and walkway is 224m2. What are the dimensions.

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: a rectangular garden has a perimeter of 44m. A walkway 2m wide surrounds the garden. The combined area of the garden and walkway is 224m2. What are the dimensions.      Log On


   

Question 188725This question is from textbook
: a rectangular garden has a perimeter of 44m. A walkway 2m wide surrounds the garden. The combined area of the garden and walkway is 224m2. What are the dimensions. This question is from textbook

Answer by jojo14344(1513) About Me  (Show Source):
You can put this solution on YOUR website!


Let's picture out the garden wit the walkway.


We know for Rectangle, ---> Perimeter+=+2%28L%2BW%29, Eqn 1
44m=2%28L%2BW%29
cross%2844%2922%2Fcross%282%29=cross%282%29%28L%2BW%29%2Fcross%282%29
red%28L=22-W%29, Eqn 2

Also we know, Area+=+L%2AW, Eqn 3

Now for the Combined Area (Garden + Walkway), we need to add "4 meters" for each Length and Width as shown.
New Length= L+2m+2m= L+4m; New Width= W+2m+2m = W+4m

Then going back Eqn 3, for the combined Area:
A=%28L%2B4%29%28W%2B4%29, substitute Eqn 2:
224=%28highlight%2822-W%29%2B4%29%28W%2B4%29
224=%2826-W%29%28W%2B4%29
Expand right terms (multiply):
First terms: %28highlight%2826%29-W%29%28highlight%28W%29%2B4%29=red%2826W%29
Outer terms: %28highlight%2826%29-W%29%28W%2Bhighlight%284%29%29=red%28104%29
Inner terms: %2826-highlight%28W%29%29%28highlight%28W%29%2B4%29=red%28-W%5E2%29
Last terms: %2826-highlight%28W%29%29%28W%2Bhighlight%284%29%29=red%28-4W%29

Adding all results in red mark:
224=highlight%2826W%2B104-W%5E2-4W%29
-W%5E2%2B22W%2B104-224=0
-W%5E2%2B22W-120=0, divide by "-1" becomes---> W%5E2-22W%2B120=0, working eqn
*Remembersystem%28a=1%2Cb=%28-22%29%2Cc=120%29
Solving discriminant: b%5E2-4ac=-22%5E2-4%281%29%28120%29=484-480=red%284%29

Then, by quadratic:
W=x=%28-b%2B-sqrt%284%29%29%2F%282a%29=%28-%28-22%29%2B-2%29%2F%282%2A1%29=%2822%2B-2%29%2F2
W=%2822%2B2%29%2F2=24%2F2=red%2812%29
W=%2822-2%29%2F2=20%2F2=red%2810%29

Width of the Garden can either be W[1]=12m or W[2]=10m.

For Length, go back Eqn 2:
L=22-W=22-12=red%2810m%29
Or
L=22-W=22-10=red%2812m%29

Length can either be L[1]=10m or L[2]=12m

Let's check with the combined Area:
A%5BG%2BW%5D=+%28L%5B1%5D%2B4%29%28W%5B1%5D%2B4%29
224m%5E2=%2810%2B4%29%2812%2B4%29
224m%5E2=%2814%29%2816%29
224m%5E2=224m%5E2

Thank you,
Jojo