SOLUTION: Aleksandra has a 4 meter by 5 meter rectangular carrot garden. She would like to add a stone path around the entire garden. She has 22 square meters of stone available. What is the

Algebra ->  Surface-area -> SOLUTION: Aleksandra has a 4 meter by 5 meter rectangular carrot garden. She would like to add a stone path around the entire garden. She has 22 square meters of stone available. What is the      Log On


   

Question 135792: Aleksandra has a 4 meter by 5 meter rectangular carrot garden. She would like to add a stone path around the entire garden. She has 22 square meters of stone available. What is the widest that she can make the path? It gives a hint that says Write an equation for the area. I have no idea what formula to use for this problem. I believe I have the correct topic listed above.
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
The formula is just the formula for the area of a rectangle, namely A=LW. The thing you have to deal with is that you actually have two rectangles, first is the 4m by 5m garden, and the second is the outside edge of the walkway around it. Presuming that Alexsandra intends to build the walkway of uniform width all the way around, you can say that the width of the walkway is x. That means that 2x is added to the dimensions of the garden to get the dimensions of the outside edge of the walkway -- x on either side, x on the top, x on the bottom. Next you have to realize that the actual surface area of the walkway is the area of the rectangle defined by the outer edge of the walkway MINUS the area of the garden.



Area of the garden A=4%2A5=20

Area of the rectangle defined by the outer edge of the walkway A=%282x%2B4%29%282x%2B5%29=4x%5E2%2B18x%2B20

Area of outer rectangle MINUS Area of garden A=4x%5E2%2B18x and this area must be equal to 22 square meters because that is the amount of walkway material available.

4x%5E2%2B18x=22

4x%5E2%2B18x-22=0, in standard form

2x%5E2%2B9x-11=0, after dividing by 2 to eliminate a common factor

%282x%2B11%29%28x-1%29=0, after factoring

2x%2B11=0 or x-1=0

x=-%282%2F11%29 or x=1. The negative root makes no sense when we are looking for a measure of length, so exclude that root as extraneous. Therefore, x=1 meaning the walkway is 1 meter wide.

Check:
If the walkway is 1 meter wide, then the outside dimensions of the walkway must be (4 + 2 = 6) and (5 + 2 = 7). 6 * 7 = 42, less the area of the actual garden, 20, leaves 22 square meters of area for the walkway.