A rectangular garden is made using a wall as one side of the rectangle. What are the dimensions if we have 16ft of fencing and area is 30 square feet?

Question

I need help on getting up and solving an equation for this word problem. 

Answer 1

So, since the wall is being used as one side, we only need to find out how the rest of the fencing is used, on the other 3 sides of the rectangle.

 

We know that the square footage is equal to 30, and the total amount of fencing we have is 16 feet. So, 2 times the width of the rectangle, plus the length of the rectangle is equal to the total amount of fencing:

 

2w + l = 16

 

We also know that square footage is equal to the length times the width, so:

 

w x l = 30

 

Let's solve for l in the first equation:

 

2w + l = 16

l = 16 - 2w

 

Now let's put the right side of the equation we just found in as l in the second equation:

 

l = 16 - 2w

w x l = 30                               Substitute

w x (16 - 2w) = 30                  Distribute

-2w² + 16w = 30                     Subtract 30 from each side

-2w² + 16w - 30 = 0                Divide by -2

w² - 8w + 15 = 0                     Factor

(w - 5)(w - 3) = 0                    Set each of the parentheses equal to zero

w = 5 feet, 3 feet

 

We have 2 answers for w, so let's plug them in to our very first equation:

 

2w + l = 16                           Use 5 for w

2(5) + l = 16                         Simplify

10 + l = 16                            Subtract 10 from both sides

l = 6

 

2w + l = 16                           Use 3 for w

2(3) + l = 16                         Simplify

6 + l = 16                             Subtract 6 from both sides

l = 10

 

So the dimensions of the rectangular garden can either be 5ft by 6ft, or 3ft by 10ft.