algebra story problem help!

Question

Bill and Ann want to enlarge their family room. Currently the family room has an area of 120 square feet. If they increase the length by 4 feet and the width by 5 feet, the new family room will have an area of 240 square feet. What are the dimensions of the family room now, and what will the new dimensions be?

 

Recall: A= L x W

L x W =120 

(L+4)(W+5)=240

Answer 1

You have the right set up.  What you need to do next is expand the last equation using FOIL.

 

(L + 4)(W + 5) = 240

 

LW + 5L + 4W + 20 = 240

 

 

Subtract 20 on both sides of the equation to simplify.

 

LW + 5L + 4W = 220

 

 

You know that  LW is 120 from the original area.  So we substitute that into the expanded equation.

 

120 + 5L + 4W = 220

 

 

Subtract 120 on both sides of the equation.

 

5L + 4W = 100

 

 

Next, substitute the first equation into the second equation once more, so that it is in terms of one variable.

 

5L + 4(120 / L) = 100

 

 

Solve for L:

 

5L² + 480 = 100L

 

 

Subtracting 100L on both sides of the equation, you get

 

5L² - 100L + 480 = 0

 

 

Then you can factor the left side of the equation extracting the GCF.

 

5(L² - 20L + 96) = 0

 

   L² - 20L + 96 = 0

 

 

Now we have a simple quadratic equation that can be easily solve using either FOIL or the quadratic formula.  I recommend using the quadratic formula. Can you continue from here?

Answer 2

 

Solve the system of equations you constructed.

L x W =120
(L+4)(W+5)=240

 

Solving the first equation for W ... W = 120/L

 

Expand the 2nd equation ... then substitute ... LW = 120 ... W = 120/L

 

LW + 5L + 4W + 20 = 240

120 + 5L + 4(120/L) = 220

5L + 480/L = 100

 

Clear the fraction by multiplying all terms by "L"

 

5L² + 480 = 100L

 

Divide all terms by 5, to simplify

 

L² + 96 = 20L

L² -20L + 96 = 0

(L - 8)(L - 12) = 0

 

L - 8 =0 ... L = 8 ... W = 15, because 8 x 15 = 120 ... discard this .... length cannot be less than the width

 

L - 12 = 0 ... L = 12 ... W = 10, because 12 x 10 = 120

 

the original dimensions were 12 ft long and 10 ft wide ... area is 120

 

Add 4 to the length and 5 to the width to obtain the new dimensions.

 

the new dimensions are 16 ft long and 15 ft wide ... area is 240