what is the width if they keep 45 square feet out of a 15 by 19 rectangle

Question

A rectangular plot, 15 ft by 19 ft, is to be used for a garden. It is decided to put a pavement inside the entire border so that 45 square feet of the plot is left for flowers. How wide should the pavement be?

Answer 1

Hi Mat:

 

If you draw a rectangle with  L = 19 and W = 15 feet

Then draw a smaller rectangle inside that, x feet wide,

you can see the new Length is 19-2x and the new width

is 15-2x

 

Since Area = LW we have:

 

(-2x+15)(-2x+19) = 45

4x² - 68x + 285 = 45

4x² - 68x + 240 = 0

We can factor a 4 out of each term so:

x² - 17x + 60 = 0

 

Now we have a standard quadratic.  Using the quadratic

formula  x = [-b ±√(b²-4a)]/2a     a=1, b=-17, c=60

 

x = [17 ±√((-17)²-4(60))]/2 = (17 ±√49)/2 = (17±7)/2

 

x = 24/2 = 12

or

x = 10/2 = 5

 

Check: 

If x = 12 our dimensions are:

19-2(12) by 15-2(12) = -5 by -9 = 45 ft² but

we cannot have negative values for length and width

 

If x=5 our dimensions are:

19-2(5) by 15-2(5) = 9 by 5 = 45 ft²

 

Answer:  x = 5 ft wide for the pavement