1. A rectangle has the width x and its length is 10 greater than the width. For what values of x the area of the rectangle is greater or equal to 24

Question

1. A rectangle has the width x and its length is 10 greater than the
width. For what values of x the area of the rectangle is greater or equal
to 24. This is a word problem for a quadratic equation.

Answer 1

We can write an equation for the area in terms of x and then solve for x. An area of a rectangle is the width times the length.

 

A = x(x + 10) 

 

which must be greater than or equal to 24.  So we have

 

x(x + 10) ≥ 24

 

Solving the related equation for x will involve simplifying and using the quadratic formula.

 

x² + 10x - 24 = 0

 

x= {2, -12}. I bet you can see that one of those solutions is extraneous (outside the realm of possibilities). Now we can test x values that are less than 2 and greater than 2. A = 3(3+10) = 39 which is greater than 24. Bingo. x must be greater than or equal to 2. Note that smaller x values don't cut it as they give areas less than 24.