SOLUTION: I really need your help in solving this problem. I appreciate your time and efforts. Have a terrific day :) Here is the question? Tubing A piece of cardboard in the shape of a par

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: I really need your help in solving this problem. I appreciate your time and efforts. Have a terrific day :) Here is the question? Tubing A piece of cardboard in the shape of a par      Log On


   

Question 917290: I really need your help in solving this problem. I appreciate your time and efforts. Have a terrific day :) Here is the question?
Tubing A piece of cardboard in the shape of a parallelogram is twisted to form the tube. The parallelogram has an area of 60 square inches. If its height h, is 7 inches more than the length of the base b, what is the length of the base? (Hint: The Formula for the area of a parallelogram is A = bh.) I Really need your help in solving this!!!!! Thank you so much.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There is a parallelogram with a "base" measuring b inches and a "height" measuring h=b%2B7 inches.
The area of that parallelogram, in square inches, calculated as A=b%2Ah is
A=b%28b%2B7%29 and we know that is 60 square inches, so our equation is
b%28b%2B7%29=60 .
That is, indeed, a quadratic equation.
It be written in many equivalent forms, like
b%28b%2B7%29=60<--->b%5E2%2B7b=60<--->b%5E2%2B7b-60=0
and infinitely many other ways.

There are many ways to solve the quadratic equation.
One of them if "by factoring", which involves knowing factors of 60.
A fifth grader that realizes that 5%2A12=60
would figure that highlight%28b=5%29 with b%2B7=5%2B7=12 is a solution.
Your teacher, however,
would expect you to start from the b%5E2%2B7b-60=0 form of the equation,
factor the left hand side into two factors to get
%28b%2B12%29%28b-5%29=0 and find that the solutions of the equation are
b=-12<--->b%2B12=0 . which makes the first factor zero, and
highlight%28b=5%29<--->b-5=0 which makes the second factor zero.
Since the measurement of the base is a positive number, b=-12 makes no sense, highlight%28b=5%29 is the only sensible answer.
The length of the base ishighlight%285inches%29 .

You could also solve the quadratic equation by "completing the square" or by using the quadratic formula.
Completing the square:
b%5E2%2B7b=60
b%5E2%2B7b%2B%287%2F2%29%5E2=60%2B%287%2F2%29%5E2
%28b%2B7%2F2%29%5E2=60%2B49%2F4
%28b%2B7%2F2%29%5E2=240%2F40%2B49%2F4
%28b%2B7%2F2%29%5E2=289%2F4
%28b%2B7%2F2%29%5E2=%2817%2F2%29%5E2
So,
either b%2B7%2F2=17%2F2-->b=17%2F2-7%2F2-->b=10%2F2-->highlight%28b=5%29 ,
or b%2B7%2F2=-17%2F2-->b=-17%2F2-7%2F2-->b=-24%2F2-->b=-12 , which is not a good length for the base of a parallelogram.
Using the quadratic formula, which says that x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ gives the solutions to ax%5E2%2Bbx%2Bc=0 .
Applied to x%5E2%2B7x-60=0 , a=1 b=7 and c=-60 .
So the solutions to x%5E2%2B7x-60=0 are given by
.
So,
either x=%28-7+%2B+17%29%2F2=10%2F2=5 ,
or x=%28-7+-+17%29%2F2=%28-24%29%2F2=-12 .
So, since the solutions to x%5E2%2B7x-60=0 are x=5 and x=-12 ,
the solutions to b%5E2%2B7b-60=0 are b=5 and b=-12 . (I just change the name of the variable used for the base so that ift would not get confused with the b in x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+and ax%5E2%2Bbx%2Bc=0 ).

NOTES:
What is the base and what is the height of a parallelogram (or a triangle) is
"in the eye of the beholder". We think of the base as what we stand something on so that it will stay stable, and the height as the perpendicular measurement to the farthest point from the base. However, for calculation purposes, we use as base whatever makes calculations easier.
Your parallelogram looks like this:
so it looks a bit unstable if you stand it on that "base".