Question

The length of my rectangular calculator lid is 11 mm more than the width. The area is 152 square

mm. Find the dimensions of the lid.

Answer 1

The dimensions of the lid are 8mm by 19mm.

w = 8

l = 19

Explanation:

`l=w+11\\l* w=152`

where l is length and w is width. This can be solved as a system of equations.

`l* w=152\\(w+11)* w=152\\w(w+11)=152\\w^2+11w=152\\w^2+11w-152=0`

At this point, it gets a little tough. I might be unnecessarily overcomplicating things, but this is the only way I see to solve the problem.

=================== Skip down below if you don't care about factoring

You need to factor the newly created trinomial.

`ax^2+bx+c`

With a trinomial in this form, you need to find 2 numbers that add together to make b and multiply together to make ac.

Here, we need 2 numbers that add to 11 and multiply to -152. First, factor 152:

1, 152

2, 76

4, 38

8, 19

Then the reverse of all of those is true too, of course:

19, 8

38, 4

etc

In our case, we're looking for -152, so one of our factors will be negative. We're also looking for factors that add up to 11. Looking at these factors, you can see that 19 - 8 = 11, so our factors are 19 and -8.

Finally, you can use those to factor our trinomial. Split up the middle number (11w) into two:

`w^2+11w-152=0\\w^2-8w+19w-152=0`

And now, you can factor by grouping:

`w^2-8w+19w-152=0\\w(w-8)+19(w-8)=0\\(w+19)(w-8)=0`

===================

Now that the number is factored, you can finally find w:

`(w+19)(w-8)=0`

Here, you can see that the equation will be true when w = -19 or w = 8. Those are our solutions, but we can't have a negative distance, so it's just

`w=8`

Going all the way back to the top, now you can use the width to find the length.

`l=w+11\\l=8+11\\l=19`

That one was much easier.

The dimensions of the lid are 8mm by 19mm.

Finally, check that with both of the original equations to make sure it's correct.

`l=w+11\\19=8+11\\19=19\\\\l* w=152\\19*8=152\\152=152`