Answer:
The dimensions of the lid are 8mm by 19mm.
w = 8
l = 19
Explanation:
![l=w+11\\l* w=152](https://img.qammunity.org/2022/formulas/mathematics/college/hrg2mer48rhs1b9emlubqw63ctav4ap3hu.png)
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](https://img.qammunity.org/2022/formulas/mathematics/college/l80adq3rbqk2tudoifh3bn5g95a5g30j1d.png)
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](https://img.qammunity.org/2022/formulas/mathematics/college/ls8sd3d86nio6ip6c65icmceeplhmx8zo2.png)
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](https://img.qammunity.org/2022/formulas/mathematics/college/anr94h01rq4fcifcjy7lvojhgu8aojcspf.png)
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](https://img.qammunity.org/2022/formulas/mathematics/college/nulw5ntalozm85m0arphrore5wi2wsfq1j.png)
===================
Now that the number is factored, you can finally find w:
![(w+19)(w-8)=0](https://img.qammunity.org/2022/formulas/mathematics/college/pq2wgnp44qp8ihyr5agly7d0ppfpuzdexq.png)
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](https://img.qammunity.org/2022/formulas/mathematics/high-school/nw89emm713mppbnxqhse6mruba1cwvd38q.png)
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](https://img.qammunity.org/2022/formulas/mathematics/college/uiq3yzufaxrzu61a0i4xlefvxq90vaffyb.png)
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](https://img.qammunity.org/2022/formulas/mathematics/college/tbtgpcsvcyxspuz2xyg52o15wjydmy9gw2.png)