1. Problem b) Determine the dimensions and volume of the box if $x=20$. Assuming the box dimensions depend on $x$, we substitute $x=20$ into the given expressions for length, width, and height (not provided explicitly, so we assume dimensions are functions of $x$). 2. Calculate the volume $V$ using the formula: $$V = \text{length} \times \text{width} \times \text{height}$$ 3. Substitute $x=20$ into each dimension and multiply to find the volume. --- 4. Problem 6: Prove that the sum of the squares of any two consecutive integers is always an odd number. 5. Let the two consecutive integers be $n$ and $n+1$. 6. Compute the sum of their squares: $$n^2 + (n+1)^2 = n^2 + n^2 + 2n + 1 = 2n^2 + 2n + 1$$ 7. Factor out 2 from the first two terms: $$2(n^2 + n) + 1$$ 8. Since $n^2 + n = n(n+1)$, which is always even because one of $n$ or $n+1$ is even, $n^2 + n$ is even. 9. Therefore, $2(n^2 + n)$ is divisible by 4, hence even. 10. Adding 1 to an even number results in an odd number. 11. Thus, the sum $n^2 + (n+1)^2$ is always odd. Final answers: - For $x=20$, dimensions depend on the box formula (not provided). - The sum of squares of any two consecutive integers is always odd.