To solve the problem shown in the image, let's break it into two parts: calculating $a$ and then using $a$ to find $b$.

Step (a): Calculate $a$

From the diagram, $a$ represents the diagonal of the rectangular face of the prism. The rectangle's dimensions are $12 \, \text{in}$ and $5 \, \text{in}$. To calculate $a$, we use the Pythagorean theorem:

$a = \sqrt{12^2 + 5^2}$

$a = \sqrt{144 + 25}$

$a = \sqrt{169}$

$a = 13 \, \text{in}$

So, $a = 13 \, \text{in}$.

Step (b): Calculate $b$

The length $b$ represents the diagonal of the triangular face formed by $a$ and the other side $9 \, \text{in}$. Again, we use the Pythagorean theorem:

$b = \sqrt{a^2 + 9^2}$

Substitute $a = 13$:

$b = \sqrt{13^2 + 9^2}$

$b = \sqrt{169 + 81}$

$b = \sqrt{250}$

$b \approx 15.8 \, \text{in}$

Final Answers:

• $a = 13 \, \text{in}$
• $b = 15.8 \, \text{in}$

Would you like additional details or have any questions about this?

Here are 5 related questions for further exploration:

1. How does the Pythagorean theorem apply in three-dimensional problems like this one?
2. Could this problem be solved using trigonometry instead? How?
3. What is the significance of rounding in practical scenarios involving measurements?
4. How would the calculation change if one of the dimensions was modified?
5. What are the practical uses of diagonal calculations in design or packaging?

Tip: When solving problems with multiple steps, always double-check intermediate calculations to avoid errors in later steps!