https://file.aiquickdraw.com/mathbot/file/17282746928126xikq6x6.jpg

To solve this problem, let's break it down step by step:

### (a) Find \( a \)

The diagram shows that we need to find the value of \( a \), which forms the base of a right triangle. The two sides of this triangle are given as 6 inches and 8 inches.

We can use the **Pythagorean theorem** to find \( a \). According to the theorem:

\[
a^2 = 6^2 + 8^2
\]
\[
a^2 = 36 + 64 = 100
\]
\[
a = \sqrt{100} = 10 \text{ inches}
\]

So, \( a = 10 \) inches.

### (b) Find \( b \)

Now, \( b \) is the diagonal of the three-dimensional shape, which means we have another right triangle where one leg is \( a = 10 \) inches (the base), and the other leg is the height of 3 inches.

Again, using the Pythagorean theorem:

\[
b^2 = a^2 + 3^2
\]
\[
b^2 = 10^2 + 3^2 = 100 + 9 = 109
\]
\[
b = \sqrt{109} \approx 10.4 \text{ inches}
\]

Thus, \( b \approx 10.4 \) inches when rounded to the nearest tenth.

Would you like further clarification on any part of this process?

Here are 5 related questions:
1. What is the Pythagorean theorem, and how is it applied in 3D geometry?
2. How do you determine the hypotenuse in a right triangle?
3. What is the relationship between a diagonal in a rectangular prism and its edges?
4. Can the Pythagorean theorem be extended to higher dimensions?
5. How would the solution change if the dimensions of the gift box were different?

**Tip:** When solving problems involving 3D shapes, breaking the problem into smaller 2D problems (using right triangles) simplifies the calculations.

A gift box has the shape of a right triangular prism. A piece of tape of length b (in inches) is placed from one corner to the opposite corner, as shown in the figure. (The figure is not drawn to scale.) (a) Find a. (b) Use your answer to part (a) to find b, the length of the tape. Round your answer to the nearest tenth of an inch.

Learn how to solve a word problem involving a triangular prism and the Pythagorean theorem. Calculate the length of the tape placed diagonally across the prism by first finding a side of the right triangle and then applying the theorem in 3D. Suitable for students in Grades 8-10.

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