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To solve the problem, we need to find the values of \( a \) and \( b \) (the dimensions of the rectangular box) using the given diagram and information. Here's a breakdown of the key details:

### Key Information from the Diagram:
- The length of each semicircular cutter is given as 6 cm, which means the diameter of the full circular shape would be 12 cm.
- The two semicircles fit into the rectangular box side by side.
- The rectangular box has dimensions \( a \) cm (length) and \( b \) cm (width).
- The distance between the outer corners of the semicircles, marked by two radii, is 1.2 cm each.

### Strategy:
We can analyze the diagram geometrically to establish relationships between \( a \) and \( b \).

#### 1. Width of the Box (\( b \)):
- The total height of the rectangle will correspond to the diameter of one semicircle (6 cm) plus two small triangular gaps at the ends, each 1.2 cm.
- Therefore, \( b = 6 + 1.2 + 1.2 = 8.4 \) cm.

#### 2. Length of the Box (\( a \)):
- For the length \( a \), we need to account for the length of both semicircles and the two triangular gaps on the sides.
- The total length is equal to the diameter of one semicircle (12 cm).

Thus, the length \( a \) and width \( b \) are calculated as:
\[
a = 12 \, \text{cm}, \quad b = 8.4 \, \text{cm}
\]

Both answers should be given to 3 significant figures:
\[
a = 12.0 \, \text{cm}, \quad b = 8.40 \, \text{cm}
\]

Let me know if you'd like more details or clarifications. Here's a set of related questions to explore further:

1. How would the dimensions of the box change if the cutter's diameter were different?
2. How do you calculate the perimeter of the rectangular box?
3. How do the two triangular gaps affect the overall dimensions of the box?
4. Could you use trigonometry to solve any part of this problem?
5. How would you calculate the area of the box, and how does it relate to the cutters inside?

**Tip:** Always check the diagram carefully for hidden geometric relations when solving such problems.

A pair of cutters are kept together in a rectangular box of length a cm and width b cm. The cutters fit into the box as shown in Figure 2. Find the value of a and the value of b, giving your answers to 3 significant figures.

Solve a geometric problem involving two cutters arranged in a rectangular box. Learn how to calculate the box's dimensions using geometric relationships from the diagram and key formulas. Ideal for Grades 10-12 students.

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