Worksheet problem involving matrix multiplication to calculate total scores in a competition.

It looks like you've uploaded an image containing a math problem related to the New England Stony and Wood Carving Fair, where judges award points for each competition, and there's a difficulty factor that affects the final score.

Let me reconstruct and solve the problem based on what is visible in your image:

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Problem Description:

There are two tables:

#### Table 1: Points awarded by judges
| Category | Wood Carving | Stone Carving |
|----------------|--------------|---------------|
| Sculpture | 18 | 25 |
| Relief | 17 | 19 |
| Carving | 15 | 16 |

#### Table 2: Degree of difficulty
| Category | Wood Carving | Stone Carving |
|----------------|--------------|---------------|
| Sculpture | 1.05 | 1.15 |
| Relief | 1.05 | 1.10 |
| Carving | 1.00 | 1.05 |

> To find total score for a contestant, multiply the points awarded by the degree of difficulty.

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Task:

a) Create matrices to organize the given information.
b) Find the total scores for each contestant.

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## ✔ Solution:

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Step 1: Part (a) – Create matrices

We need to represent both the points and the difficulty as matrices.

Let’s define:

- Matrix P: Points awarded
- Matrix D: Difficulty factors

Since we have 3 categories (Sculpture, Relief, Carving) and 2 types of carving (Wood, Stone), we can arrange them as 3 × 2 matrices.

#### Matrix P (Points):
$$
P =
\begin{bmatrix}
18 & 25 \\
17 & 19 \\
15 & 16 \\
\end{bmatrix}
$$

Rows: Categories (Sculpture, Relief, Carving)
Columns: Types (Wood Carving, Stone Carving)

#### Matrix D (Difficulty):
$$
D =
\begin{bmatrix}
1.05 & 1.15 \\
1.05 & 1.10 \\
1.00 & 1.05 \\
\end{bmatrix}
$$

Same structure: Rows = Categories, Columns = Types

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Step 2: Part (b) – Find total scores for each contestant

To get the total score, we multiply each point by its corresponding difficulty factor.

So, we compute:
$$
\text{Total Score} = P \times D \quad \text{(element-wise multiplication)}
$$

But note: This is not matrix multiplication in the traditional sense — it's Hadamard product (element-wise). Since both matrices are the same size (3×2), we multiply corresponding entries.

Let’s compute each element:

#### Sculpture – Wood Carving:
$ 18 \times 1.05 = 18.9 $

#### Sculpture – Stone Carving:
$ 25 \times 1.15 = 28.75 $

#### Relief – Wood Carving:
$ 17 \times 1.05 = 17.85 $

#### Relief – Stone Carving:
$ 19 \times 1.10 = 20.9 $

#### Carving – Wood Carving:
$ 15 \times 1.00 = 15.0 $

#### Carving – Stone Carving:
$ 16 \times 1.05 = 16.8 $

Now, put these into a matrix:

$$
\text{Total Score Matrix} =
\begin{bmatrix}
18.9 & 28.75 \\
17.85 & 20.9 \\
15.0 & 16.8 \\
\end{bmatrix}
$$

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✔ Final Answer:

#### (a) Matrices:

- Points Matrix $ P $:
$$
\begin{bmatrix}
18 & 25 \\
17 & 19 \\
15 & 16 \\
\end{bmatrix}
$$

- Difficulty Matrix $ D $:
$$
\begin{bmatrix}
1.05 & 1.15 \\
1.05 & 1.10 \\
1.00 & 1.05 \\
\end{bmatrix}
$$

#### (b) Total Scores (after multiplying):

$$
\begin{bmatrix}
18.9 & 28.75 \\
17.85 & 20.9 \\
15.0 & 16.8 \\
\end{bmatrix}
$$

This means:

- Sculpture: Wood Carving → 18.9, Stone Carving → 28.75
- Relief: Wood Carving → 17.85, Stone Carving → 20.9
- Carving: Wood Carving → 15.0, Stone Carving → 16.8

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📌 Summary:

- We used matrices to organize data.
- We performed element-wise multiplication (Hadamard product) to compute total scores.
- Each total score = (Points) × (Difficulty).

Let me know if you'd like this formatted as a table or need further explanation!