Question
Question 3 Calculate the areas of both paintings. How are their areas to the scale factor?
Answers
GPT-4o mini
To calculate the areas of two paintings and how they relate to their scale factor, you'll need the dimensions of each painting and the scale factor itself.
1. **Calculate the Area of Each Painting**:
- For a rectangle (common shape for paintings), the area \( A \) can be calculated with the formula:
\[
A = \text{length} \times \text{width}
\]
- If you have paintings with different shapes, you'll need to use the appropriate formula (for example, area of a circle \( A = \pi \times r^2 \) or area of a triangle \( A = \frac{1}{2} \times \text{base} \times \text{height} \)).
2. **Calculate the Scale Factor**:
- The scale factor \( k \) is generally defined as the ratio of the dimensions of the larger painting to the smaller painting. It can be represented as:
\[
k = \frac{\text{dimension of larger painting}}{\text{dimension of smaller painting}}
\]
3. **Relationship of Areas to Scale Factor**:
- If the dimensions are scaled by a factor of \( k \), then the area scales by a factor of \( k^2 \). This means that if the dimensions of one painting are \( k \) times larger than the other, the area of that painting would be \( k^2 \) times larger than the area of the smaller painting.
4. **Example Calculation**:
Suppose Painting A has dimensions 10 cm by 20 cm and Painting B has dimensions 5 cm by 10 cm.
- Area of Painting A:
\[
A_A = 10 \, \text{cm} \times 20 \, \text{cm} = 200 \, \text{cm}^2
\]
- Area of Painting B:
\[
A_B = 5 \, \text{cm} \times 10 \, \text{cm} = 50 \, \text{cm}^2
\]
- Scale Factor:
\[
k = \frac{10}{5} = 2 \quad (\text{or using any corresponding dimension})
\]
- Relationship of Areas:
\[
\text{Area of Painting A} = k^2 \times \text{Area of Painting B} \implies 200 \, \text{cm}^2 = 2^2 \times 50 \, \text{cm}^2 \implies 200 \, \text{cm}^2 = 4 \times 50 \, \text{cm}^2
\]
If you provide the specific dimensions and details about the paintings, I can help you with the exact calculations.
1. **Calculate the Area of Each Painting**:
- For a rectangle (common shape for paintings), the area \( A \) can be calculated with the formula:
\[
A = \text{length} \times \text{width}
\]
- If you have paintings with different shapes, you'll need to use the appropriate formula (for example, area of a circle \( A = \pi \times r^2 \) or area of a triangle \( A = \frac{1}{2} \times \text{base} \times \text{height} \)).
2. **Calculate the Scale Factor**:
- The scale factor \( k \) is generally defined as the ratio of the dimensions of the larger painting to the smaller painting. It can be represented as:
\[
k = \frac{\text{dimension of larger painting}}{\text{dimension of smaller painting}}
\]
3. **Relationship of Areas to Scale Factor**:
- If the dimensions are scaled by a factor of \( k \), then the area scales by a factor of \( k^2 \). This means that if the dimensions of one painting are \( k \) times larger than the other, the area of that painting would be \( k^2 \) times larger than the area of the smaller painting.
4. **Example Calculation**:
Suppose Painting A has dimensions 10 cm by 20 cm and Painting B has dimensions 5 cm by 10 cm.
- Area of Painting A:
\[
A_A = 10 \, \text{cm} \times 20 \, \text{cm} = 200 \, \text{cm}^2
\]
- Area of Painting B:
\[
A_B = 5 \, \text{cm} \times 10 \, \text{cm} = 50 \, \text{cm}^2
\]
- Scale Factor:
\[
k = \frac{10}{5} = 2 \quad (\text{or using any corresponding dimension})
\]
- Relationship of Areas:
\[
\text{Area of Painting A} = k^2 \times \text{Area of Painting B} \implies 200 \, \text{cm}^2 = 2^2 \times 50 \, \text{cm}^2 \implies 200 \, \text{cm}^2 = 4 \times 50 \, \text{cm}^2
\]
If you provide the specific dimensions and details about the paintings, I can help you with the exact calculations.