Question 3 Calculate the areas of both paintings. How are their areas to the scale factor?

1 answer

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.