To find the dimensions of Tiana's painting, we first need to determine the area of Margaret's painting:
- Margaret's painting dimensions: 12 inches long and 18 inches wide.
- Area of Margaret's painting = length × width = 12 inches × 18 inches = 216 square inches.
Tiana's painting has a total area of 600 square inches and is similar in shape to Margaret's painting. Since the two paintings are similar, the ratio of their areas will equal the square of the ratio of their corresponding dimensions.
Let's denote the ratio of the dimensions between Tiana's painting and Margaret's painting as \( k \). Then the area ratio can be expressed as:
\[ \frac{\text{Area of Tiana's painting}}{\text{Area of Margaret's painting}} = k^2 \]
Substituting the known areas:
\[ \frac{600}{216} = k^2 \]
Now, simplifying \( \frac{600}{216} \):
\[ \frac{600 \div 12}{216 \div 12} = \frac{50}{18} = \frac{25}{9} \]
So:
\[ k^2 = \frac{25}{9} \]
Taking the square root of both sides gives us:
\[ k = \frac{5}{3} \]
Now, we can find the dimensions of Tiana's painting by multiplying the dimensions of Margaret's painting by this ratio \( k \):
- Length of Tiana's painting = \( 12 \times \frac{5}{3} = 20 \) inches
- Width of Tiana's painting = \( 18 \times \frac{5}{3} = 30 \) inches
Therefore, the dimensions of Tiana's painting are:
\[ \text{Length: } 20 \text{ inches, Width: } 30 \text{ inches} \]
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