To find the area of Magnolia’s scale drawing of the triangular flag, we first need to calculate the actual area of the flag before scaling it down. The formula for the area of a triangle is:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Step 1: Calculate the Area of the Actual Flag
The actual flag has a height of 15 inches and a base length of 25 inches. Plugging these values into the formula:
\[ \text{Area} = \frac{1}{2} \times 25 , \text{inches} \times 15 , \text{inches} \]
Calculating this step-by-step:
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Multiply the base and height: \[ 25 \times 15 = 375 , \text{square inches} \]
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Now, divide by 2: \[ \text{Area} = \frac{375}{2} = 187.5 , \text{square inches} \]
Step 2: Find the Scale Factor
Now, we need to determine the scale factor based on the base length in the scale drawing. The actual base length is 25 inches, and the scale drawing base length is 10 inches. The scale factor can be found by dividing the scale drawing base length by the actual base length:
\[ \text{Scale Factor} = \frac{10}{25} = \frac{2}{5} \]
Step 3: Calculate the Area of the Scale Drawing
The area of a triangle scales with the square of the scale factor. So, to find the area of the scale drawing, we apply the scale factor squared to the actual area:
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First, calculate the square of the scale factor: \[ \left(\frac{2}{5}\right)^2 = \frac{4}{25} \]
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Now, multiply the actual area by this scaled factor: \[ \text{Area of Scale Drawing} = 187.5 \times \frac{4}{25} \]
Calculating \( 187.5 \times \frac{4}{25} \) step-by-step:
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Calculate \( 187.5 \div 25 \): \[ 187.5 \div 25 = 7.5 \]
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Now multiply by 4: \[ 7.5 \times 4 = 30 , \text{square inches} \]
Final Answer
So, the area of Magnolia's scale drawing of the flag is:
\[ \boxed{30 , \text{square inches}} \]