Question
A triangular flag has a base length of 21 inches and a height of 49 inches. Dillon makes a scale drawing of the flag in which the height is 14 inches. Dillon needs to find the area of the drawing.
Find the area of the scale drawing.
Answers
GPT-4o mini
To find the area of the triangular flag, we start by using the formula for the area of a triangle:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the original flag, the base is 21 inches and the height is 49 inches. Thus, the area of the original flag is:
\[
\text{Area} = \frac{1}{2} \times 21 \times 49
\]
Calculating the area step-by-step:
1. Calculate \( 21 \times 49 \):
\[
21 \times 49 = 1029
\]
2. Now, calculate half of 1029:
\[
\text{Area} = \frac{1}{2} \times 1029 = 514.5 \text{ square inches}
\]
Next, we need to find the area of Dillon's scale drawing. The scale factor can be determined by comparing the heights of the original flag and the scale drawing.
The original flag's height is 49 inches, and Dillon's scale drawing height is 14 inches. The scale factor \( k \) is:
\[
k = \frac{\text{height of scale drawing}}{\text{height of original}} = \frac{14}{49} = \frac{2}{7}
\]
The area of similar figures (like the flag and its scale drawing) scales with the square of the scale factor. Therefore, the area of the scale drawing \( A_d \) can be found using the formula:
\[
A_d = k^2 \times \text{Area of original}
\]
Calculating \( k^2 \):
\[
k^2 = \left(\frac{2}{7}\right)^2 = \frac{4}{49}
\]
Now substitute the area of the original flag (514.5 square inches) into the area formula for the scale drawing:
\[
A_d = \frac{4}{49} \times 514.5
\]
Calculating this:
1. Multiply:
\[
A_d = \frac{4 \times 514.5}{49}
\]
\[
4 \times 514.5 = 2058
\]
2. Now divide by 49:
\[
A_d = \frac{2058}{49} \approx 42
\]
Finally, the area of Dillon's scale drawing is approximately:
\[
\boxed{42} \text{ square inches}
\]
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the original flag, the base is 21 inches and the height is 49 inches. Thus, the area of the original flag is:
\[
\text{Area} = \frac{1}{2} \times 21 \times 49
\]
Calculating the area step-by-step:
1. Calculate \( 21 \times 49 \):
\[
21 \times 49 = 1029
\]
2. Now, calculate half of 1029:
\[
\text{Area} = \frac{1}{2} \times 1029 = 514.5 \text{ square inches}
\]
Next, we need to find the area of Dillon's scale drawing. The scale factor can be determined by comparing the heights of the original flag and the scale drawing.
The original flag's height is 49 inches, and Dillon's scale drawing height is 14 inches. The scale factor \( k \) is:
\[
k = \frac{\text{height of scale drawing}}{\text{height of original}} = \frac{14}{49} = \frac{2}{7}
\]
The area of similar figures (like the flag and its scale drawing) scales with the square of the scale factor. Therefore, the area of the scale drawing \( A_d \) can be found using the formula:
\[
A_d = k^2 \times \text{Area of original}
\]
Calculating \( k^2 \):
\[
k^2 = \left(\frac{2}{7}\right)^2 = \frac{4}{49}
\]
Now substitute the area of the original flag (514.5 square inches) into the area formula for the scale drawing:
\[
A_d = \frac{4}{49} \times 514.5
\]
Calculating this:
1. Multiply:
\[
A_d = \frac{4 \times 514.5}{49}
\]
\[
4 \times 514.5 = 2058
\]
2. Now divide by 49:
\[
A_d = \frac{2058}{49} \approx 42
\]
Finally, the area of Dillon's scale drawing is approximately:
\[
\boxed{42} \text{ square inches}
\]