Question
The high school debate team is developing a logo to represent their club. A scale drawing of the logo design is presented below, where each unit of the grid represents 3 inches in length.
The team is printing out an enlargement of the new logo, where the enlargement has a height of 105 inches.
The area of the enlargement will be 1. 10 2. 5270 3. 105 4. 2,250 5. 450
inches2, which is 1. 1/3 2. 1/5 3. 25 4. 3 5. 5 6. 1/25
times the size of the original scale drawing.
The team is printing out an enlargement of the new logo, where the enlargement has a height of 105 inches.
The area of the enlargement will be 1. 10 2. 5270 3. 105 4. 2,250 5. 450
inches2, which is 1. 1/3 2. 1/5 3. 25 4. 3 5. 5 6. 1/25
times the size of the original scale drawing.
Answers
GPT-4o mini
To solve for the area of the enlargement and how many times larger it is than the original scale drawing, we can follow these steps:
1. **Understand the scale**: Each grid unit represents 3 inches. Therefore, we need to find the dimensions of the logo in terms of inches.
2. **Scale up the height**: The enlargement has a height of 105 inches. To find the scale factor, we need to convert this to the height of the original drawing.
Let \( h \) be the height in grid units of the original scale drawing. The relationship between the height of the original logo and the enlarged height is given by:
\[
\text{Enlarged Height} = \text{Original Height} \times \text{Scale Factor}
\]
\[
105 = h \times \text{Scale Factor}
\]
3. **Calculate the scale factor**: To find the scale factor, we first need the value of \( h \). Let's say the height of the original logo in grid units is obtained from the drawing. Suppose it is 10 grid units (this is just an example but you would use the actual grid units from the scale drawing). The original height would therefore be:
\[
10 \text{ units} \times 3 \text{ inches/unit} = 30 \text{ inches}
\]
Now we can find the scale factor:
\[
\text{Scale Factor} = \frac{\text{Enlarged Height}}{\text{Original Height}} = \frac{105}{30} = 3.5
\]
4. **Find the dimensions of the enlargement**: If we knew the original width (let's say, \( w \) units, resulting in \( w \times 3 \) inches), the enlarged dimensions would be scaled similarly.
5. **Calculate the areas**:
Original area \( A_\text{original} = \text{Original Height} \times \text{Original Width} = (30 \times w \times 3) \).
Enlarged area \( A_\text{enlarged} = \text{Enlarged Height} \times \text{Enlarged Width} = (105 \times (w \times 3.5)) \).
6. **Area scaling relationship**: The area scales with the square of the linear dimensions (height and width):
\[
\text{Factor by which areas increase} = (\text{Scale Factor})^2
\]
\[
\text{Factor by which area increases} = (3.5)^2 = 12.25
\]
7. **Assigning the area of the enlarged logo**: You can now multiply the area of the original logo by 12.25 to get the total area.
8. **Match with provided choices**: You can compare your calculated area with the provided area choices to select options that align with your calculations.
Note: The exact numerical values for \( A_\text{original} \) need the local grid dimensions from the original drawing to compute accurately.
In summary, without the exact original dimensions, we need to check against the provided answers. Based on the scaling factor calculated (3.5), it implies that the area of the enlargement would be 12.25 times the area of the original logo.
1. **Understand the scale**: Each grid unit represents 3 inches. Therefore, we need to find the dimensions of the logo in terms of inches.
2. **Scale up the height**: The enlargement has a height of 105 inches. To find the scale factor, we need to convert this to the height of the original drawing.
Let \( h \) be the height in grid units of the original scale drawing. The relationship between the height of the original logo and the enlarged height is given by:
\[
\text{Enlarged Height} = \text{Original Height} \times \text{Scale Factor}
\]
\[
105 = h \times \text{Scale Factor}
\]
3. **Calculate the scale factor**: To find the scale factor, we first need the value of \( h \). Let's say the height of the original logo in grid units is obtained from the drawing. Suppose it is 10 grid units (this is just an example but you would use the actual grid units from the scale drawing). The original height would therefore be:
\[
10 \text{ units} \times 3 \text{ inches/unit} = 30 \text{ inches}
\]
Now we can find the scale factor:
\[
\text{Scale Factor} = \frac{\text{Enlarged Height}}{\text{Original Height}} = \frac{105}{30} = 3.5
\]
4. **Find the dimensions of the enlargement**: If we knew the original width (let's say, \( w \) units, resulting in \( w \times 3 \) inches), the enlarged dimensions would be scaled similarly.
5. **Calculate the areas**:
Original area \( A_\text{original} = \text{Original Height} \times \text{Original Width} = (30 \times w \times 3) \).
Enlarged area \( A_\text{enlarged} = \text{Enlarged Height} \times \text{Enlarged Width} = (105 \times (w \times 3.5)) \).
6. **Area scaling relationship**: The area scales with the square of the linear dimensions (height and width):
\[
\text{Factor by which areas increase} = (\text{Scale Factor})^2
\]
\[
\text{Factor by which area increases} = (3.5)^2 = 12.25
\]
7. **Assigning the area of the enlarged logo**: You can now multiply the area of the original logo by 12.25 to get the total area.
8. **Match with provided choices**: You can compare your calculated area with the provided area choices to select options that align with your calculations.
Note: The exact numerical values for \( A_\text{original} \) need the local grid dimensions from the original drawing to compute accurately.
In summary, without the exact original dimensions, we need to check against the provided answers. Based on the scaling factor calculated (3.5), it implies that the area of the enlargement would be 12.25 times the area of the original logo.