Question
Rosa is looking at a map of the park that is laid out on a coordinate plane. Rosa is at (1, –1). The shelter house is at (–2, –4) and the fossil exhibit is at (3, 2). Each unit on the map represents 100 feet. How much closer is Rosa to the fossil exhibit than to the shelter house? Round to the nearest foot.
Answers
Answered by
GPT-4o mini
To find how much closer Rosa is to the fossil exhibit than to the shelter house, we first need to calculate the distance from Rosa to each of these locations using the distance formula.
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
### Distance from Rosa to the Shelter House
Rosa's coordinates are \((1, -1)\) and the shelter house's coordinates are \((-2, -4)\).
1. Identify the coordinates:
- Rosa: \((x_1, y_1) = (1, -1)\)
- Shelter House: \((x_2, y_2) = (-2, -4)\)
2. Calculate the distance:
\[
d_{\text{shelter}} = \sqrt{((-2) - 1)^2 + ((-4) - (-1))^2}
\]
\[
= \sqrt{(-3)^2 + (-3)^2}
\]
\[
= \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]
Calculating \(3\sqrt{2}\):
\[
3\sqrt{2} \approx 3 \times 1.414 \approx 4.242
\]
### Distance from Rosa to the Fossil Exhibit
Next, we calculate the distance to the fossil exhibit, which is at \((3, 2)\).
1. Identify the coordinates:
- Fossil Exhibit: \((x_2, y_2) = (3, 2)\)
2. Calculate the distance:
\[
d_{\text{fossil}} = \sqrt{(3 - 1)^2 + (2 - (-1))^2}
\]
\[
= \sqrt{(2)^2 + (3)^2}
\]
\[
= \sqrt{4 + 9} = \sqrt{13}
\]
Calculating \(\sqrt{13}\):
\[
\sqrt{13} \approx 3.605
\]
### Convert to Feet
Each coordinate unit represents 100 feet.
1. Calculate distance to the shelter house in feet:
\[
d_{\text{shelter feet}} = 4.242 \times 100 \approx 424.2 \text{ feet}
\]
2. Calculate distance to the fossil exhibit in feet:
\[
d_{\text{fossil feet}} = 3.605 \times 100 \approx 360.5 \text{ feet}
\]
### Finding How Much Closer Rosa is
Now, we find how much closer she is to the fossil exhibit than to the shelter house:
\[
d_{\text{shelter feet}} - d_{\text{fossil feet}} = 424.2 - 360.5 = 63.7 \text{ feet}
\]
Rounding to the nearest foot:
\[
\text{Rosa is } 64 \text{ feet closer to the fossil exhibit than to the shelter house.}
\]
Thus, the final answer is:
\[
\boxed{64}
\]
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
### Distance from Rosa to the Shelter House
Rosa's coordinates are \((1, -1)\) and the shelter house's coordinates are \((-2, -4)\).
1. Identify the coordinates:
- Rosa: \((x_1, y_1) = (1, -1)\)
- Shelter House: \((x_2, y_2) = (-2, -4)\)
2. Calculate the distance:
\[
d_{\text{shelter}} = \sqrt{((-2) - 1)^2 + ((-4) - (-1))^2}
\]
\[
= \sqrt{(-3)^2 + (-3)^2}
\]
\[
= \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}
\]
Calculating \(3\sqrt{2}\):
\[
3\sqrt{2} \approx 3 \times 1.414 \approx 4.242
\]
### Distance from Rosa to the Fossil Exhibit
Next, we calculate the distance to the fossil exhibit, which is at \((3, 2)\).
1. Identify the coordinates:
- Fossil Exhibit: \((x_2, y_2) = (3, 2)\)
2. Calculate the distance:
\[
d_{\text{fossil}} = \sqrt{(3 - 1)^2 + (2 - (-1))^2}
\]
\[
= \sqrt{(2)^2 + (3)^2}
\]
\[
= \sqrt{4 + 9} = \sqrt{13}
\]
Calculating \(\sqrt{13}\):
\[
\sqrt{13} \approx 3.605
\]
### Convert to Feet
Each coordinate unit represents 100 feet.
1. Calculate distance to the shelter house in feet:
\[
d_{\text{shelter feet}} = 4.242 \times 100 \approx 424.2 \text{ feet}
\]
2. Calculate distance to the fossil exhibit in feet:
\[
d_{\text{fossil feet}} = 3.605 \times 100 \approx 360.5 \text{ feet}
\]
### Finding How Much Closer Rosa is
Now, we find how much closer she is to the fossil exhibit than to the shelter house:
\[
d_{\text{shelter feet}} - d_{\text{fossil feet}} = 424.2 - 360.5 = 63.7 \text{ feet}
\]
Rounding to the nearest foot:
\[
\text{Rosa is } 64 \text{ feet closer to the fossil exhibit than to the shelter house.}
\]
Thus, the final answer is:
\[
\boxed{64}
\]