Question
Nicholas is deep-sea diving with Janelle, who is to the south of him. A fish emerges from a hole to the east of Janelle. The fish swims 40 feet to Janelle and then swims 75 feet to Nicholas. The fish then swims back to its hole and disappears. How far did the fish travel in all?
feet
feet
Answers
GPT-4o mini
To find the total distance traveled by the fish, we can break the journey into three parts.
1. **Distance from the hole to Janelle**: This is given as 40 feet.
2. **Distance from Janelle to Nicholas**: This is given as 75 feet.
3. **Distance from Nicholas back to the hole**: To calculate this, we need to determine the coordinates.
Let's set a coordinate system:
- Assume Janelle is located at coordinates (0, 0).
- Since Nicholas is to the north of Janelle, we can set Nicholas at (0, y), where y is any positive value.
- The fish emerges from the hole to the east (so horizontally from Janelle), let's set its hole at (x, 0) where x is greater than 0.
From the information given, the fish swims:
- From its hole at (x, 0) to Janelle at (0, 0), which is 40 feet.
- Note that the distance formula tells us that \(\sqrt{(x - 0)^2 + (0 - 0)^2} = 40\) means \(x = 40\). So, the hole's coordinates are (40, 0).
3. **Finding the coordinates of Nicholas**: We do not have specific information on how far north Nicholas is, so we represent Nicholas' position as (0, y) without needing the exact value for y. However, we know the distance from Janelle to Nicholas is 75 feet.
The distance from Janelle (0, 0) to Nicholas (0, y) is \(y = 75\), since they are vertically aligned.
4. **Finding the distance from Nicholas back to the hole**: The coordinates for Nicholas are (0, 75) and for the hole are (40, 0). Using the distance formula now gives us:
\[
\text{Distance} = \sqrt{(40 - 0)^2 + (0 - 75)^2}
\]
\[
= \sqrt{(40)^2 + (-75)^2}
\]
\[
= \sqrt{1600 + 5625} = \sqrt{7225} = 85 \text{ feet}
\]
Now, we sum all distances the fish swam:
1. From hole to Janelle: 40 feet
2. From Janelle to Nicholas: 75 feet
3. From Nicholas back to the hole: 85 feet
So, the total distance traveled by the fish is:
\[
40 + 75 + 85 = 200 \text{ feet}
\]
Thus, the fish traveled a total of **200 feet**.
1. **Distance from the hole to Janelle**: This is given as 40 feet.
2. **Distance from Janelle to Nicholas**: This is given as 75 feet.
3. **Distance from Nicholas back to the hole**: To calculate this, we need to determine the coordinates.
Let's set a coordinate system:
- Assume Janelle is located at coordinates (0, 0).
- Since Nicholas is to the north of Janelle, we can set Nicholas at (0, y), where y is any positive value.
- The fish emerges from the hole to the east (so horizontally from Janelle), let's set its hole at (x, 0) where x is greater than 0.
From the information given, the fish swims:
- From its hole at (x, 0) to Janelle at (0, 0), which is 40 feet.
- Note that the distance formula tells us that \(\sqrt{(x - 0)^2 + (0 - 0)^2} = 40\) means \(x = 40\). So, the hole's coordinates are (40, 0).
3. **Finding the coordinates of Nicholas**: We do not have specific information on how far north Nicholas is, so we represent Nicholas' position as (0, y) without needing the exact value for y. However, we know the distance from Janelle to Nicholas is 75 feet.
The distance from Janelle (0, 0) to Nicholas (0, y) is \(y = 75\), since they are vertically aligned.
4. **Finding the distance from Nicholas back to the hole**: The coordinates for Nicholas are (0, 75) and for the hole are (40, 0). Using the distance formula now gives us:
\[
\text{Distance} = \sqrt{(40 - 0)^2 + (0 - 75)^2}
\]
\[
= \sqrt{(40)^2 + (-75)^2}
\]
\[
= \sqrt{1600 + 5625} = \sqrt{7225} = 85 \text{ feet}
\]
Now, we sum all distances the fish swam:
1. From hole to Janelle: 40 feet
2. From Janelle to Nicholas: 75 feet
3. From Nicholas back to the hole: 85 feet
So, the total distance traveled by the fish is:
\[
40 + 75 + 85 = 200 \text{ feet}
\]
Thus, the fish traveled a total of **200 feet**.