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Answer :
Let's solve this problem step-by-step:
1. Determine the distance traveled from 2 PM to 3 PM:
Patrick traveled to a point that is 6 miles west and 8 miles south from the dock. To find the straight-line distance, we use the Pythagorean theorem:
[tex]\[
\text{Distance} = \sqrt{(6^2 + 8^2)} = \sqrt{(36 + 64)} = \sqrt{100} = 10 \text{ miles}
\][/tex]
2. Calculate the boat's speed:
From 2 PM to 3 PM, Patrick traveled 10 miles. Since this happened over 1 hour, the boat's speed is:
[tex]\[
\text{Speed} = \frac{10 \text{ miles}}{1 \text{ hour}} = 10 \text{ miles per hour}
\][/tex]
3. Determine the total distance the boat can travel with a full tank:
The boat averages 5 miles per gallon, and the tank holds 4 gallons. Therefore, the total distance the boat can travel on a full tank is:
[tex]\[
\text{Total Distance} = 5 \text{ miles/gallon} \times 4 \text{ gallons} = 20 \text{ miles}
\][/tex]
4. Calculate the remaining distance the boat can travel:
By 3 PM, Patrick has already traveled 10 miles, so the distance left before the boat runs out of gasoline is:
[tex]\[
\text{Remaining Distance} = 20 \text{ miles} - 10 \text{ miles} = 10 \text{ miles}
\][/tex]
5. Determine how many more hours the boat can travel:
With a speed of 10 miles per hour and 10 miles left before running out of gasoline, the remaining travel time is:
[tex]\[
\text{Remaining Hours} = \frac{10 \text{ miles}}{10 \text{ miles per hour}} = 1 \text{ hour}
\][/tex]
Thus, Patrick's boat can continue for 1 more hour before the tank is out of gasoline.
1. Determine the distance traveled from 2 PM to 3 PM:
Patrick traveled to a point that is 6 miles west and 8 miles south from the dock. To find the straight-line distance, we use the Pythagorean theorem:
[tex]\[
\text{Distance} = \sqrt{(6^2 + 8^2)} = \sqrt{(36 + 64)} = \sqrt{100} = 10 \text{ miles}
\][/tex]
2. Calculate the boat's speed:
From 2 PM to 3 PM, Patrick traveled 10 miles. Since this happened over 1 hour, the boat's speed is:
[tex]\[
\text{Speed} = \frac{10 \text{ miles}}{1 \text{ hour}} = 10 \text{ miles per hour}
\][/tex]
3. Determine the total distance the boat can travel with a full tank:
The boat averages 5 miles per gallon, and the tank holds 4 gallons. Therefore, the total distance the boat can travel on a full tank is:
[tex]\[
\text{Total Distance} = 5 \text{ miles/gallon} \times 4 \text{ gallons} = 20 \text{ miles}
\][/tex]
4. Calculate the remaining distance the boat can travel:
By 3 PM, Patrick has already traveled 10 miles, so the distance left before the boat runs out of gasoline is:
[tex]\[
\text{Remaining Distance} = 20 \text{ miles} - 10 \text{ miles} = 10 \text{ miles}
\][/tex]
5. Determine how many more hours the boat can travel:
With a speed of 10 miles per hour and 10 miles left before running out of gasoline, the remaining travel time is:
[tex]\[
\text{Remaining Hours} = \frac{10 \text{ miles}}{10 \text{ miles per hour}} = 1 \text{ hour}
\][/tex]
Thus, Patrick's boat can continue for 1 more hour before the tank is out of gasoline.
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