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Answer:
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Step-by-step explanation:
Let's denote the distance from Jack's home to the nearby park as \(d\) kilometers.
When Jack jogs to the park at an average speed of 12 kilometers per hour, the time it takes him is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{d}{12} \]
When he jogs back from the park at an average speed of 7 kilometers per hour, the time it takes him is:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{d}{7} \]
We know that the total time taken is 57 minutes, which is \(\frac{57}{60}\) hours.
So, we can set up the equation:
\[ \frac{d}{12} + \frac{d}{7} = \frac{57}{60} \]
To solve this equation, we need to find a common denominator:
\[ \frac{7d}{84} + \frac{12d}{84} = \frac{57}{60} \]
\[ \frac{19d}{84} = \frac{57}{60} \]
Now, we can cross multiply:
\[ 19d \times 60 = 57 \times 84 \]
\[ 1140d = 4788 \]
Now, let's solve for \(d\):
\[ d = \frac{4788}{1140} \]
\[ d = 4.19 \]
So, the distance from Jack's home to the nearby park is approximately 4.19 kilometers.
To find the time it took him to jog from his home to the nearby park, we can use the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
\[ \text{Time} = \frac{4.19}{12} \]
\[ \text{Time} \approx 0.35 \text{ hours} \]
Now, let's convert the time from hours to minutes:
\[ 0.35 \times 60 = 21 \text{ minutes} \]
So, Jack jogged from his home to the nearby park for approximately 21 minutes.
