Calculating Jerry's Ice Cream Stop Duration: Speed and Time Relation

To determine how long Jerry stayed at the ice cream shop, we need to analyze his overall journey using his average speed.

Step 1: Calculate the expected total travel time.
His average speed is given as $1.567$ meters per second. Convert this speed into meters per minute to match the problem's time format: $1.567 \times 60 = 94.02$ meters per minute.

Step 2: Calculate total travel time using the average speed.
He needs to cover a total distance of $4700$ meters (school to shop to house). Using the formula for total time based on average speed:
$\text{Total time} = \frac{\text{Total distance}}{\text{Average speed}} = \frac{4700}{94.02} \approx 50$ minutes

Step 3: Determine the actual travel time.
The actual time taken to travel without the shop stay is the sum of the travel times for the two legs of his journey: $17 + 20 = 37$ minutes.

Step 4: Calculate the shop stay time.
The difference between the expected travel time $50$ minutes and the actual travel time $37$ minutes is the duration Jerry spent at the ice cream shop:
$\text{Shop stay time} = 50 - 37 = 13$ minutes

Therefore, Jerry stayed at the ice cream shop for 13 minutes.