Hannah runs down the street to catch her bus. She runs for a total of 12 seconds. On the sidewalk, she runs 16 feet per second. When she crosses the street, she runs 3 feet per second. Hannah runs a total distance of 127 feet

Question

Hannah runs down the street to catch her bus. She runs for a total of 12 seconds. On the sidewalk, she runs 16 feet per second. When she crosses the street, she runs 3 feet per second. Hannah runs a total distance of 127 feet
How many seconds did Hannah spend running on the sidewalk? Enter your answer in the box.

GPT-4o mini · Answer

Let $ t $ be the time Hannah spends running on the sidewalk in seconds. Therefore, the time she spends running in the street will be $ 12 - t $ seconds, since the total running time is 12 seconds. The distance Hannah covers while running on the sidewalk can be calculated using the formula: $$ \text{Distance} = \text{Speed} \times \text{Time} $$ For the sidewalk: $$ \text{Distance on sidewalk} = 16 \, \text{feet/second} \times t \, \text{seconds} = 16t \, \text{feet} $$ For the street: $$ \text{Distance in street} = 3 \, \text{feet/second} \times (12 - t) \, \text{seconds} = 3(12 - t) \, \text{feet} = 36 - 3t \, \text{feet} $$ The total distance that Hannah runs is the sum of the distances she runs on the sidewalk and in the street: $$ \text{Total distance} = \text{Distance on sidewalk} + \text{Distance in street} $$ Given that the total distance is 127 feet, we set up the equation: $$ 16t + (36 - 3t) = 127 $$ Now, simplify and solve for $ t $: $$ 16t + 36 - 3t = 127 $$ $$ 13t + 36 = 127 $$ $$ 13t = 127 - 36 $$ $$ 13t = 91 $$ $$ t = \frac{91}{13} $$ $$ t = 7 $$ Hannah spent $ \boxed{7} $ seconds running on the sidewalk.