Emily leaves her house at exactly 8:25am to bike to her school, which is 3.42 miles away. When she passes the post office, which is 3/4 mile away from her home, she looks at her watch and sees that it is 30 seconds past 8:29am. If Emily's school starts at 8:50am can Emily make it to school on time without increasing her rate of speed? Show/ explain the work necessary
To determine if Emily can make it to school on time without increasing her rate of speed, we need to calculate the time it will take her to reach the school at her current speed.
First, we need to calculate the time it took Emily to reach the post office from her house:
Distance to the post office = 3/4 mile
Time taken = (3/4) / (speed) hours
Next, we need to calculate the time it took Emily to reach the school from the post office:
Distance to the school = 3.42 - 3/4 = 3.42 - 0.75 = 2.67 miles
Time taken = 2.67 / (speed) hours
Adding the time taken to reach the post office to the time taken to reach the school will give us the total time it will take Emily to reach school from her house. This time must be less than or equal to 21 minutes (the time between when Emily left home and the start of the school).
Let's calculate the total time it will take Emily to reach school at her current speed:
Time to reach post office = (3/4) / (speed) hours
Time to reach school = 2.67 / (speed) hours
Total time to reach school = (3/4) / (speed) + 2.67 / (speed) = 2.67 + 0.75 / speed
Since Emily reached the post office at 8:29:30am, she has 20 minutes and 30 seconds to reach school. Converting this time to hours, we get:
Time remaining = 20 minutes + 30 seconds = 20.5 minutes = 20.5 / 60 hours
Therefore, we need to have:
Total time to reach school <= Time remaining
2.67 + 0.75 / speed <= 20.5 / 60
Now we just need to solve for speed:
2.67 + 0.75 / speed <= 20.5 / 60
2.67 * speed + 0.75 <= 20.5 / 60 * speed
2.67 * speed <= 20.5 / 60 * speed - 0.75
(2.67 - 20.5 / 60) * speed <= -0.75
(2.67 - 20.5 / 60) <= -0.75 / speed
speed >= (2.67 - 20.5 / 60) / -0.75
Calculate the value within the parentheses and you will find the minimum speed required for Emily to reach school before 8:50am. If her current speed is greater than this value, then she will make it to school on time without increasing her speed.
Can you finish calculating everything?
Of course! Let's finish the calculations:
Calculate the value within the parentheses:
(2.67 - 20.5 / 60) = 2.67 - 0.3417 = 2.3283
Now, calculate the minimum speed required:
Minimum speed = 2.3283 / -0.75 = -3.1044
Since speed cannot be negative, this means that Emily cannot make it to school on time without increasing her rate of speed. She would need to bike at a speed greater than 3.1044 mph in order to reach her school by 8:50am. So, she would have to increase her rate of speed to reach school on time.
Are you sure that's all correct?
Let's re-evaluate the calculations to ensure accuracy:
Given:
- Distance to the post office = 3/4 mile
- Distance to the school from the post office = 3.42 - 3/4 = 3.42 - 0.75 = 2.67 miles
- Time to reach the post office = 30 seconds past 8:29am
- Time to reach the school from the post office = 20 minutes and 30 seconds
Calculating the total time, we have:
Time to reach post office = (3/4) / speed
Time to reach school = 2.67 / speed
Total time = (3/4) / speed + 2.67 / speed = 2.67 + 0.75 / speed
Given time remaining to reach school is 20.5 minutes (0.3417 hours), we have:
Total time <= Time remaining
2.67 + 0.75 / speed <= 0.3417
Now, solve for speed:
2.67 + 0.75 / speed <= 0.3417
2.67 * speed + 0.75 <= 0.3417 * speed
2.67 * speed <= 0.3417 * speed - 0.75
(2.67 - 0.3417) * speed <= -0.75
2.3283 * speed <= -0.75
speed >= -0.75 / 2.3283
speed >= -0.3225
As speed cannot be negative, the minimum speed required is 0.3225 mph. This means that Emily can make it to school on time without increasing her speed, as long as she is biking at a speed greater than 0.3225 mph. Thank you for your patience.
Can you put the answer in simple terms
Apologies for the confusion. Here is the simplified answer:
Emily needs to bike at a speed greater than 0.3225 mph to reach school on time without increasing her speed. As long as she maintains a speed faster than 0.3225 mph, she will arrive at school before 8:50am. This means Emily does not need to increase her speed to make it to school on time. Thank you for your understanding.