Let's break down the task and solve it step by step:

Part (a): Determine the algebraic model

Given that Sophie is traveling 50 km to her grandfather’s house, we need to create a model for speed, which is a function of distance and time.

The general formula for speed is:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

In this case, the distance Sophie travels is 50 km, and time (in hours) is what we need to find for different speeds.

This can be rewritten as:

$\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{50}{\text{Speed}}$

Part (b): Determine the required speed

• Sophie usually drives at 95 km/h.
• To find the speed she needs to drive if she wants to arrive earlier, we can use the same formula.

If she wants to arrive 15 minutes earlier:

15 minutes = 0.25 hours, so the new time would be:

$\text{New time} = \frac{50}{95} - 0.25$

Now, we can rearrange the formula to find the new speed:

$\text{New speed} = \frac{50}{\text{New time}}$

If she wants to arrive 30 minutes earlier:

30 minutes = 0.5 hours, so the new time would be:

$\text{New time} = \frac{50}{95} - 0.5$

Now, calculate the new speed with:

$\text{New speed} = \frac{50}{\text{New time}}$

Let me compute the exact speeds for these two scenarios. Given data distance = 50 # km current_speed = 95 # km/h current_time = distance / current_speed # Time taken at current speed

Time changes (15 minutes = 0.25 hours, 30 minutes = 0.5 hours)

time_change_15min = 0.25 time_change_30min = 0.5

New time after arriving 15 and 30 minutes earlier

new_time_15min = current_time - time_change_15min new_time_30min = current_time - time_change_30min

Calculate the new speeds required

new_speed_15min = distance / new_time_15min new_speed_30min = distance / new_time_30min

new_speed_15min, new_speed_30minThe required speeds are as follows:

• To arrive 15 minutes earlier, Sophie would need to drive at approximately 181 km/h.
• To arrive 30 minutes earlier, Sophie would need to drive at a very high speed of about 1900 km/h, which is impractical for everyday driving.

Would you like me to explain more or assist you further with any of the calculations? Here are some additional questions to deepen your understanding:

1. How does the time taken relate to the speed and distance in the context of a trip?
2. What are the limitations when trying to drive at these calculated speeds?
3. How can the graph for this problem be drawn based on the algebraic model?
4. What other factors can influence the time taken for a journey (other than speed)?
5. How can we apply similar concepts to other real-life problems, such as traveling by train or plane?

Tip: Always ensure the units are consistent when solving such problems (in this case, time in hours and speed in km/h).