Calculate the Coffee Stop Duration with Speed Factor Y

To solve the problem, we need to calculate the time Gerard stopped to have coffee, given his driving speeds and average speed. We'll follow these steps:

• Step 1: Calculate the distance covered during the first segment of his journey.
• Step 2: Calculate the distance covered during the second segment after his break.
• Step 3: Write the average speed equation and solve for the stop time.

Step 1: Gerard drives for 40 minutes, which is $\frac{2}{3}$ of an hour. His speed is 90 km/h. The distance covered, $D_1$, is:

$D_1 = 90 \times \frac{2}{3} = 60 \text{ km}$.

Step 2: After stopping, he drives for 30 minutes, which is $\frac{1}{2}$ of an hour, at a speed $Y$ times greater than 90 km/h, which translates to $90Y$ km/h. The distance covered, $D_2$, is:

$D_2 = 90Y \times \frac{1}{2} = 45Y \text{ km}$.

Step 3: His average speed over the entire trip, including the stop, is given as $\frac{x}{4}$ km/h. Let $t$ be the time he stops. The total distance is $60 + 45Y$ km, and the total time is $(\frac{2}{3} + \frac{1}{2} + t)$ hours. The average speed equation is:

$\frac{60 + 45Y}{\frac{2}{3} + \frac{1}{2} + t} = \frac{x}{4}$.

Solving for $t$, we get:

$4(60 + 45Y) = x(\frac{7}{6} + t)$.

Simplifying gives:

$240 + 180Y = \frac{7}{6}x + xt$.

Rearranging for $t$, we have:

$t = \frac{240+180Y-\frac{7}{6}x}{x}$.

Therefore, Gerard stops for $\frac{240+180y-\frac{7}{6}x}{x}$ hours.