Question

alonso’s mother drives to the gas station and fills up her tank. then she drives to the market. sketch the graph that shows the relationship between the amount of fuel in the gas tank of her car and time.

Explanation:

Step1: Analyze driving to gas station

When driving to the gas station, the car uses fuel, so the fuel level (y - axis) decreases as time (x - axis) increases. This is a line with a negative slope. Let's assume the initial fuel is \( F_0 \), and the rate of fuel consumption is \( r_1 \), so the equation is \( y = F_0 - r_1x \) for this interval.

Step2: Analyze filling the tank

At the gas station, while filling the tank, time is increasing (x increases), and fuel is being added, so the fuel level (y) increases rapidly (steeper positive slope) as time passes. Let the rate of fuel filling be \( r_2 \) ( \( r_2>>r_1 \) ), and the time spent filling is \( t_f \). The equation here is \( y = F_1 + r_2(x - x_1) \), where \( F_1 \) is the fuel level when arriving at the gas station, and \( x_1 \) is the time when arriving at the gas station.

Step3: Analyze driving to the market

After filling the tank, driving to the market, so fuel is consumed again, fuel level (y) decreases as time (x) increases, with a negative slope (similar to the first segment, maybe same or different rate \( r_3 \)), equation \( y = F_2 - r_3(x - x_2) \), where \( F_2 \) is the fuel level after filling, and \( x_2 \) is the time when leaving the gas station.

To sketch the graph:

1. Start with a line decreasing from the initial fuel level (let's say at \( x = 0 \), \( y = F_0 \)) until the time \( x_1 \) when reaching the gas station.
2. Then a vertical (or steep) increase from \( (x_1,F_1) \) to \( (x_2,F_2) \) (where \( F_2>F_1 \)) during the filling time.
3. Then a line decreasing from \( (x_2,F_2) \) as time goes on (driving to market) with a slope similar or different (depending on driving conditions) to the first decreasing line.

Answer:

The graph will have three segments:

1. A decreasing line (fuel consumption) from the initial time - fuel point.
2. A steeply increasing line (fuel filling) at the gas station.
3. A decreasing line (fuel consumption) while driving to the market.

(Note: Since it's a sketch, we can represent it with these three parts. The x - axis is time (min), y - axis is fuel (gal). The key is to show the decrease - increase - decrease pattern of fuel with respect to time.)