Question

a car is driving down a straight road. its position (in meters) at time t (in seconds) is given by s(t)=at + b where a and b are constants. how would you interpret the slope of this line, a? for whatever answer you choose, think about what the units of a would be and how you know. the velocity of the car. the acceleration of the car. the distance travelled by the car. the position of the car. a second car is driving along the same straight highway. the distance between the two cars at time t is given by the line d(t)=ct + d where c and d are constants. which scenarios below are consistent with c < 0? check all that apply. the cars are driving in the same direction, and the front car is faster than the back car. the cars are driving in the same direction, and the front car is slower than the back car. the cars are driving in opposite directions, towards one another. the cars are driving in the same direction, at exactly the same speed. the cars are driving in opposite directions, away from one another. note: in order to get credit for this problem all answers must be correct. preview my answers submit answers you have attempted this problem 4 times. your overall recorded score is 0%. you have unlimited attempts remaining.

Explanation:

Step1: Analyze the first - part

The position - time function of the car is $s(t)=at + b$. The derivative of the position function with respect to time gives the velocity. For a linear position - time function $s(t)=at + b$, the rate of change of position (slope) with respect to time is $a$. The units of position $s$ are meters and of time $t$ are seconds. The rate of change $\frac{\Delta s}{\Delta t}$ has units of $\frac{\text{meters}}{\text{second}}$, which is the unit of velocity. So, $a$ is the velocity of the car.

Step2: Analyze the second - part

The distance between two cars is given by $d(t)=ct + d$. The slope $c$ represents the rate of change of the distance between the two cars with respect to time. If $c\lt0$, it means the distance between the two cars is decreasing over time. When the cars are driving in the same direction and the front car is faster than the back car, the distance between them is increasing ($c\gt0$). When the cars are driving in the same direction and the front car is slower than the back car, the distance between them is decreasing ($c\lt0$). When the cars are driving in opposite directions towards one another, the distance between them is decreasing ($c\lt0$). When the cars are driving in the same direction at exactly the same speed, the distance between them is constant ($c = 0$). When the cars are driving in opposite directions away from one another, the distance between them is increasing ($c\gt0$).

Answer:

The velocity of the car.
The cars are driving in the same direction, and the front car is slower than the back car.
The cars are driving in opposite directions, towards one another.