Question

ww3: problem 6
(1 point)
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a car is driving down a straight road. its position (in meters) at time t (in seconds) is given by (x(t)=at + b) where a and b are constants.

how would you interpret the slope of the 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.
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Explanation:

Step1: Analyze the position - time function

The position of the first car is given by $x(t)=at + b$. The derivative of the position function with respect to time gives the velocity. The derivative of $x(t)$ with respect to $t$ is $\frac{dx}{dt}=a$. The units of $x$ are meters and the units of $t$ are seconds, so the units of $a$ are $\frac{m}{s}$, which is the unit of velocity. So, $a$ represents the velocity of the car.

Step2: Analyze the distance - time function between two cars

The distance between two cars is given by $d(t)=ct + d$. The coefficient $c$ of $t$ in the function $d(t)$ represents the rate of change of the distance between the two cars with respect to time. If $c<0$, it means the distance between the two cars is decreasing over time.

• If the cars are moving in the same direction, for the distance between them to decrease, the front - car must be slower than the back - car.
• If the cars are moving in opposite directions towards one another, the distance between them will also decrease.

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.