Question

Students studying motion observed a cart rolling at a constant rate along a straight line. The table below gives the distance, $d$ feet, the cart was from a reference point at $1$-second intervals from $t=0$ seconds to $t=5$ seconds.
Which of the following equations represents this relationship between $d$ and $t$?

• A. $d=t+14$
• B. $d=6t+8$
• C. $d=6t+14$
• D. $d=14t+6$
• E. $d=34t$

Answer 1

The correct option is C $d=6t+14$

Cart rollling along straight line.

So graph of the $d$ is line.

Let the equation of the line is

$d\left(t\right)=at+b$ where $a$ and $b$ are some constants.

So when $t=0$, $f=14$

$\Rightarrow d\left(0\right)=14$

$\Rightarrow a\left(0\right)+b=14$

$\Rightarrow b=14$

So when $t=1$, $d=20$

$\Rightarrow d\left(1\right)=20$

$\Rightarrow a\left(1\right)+b=20$

we know that $b=14$.so

$\Rightarrow a+14=20$

$\Rightarrow a=6$

Checking:

When $t=2$, $d=26$

$\Rightarrow d\left(2\right)=26$

$\Rightarrow 6t+14=20$

$\Rightarrow 6\times 2+14=26$

When $t=3$, $d=32$

$\Rightarrow d\left(3\right)=32$

$\Rightarrow 6\times 3+14=32$

$\Rightarrow 32=32$

The equation $d\left(t\right)=6t+14$ is satisfying the table.

So $\Rightarrow d\left(t\right)=6t+14$