## Roller Coasters

Consider a roller coaster on a section of a track that is in a straight line. It travels a distance (D). After a certain amount of time (t), D = 10t^4-t^3. Calculate the roller coaster's velocity when t = 5.

##
__
__**Hint**

**Hint**

Velocity is the derivative of distance.

##
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__**Hint 2**

**Hint 2**

The power rule for the first derivative:

$$$\frac{d}{dx}[x^n]=n\cdot x^{n-1}$$$

Velocity is a derivative of distance:

$$$v=\frac{dr}{dt}$$$

where
$$v$$
is the instantaneous velocity,
$$t$$
is time, and
$$r$$
is position.

The power rule for the first derivative:

$$$\frac{d}{dx}[x^n]=n\cdot x^{n-1}$$$

Thus,

$$$D=10t^4-t^3$$$

$$$v=(4)10t^{4-1}-3t^{3-1}=40t^3-3t^2$$$

For
$$t=5$$
, substitute "5" in the equation for time:

$$$v=40(5)^3-3(5)^2=5000-75=4,925$$$

4,925