A snowboarder starts from rest at the top of a double black diamond hill. As he rides down the slope, GPS coordinates are used to determine his displacement as a function of time: x= 0.5t^3 + t^2 + 2t where x and t are expressed in ft and seconds, respectively. Determine the position, velocity, and acceleration of the boarder when t=5 seconds.

#### Solution:

Show me the final answerâ†“

We can figure out the position by substituting 5 seconds into our position equation.

x= 0.5(5)^3 + (5)^2 + 2(5)=97.5 ft

To figure out the velocity, we will take the derivative of our position equation.

Again, substitute t=5 into our velocity equation to figure out the velocity at 5 seconds.

v=1.5(5)^2+2(5)+2=49.5 ft/s

To find the acceleration, we will take the derivative of the velocity equation.

As before, substitute t=5 into our acceleration equation to figure out the acceleration at 5 seconds.

#### Final Answers:

v=49.5 ft/s

a=17\dfrac{\text{ft}}{s^2}