Question

Sandra who is a Level 200 student of SoE and also a snowboarder starts from rest at the top of a double black diamond hill. As she rides down the slope, GPS coordinates are used to determine her displacement as a function of time: x=0.5t3 + 6t2 +3t where x is in metres and t is in seconds. where x and t are expressed in feet and seconds, respectively. a) Determine the position of the boarder when t = 4 s b) Determine the velocity of the boarder when t = 4s c) Determine the acceleration of the boarder when t = 4s 2021/22​

Answer 1

Step-by-step explanation:

a) To determine the position of the snowboarder when t = 4 seconds, we can substitute t = 4 into the equation x = 0.5t^3 + 6t^2 + 3t:

x = 0.5 * 4^3 + 6 * 4^2 + 3 * 4

x = 64 + 96 + 12

x = 172

So when t = 4 seconds, the snowboarder's position is 172 meters.

b) To determine the velocity of the snowboarder when t = 4 seconds, we'll need to find the first derivative of the displacement function x = 0.5t^3 + 6t^2 + 3t with respect to time:

dx/dt = 3 * 0.5 * t^2 + 2 * 6 * t + 3

Next, we can substitute t = 4 into this expression to find the velocity when t = 4 seconds:

dx/dt = 3 * 0.5 * 4^2 + 2 * 6 * 4 + 3

dx/dt = 72 + 48 + 3

dx/dt = 123

So the velocity of the snowboarder when t = 4 seconds is 123 meters per second.

c) To determine the acceleration of the snowboarder when t = 4 seconds, we'll need to find the second derivative of the displacement function x = 0.5t^3 + 6t^2 + 3t with respect to time:

d^2x/dt^2 = 6 * 0.5 * t + 2 * 6

Next, we can substitute t = 4 into this expression to find the acceleration when t = 4 seconds:

d^2x/dt^2 = 6 * 0.5 * 4 + 2 * 6

d^2x/dt^2 = 24 + 12

d^2x/dt^2 = 36

So the acceleration of the snowboarder when t = 4 seconds is 36 meters per second squared.