Explain how parametric equations can be used to model the motion of a projectile launched at an angle. Include the general form of the parametric equations for the horizontal and vertical positions.

Given the parametric equations $x(t)=3t+2$ and $y(t)=4t-1$, describe the path of the object and determine its position at $t=2$.

A car is moving along a path described by the parametric equations $x(t)=5t^2$ and $y(t)=3t+1$. Determine the car's speed at $t=2$.

A particle moves along a path given by the parametric equations $x(t)=\cos (t)$ and $y(t)=\sin (t)$. Describe the shape of the path and determine the position of the particle at $t=\frac{\pi }{2}$.

Consider a parametric curve defined by $x(t)=t^2-4t$ and $y(t)=2t+3$. Find the points where the curve intersects the x-axis.

Given the parametric equations $x(t)=3t^2+2t$ and $y(t)=2t^3-t$, determine the path of the object by eliminating the parameter $t$.

Given the parametric equations $x(t)=4\cos (t)$ and $y(t)=4\sin (t)$, describe the path of the object.

Given the parametric equations $x(t)=t^2$ and $y(t)=t^3$, find the speed of the object at $t=2$.

Given the parametric equations $x(t)=5t$ and $y(t)=3t+2$, determine the direction of the object's motion.

A ball is thrown from the top of a building with an initial velocity of $20\text{m/s}$ at an angle of ${30}^{\circ }$ above the horizontal. Write the parametric equations that describe the horizontal and vertical positions of the ball as functions of time. Assume the height of the building is $50\text{m}$ and ignore air resistance.

A particle moves in a circle of radius $5\text{m}$ with a constant speed of $2\text{m/s}$. Write the parametric equations for the position of the particle as a function of time, assuming it starts at the point $(5,0)$.

A car travels along a path described by the parametric equations $x(t)=3t^2$ and $y(t)=2t^3$. Find the speed of the car at $t=2$.

A drone flies in a path given by the parametric equations $x(t)=4t-t^2$ and $y(t)=3t-\frac{1}{2}{t}^2$. Determine the time at which the drone reaches its maximum height.

A cyclist rides along a path described by the parametric equations $x(t)=2t^2$ and $y(t)=3t$. Find the distance traveled by the cyclist from $t=0$ to $t=2$.