Lesson Worksheet: Applications on Parametric Equations Mathematics

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In this worksheet, we will practice using parametric equations, with time as a parameter, to model motion in two dimensions.

Q1:

A body is moving in the coordinate plane such that at time 𝑡 its 𝑥-coordinate is given by the equation 𝑥=3𝑡+(𝜋𝑡)sin and its 𝑦-coordinate is given by the equation 𝑦=1−4𝑡(𝜋𝑡)cos, for 0≤𝑡≤2.

Find the distance between the start point and endpoint of the motion of the body.

  • A√85 units
  • B2 units
  • C10 units
  • D100 units
  • E√14 units

Q2:

A car is traveling up a slope of angle 𝜃 at a constant velocity of 10 m⋅s−1. If we call the height gained by the car ℎ m and the horizontal displacement of the car 𝑥 m, then we can model the values of 𝑥 and ℎ using the pair of parametric equations 𝑥=10𝑡𝜃cos and ℎ=10𝑡𝜃sin, where 𝑡 is the time in seconds and 𝑡≥0.

Given that for every meter that the car travels horizontally, it travels 0.17 m vertically, determine how long it will take the car to gain a height of 3 m. Give your answer to two decimal places.

Q3:

A swimmer is trying to cross a river. The distance 𝑑 m across the river the swimmer has crossed after 𝑡 s is given by the equation 𝑑=3𝑡2. The current of the river moves the swimmer downstream. We can model the distance 𝑥 m the current moves the swimmer downstream with the equation 𝑥=𝑡2. Given that the river is 20 m wide, find the speed of the swimmer as they swim from one side of the river to the other.

  • A2√10 m⋅s−1
  • B2√5 m⋅s−1
  • C√52 m⋅s−1
  • D√102 m⋅s−1
  • E√10 m⋅s−1

Q4:

A ball is kicked off a flat ground such that, after 𝑡 seconds, the height, ℎ m, of the ball off the ground is given by the equation ℎ=14𝑡−4.9𝑡 and the distance the ball moved from its initial position along the ground, 𝑑 m, is given by the equation 𝑑=2𝑡, 0≤𝑡≤𝑘. Find the maximum height of the ball and its distance along the ground at this point.

  • Aℎ=10033m, 𝑑=107m
  • Bℎ=3,56449m, 𝑑=207m
  • Cℎ=10m, 𝑑=107m
  • Dℎ=710m, 𝑑=207m
  • Eℎ=10m,𝑑=207m

Q5:

A standing fan is spinning such that at a time of 𝑡 seconds the position of the end of one of its blades is given by the equations 𝑥=0.15(50𝜋𝑡)sin and 𝑦=1+0.15(50𝜋𝑡)cos, 𝑡≥0, where 𝑦 is the distance of the fan blade from the ground in meters and 𝑥 is the horizontal distance of the fan blade from its center of rotation in meters.

By showing that the rotation is a circle, find the radius of the rotation and the number of revolutions the fan makes per minute.

  • A15 cm, 750 rpm
  • B225 cm, 3,000 rpm
  • C225 cm, 1,500 rpm
  • D15 cm, 3,000 rpm
  • E15 cm, 1,500 rpm

Q6:

A clay pigeon is traveling through the air behind a wall of height 3 meters. The height of the clay pigeon in meters after 𝑡 seconds is given by the equation ℎ=16𝑡−5𝑡, and the horizontal distance 𝑥 in meters from the thrower is given by the equation 𝑥=5𝑡.

Determine the values of 𝑡 and 𝑥 for which the clay pigeon is above the wall.

  • A15<𝑡<3, 2<𝑥<30
  • B15<𝑡<3, 1<𝑥<15
  • C15≤𝑡≤3, 1≤𝑥≤15
  • D2<𝑡<30, 25<𝑥<6
  • E25<𝑡<6, 2<𝑥<30

Q7:

A cannon is fired from the top of a hill. After 𝑡 s, the height of the cannonball above the ground in meters is given by the equation ℎ=−5𝑡+5√3𝑡+35 and the horizontal displacement 𝑥 of the cannonball in meters is given by the equation 𝑥=10𝑡, where 0≤𝑡≤𝑘 and 𝑘 is the time taken for the cannonball to hit the ground.

Find the height of the hill.

Find the value of 𝑘 and the horizontal displacement of the cannonball when it hits the ground. Give your answers to two decimal places.

  • A𝑘=1.92s, 19.20 m
  • B𝑘=3.84s, 38.40 m
  • C𝑘=7.30s, 73.00 m
  • D𝑘=2.85s, 28.50 m
  • E𝑘=3.65s, 36.50 m

Q8:

The trajectory of a stunt plane can be modeled by the pair of parametric equations 𝑎=3𝑡+600sin and 𝑑=𝑡𝑡cos, 0≤𝑡≤2𝜋, where 𝑎 and 𝑑 are the altitude and distance along the ground covered by the plane in meters after 𝑡 seconds respectively.

Find the average gradient of the section of the maneuver where the plane travels from the point of minimum altitude to the end of the maneuver.

  • A3𝜋
  • B3𝜋
  • C6𝜋
  • D6𝜋
  • E32𝜋

Q9:

Pictured below is a cake.

A cross section of the cake is described by the parametric equations ℎ=10𝜃−1sin and 𝑤=15𝜃cos, 𝜋6≤𝜃≤5𝜋6, where ℎ cm is the height of the cake and 𝑤 cm is the width of the cake from the center.

By finding the maximum height and width of the cake, find the cross-sectional area of the smallest rectangular box that can contain the cake.

  • A300√3 cm2
  • B270√3 cm2
  • C67√3 cm2
  • D135√3 cm2
  • E150√3 cm2

Q10:

The position of an ant on a table after 𝑡 s is given by the pair of parametric equations 𝑥=33𝑡+𝜋3cos and 𝑦=3(6𝑡)sin, where 0≤𝑡≤𝑘 and 𝑘 is the time taken for the ant to return to its starting position.

Determine the time taken for the ant to cross over its path for the first time and the time taken for the ant to return to its starting position.

  • A𝜋18 s, 2𝜋3 s
  • B7𝜋18 s, 𝜋3 s
  • C𝜋18 s, 4𝜋9 s
  • D7𝜋18 s, 2𝜋3 s
  • E7𝜋18 s, 4𝜋9 s

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