Worksheet: Using Integration to Find Power

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In this worksheet, we will practice using integration to solve problems involving power as a function of time to find work, speed, or time.

Q1:

A car has mass 1 066 kg. At time 𝑑 seconds, its engine works at a rate of 𝑃 = ο€Ή 𝑑 + 9 𝑑  2 W . Given that at 𝑑 = 6 s the car’s speed is 78 km/h , find its speed at 𝑑 = 1 0 s . Give your answer to the nearest m/s.

Q2:

Find the time taken for a car of 1 236 kg to reach a speed of 126 km/h, given that the car started from rest and that the power of the engine is constant and equal to 103 horsepowers.

Q3:

The power of a machine is given by the relation 𝑃 = 3 𝑑 + 7 , where 𝑑 is the time elapsed in seconds. Find the work done by the machine in the first 8 seconds.

Q4:

The power of an engine is given by ο€Ό 8 𝑑 βˆ’ 1 1 5 𝑑   hp, where 𝑑 ∈ [ 0 , 1 0 3 ] is the time in seconds. Find the power of the engine 𝑃  when 𝑑 = 3 6 s e c o n d s , the work done π‘Š over the time interval [ 0 , 1 2 ] , and the maximum power 𝑃 m a x of the engine.

  • A 𝑃  = 2 0 1 . 6 h p , π‘Š = 5 3 7 . 6 k g - w t β‹… m , 𝑃 m a x = 1 , 2 0 0 h p
  • B 𝑃  = 4 8 9 . 6 h p , π‘Š = 5 3 7 . 6 k g - w t β‹… m , 𝑃 m a x = 7 2 0 h p
  • C 𝑃  = 5 4 7 . 2 h p , π‘Š = 7 7 , 7 6 0 k g - w t β‹… m , 𝑃 m a x = 8 4 0 h p
  • D 𝑃  = 2 0 1 . 6 h p , π‘Š = 4 0 , 3 2 0 k g - w t β‹… m , 𝑃 m a x = 2 4 0 h p

Q5:

The power of an engine at time 𝑑 seconds is given by 𝑃 ( 𝑑 ) = ο€Ή 2 1 𝑑 + 1 8 𝑑   W . Find the work done by the engine between 𝑑 = 6 s and 𝑑 = 7 s .

Q6:

Find the time taken for a car of 1 188 kg to reach a speed of 126 km/h, given that the car started from rest and that the power of the engine is constant and equal to 110 horsepowers.

Q7:

Find the time taken for a car of 972 kg to reach a speed of 126 km/h, given that the car started from rest and that the power of the engine is constant and equal to 90 horsepowers.

Q8:

The power of a machine is given by the relation 𝑃 = 8 𝑑 βˆ’ 2 , where 𝑑 is the time elapsed in seconds. Find the work done by the machine in the first 8 seconds.

Q9:

The power of a machine is given by the relation 𝑃 = 7 𝑑 + 3 , where 𝑑 is the time elapsed in seconds. Find the work done by the machine in the first 6 seconds.

Q10:

The power of an engine is given by ο€Ό 8 𝑑 βˆ’ 1 2 0 𝑑   hp, where 𝑑 ∈ [ 0 , 1 1 0 ] is the time in seconds. Find the power of the engine 𝑃  when 𝑑 = 3 4 s e c o n d s , the work done π‘Š over the time interval [ 0 , 1 9 ] , and the maximum power 𝑃 m a x of the engine.

  • A 𝑃  = 2 1 4 . 2 h p , π‘Š = 1 , 3 2 9 . 6 8 k g - w t β‹… m , 𝑃 m a x = 1 , 6 0 0 h p
  • B 𝑃  = 4 8 6 . 2 h p , π‘Š = 1 , 3 2 9 . 6 8 k g - w t β‹… m , 𝑃 m a x = 9 6 0 h p
  • C 𝑃  = 5 2 4 . 7 3 h p , π‘Š = 1 9 0 , 8 7 8 . 7 5 k g - w t β‹… m , 𝑃 m a x = 1 , 1 2 0 h p
  • D 𝑃  = 2 1 4 . 2 h p , π‘Š = 9 9 , 7 2 6 . 2 5 k g - w t β‹… m , 𝑃 m a x = 3 2 0 h p

Q11:

The power of an engine is given by ο€Ό 5 𝑑 βˆ’ 1 2 0 𝑑   hp, where 𝑑 ∈ [ 0 , 1 0 2 ] is the time in seconds. Find the power of the engine 𝑃  when 𝑑 = 6 8 s e c o n d s , the work done π‘Š over the time interval [ 0 , 2 1 ] , and the maximum power 𝑃 m a x of the engine.

  • A 𝑃  = 1 0 8 . 8 h p , π‘Š = 9 4 8 . 1 5 k g - w t β‹… m , 𝑃 m a x = 6 2 5 h p
  • B 𝑃  = 4 4 8 . 8 h p , π‘Š = 9 4 8 . 1 5 k g - w t β‹… m , 𝑃 m a x = 3 7 5 h p
  • C 𝑃  = 6 0 2 . 9 3 h p , π‘Š = 1 3 0 , 6 4 6 . 2 5 k g - w t β‹… m , 𝑃 m a x = 4 3 7 . 5 h p
  • D 𝑃  = 1 0 8 . 8 h p , π‘Š = 7 1 , 1 1 1 . 2 5 k g - w t β‹… m , 𝑃 m a x = 1 2 5 h p

Q12:

A car has mass 951 kg. At time 𝑑 seconds, its engine works at a rate of 𝑃 = ο€Ή 3 𝑑 + 9 𝑑  2 W . Given that at 𝑑 = 6 s the car’s speed is 73 km/h , find its speed at 𝑑 = 1 0 s . Give your answer to the nearest m/s.

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