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Worksheet: Inverse Variation and Its Applications

Q1:

Which of the following relations represents an inverse variation between the two variables π‘₯ and 𝑦 ?

  • A 𝑦 = 6 π‘₯
  • B π‘₯ 𝑦 = 7 2
  • C 𝑦 = π‘₯ + 3
  • D π‘₯ 𝑦 = 1 4

Q2:

The number of hours needed for carrying out a certain task varies inversely with the number of workers who carry out the task. If the task is carried out by 23 workers in 35 hours, what is the time needed for 115 workers to carry out the task?

Q3:

The current in a circuit varies inversely with the circuit’s resistance, measured in ohms. When the current is 40 amperes, the resistance is 10 ohms. Find the current if the resistance is 12 ohms. Round your answer to two decimal places if necessary.

Q4:

The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?

Q5:

A group of scouts receives a donation of $ 1 0 0 0 to fund places on an international jamboree. The amount each scout receives for their trip varies inversely with the number of scouts from the group going to the jamboree.

Write an equation for π‘š , the amount each scout receives, in terms of 𝑛 , the number of scouts from the group who are going to the jamboree.

  • A π‘š = 𝑛 1 0 0 0
  • B π‘š = 1 0 0 𝑛
  • C π‘š = 1 0 𝑛
  • D π‘š = 1 0 0 0 𝑛
  • E π‘š = 𝑛 1 0 0

If 25 scouts from the group are going to the jamboree, how much will each scout receive from the donation?

Q6:

Suppose that 𝑦 is inversely proportional to π‘₯ and that 𝑦 = βˆ’ 4 when π‘₯ = βˆ’ 3 0 . Write the equation, giving 𝑦 in terms of π‘₯ .

  • A 𝑦 = 4 π‘₯ 3 0
  • B 𝑦 = 7 . 5 π‘₯
  • C 𝑦 = 7 . 5 π‘₯
  • D 𝑦 = 1 2 0 π‘₯
  • E 𝑦 = 6 0 π‘₯

Q7:

Given that π‘Ž varies inversely with 𝑏 , and π‘Ž = 5 when 𝑏 = 3 4 , write an equation for π‘Ž in terms of 𝑏 .

  • A π‘Ž = 4 𝑏 1 5
  • B π‘Ž = 1 5 𝑏
  • C π‘Ž = 𝑏 1 5
  • D π‘Ž = 1 5 4 𝑏
  • E π‘Ž = 2 0 3 𝑏

Q8:

A copy center is printing leaflets for a local school. The time it takes for the job to be completed varies inversely with the number of copiers used, and if 4 copiers are used, the job will take 3 4 of an hour.

How many copiers are required to complete the job in half an hour?

Letting 𝑑 be the time in hours to complete the job using 𝑛 copiers, what is the value of the product, 𝑑 𝑛 ?

Write an equation for 𝑑 in terms of 𝑛 .

  • A 𝑑 = 1 2 𝑛
  • B 𝑑 = 3 𝑛
  • C 𝑑 = 3 𝑛
  • D 𝑑 = 𝑛 3
  • E 𝑑 = 1 2 𝑛

Q9:

Community groups have been invited to apply for a grant from a fund. The amount a group will receive varies inversely with the number of successful applications.

If there are 45 successful applications, each group will receive $800.

How much will each group receive if there are 360 successful applications?

How many successful applications mean that each group receives $ 3 6 0 0 ?

How much will each group receive if there are 150 successful applications?

What is the total value of the fund?

Write an equation for π‘š , the number of dollars each group receives when 𝑛 groups make successful applications.

  • A π‘š = 𝑛 3 6 0 0 0
  • B π‘š = 3 6 0 0 𝑛
  • C π‘š = 3 6 0 0 0 0 𝑛
  • D π‘š = 3 6 0 0 0 𝑛
  • E π‘š = 𝑛 3 6 0 0

Q10:

The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1 2 0 0 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?

Q11:

The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 foot-candles at a distance of 3 meters. Find the intensity level at 8 meters.

Q12:

A building contractor is constructing part of the perimeter wall for a school. The wall will be the same height all along its length. The length of time it will take to build the wall varies inversely with the number of bricklayers. A team of 12 bricklayers is expected to take 30 days to build the wall.

Write an equation for the number of days, 𝑑 , it would take a team of 𝑏 bricklayers to build the wall.

  • A 𝑑 = 3 6 0 𝑏
  • B 𝑑 = 3 0 𝑏
  • C 𝑑 = 1 2 𝑏
  • D 𝑑 = 3 6 0 𝑏
  • E 𝑑 = 3 0 𝑏

How many bricklayers are required if the wall is to be finished in 20 days?

Q13:

Scarlett buys 5 liters of soda to share between the children at a party. The amount of soda each child will receive varies inversely with the number of children at the party. Write an equation for 𝑠 , the amount of soda in milliliters each child will receive, in terms of 𝑛 , the number of children at the party.

  • A 𝑠 = 𝑛 5 0 0 0
  • B 𝑠 = 𝑛 5
  • C 𝑠 = 5 0 0 𝑛
  • D 𝑠 = 5 0 0 0 𝑛
  • E 𝑠 = 5 0 𝑛

Q14:

The time taken to complete a car journey at a steady speed varies inversely with the speed of the car.

Most of the quickest route from Justin’s Corner to Round Rock is on the I-35. At 45 mph, this part of the journey takes 2 hours and 40 minutes.

How long would the journey take at 75 mph?

  • A 1 hour and 55 minutes
  • B 1 hour and 15 minutes
  • C 2 hours and 15 minutes
  • D 1 hour and 36 minutes
  • E 2 hours and 36 minutes

Write an equation for , the time it takes to cover this part of the journey at speed .

  • A
  • B
  • C
  • D
  • E

How much of the route is on the I-35?

Q15:

If 𝑦 = π‘Ž βˆ’ 8 , 𝑦 ∝ 1 π‘₯ 2 , and π‘Ž = 2 0 when π‘₯ = 8 , find 𝑦 when π‘₯ = 6 .

  • A 3 6 4
  • B 16
  • C 1 1 6
  • D 6 4 3

Q16:

Determine the correct relation from the following choices, given that π‘Ž 𝑏 βˆ’ 4 0 π‘Ž 𝑏 = βˆ’ 4 0 0 2 4 2 , π‘Ž ∈ ℝ , and 𝑏 ∈ ℝ .

  • A π‘Ž ∝ 𝑏 2
  • B π‘Ž ∝ 1 𝑏
  • C π‘Ž ∝ 𝑏
  • D π‘Ž ∝ 1 𝑏 2

Q17:

Given that 𝑦 = 1 0 + 𝑏 , where 𝑏 varies inversely with π‘₯ 2 , and 𝑦 = 1 3 0 when π‘₯ = 1 2 , determine the value of 𝑦 when π‘₯ = βˆ’ 1 0 .

  • A 7 4 5
  • B 1 0 1 0 3
  • C 5 7 4
  • D 1 0 3 1 0

Q18:

If 𝑦 = π‘Ž βˆ’ 6 , 𝑦 ∝ 1 π‘₯ 2 , and π‘Ž = 1 0 when π‘₯ = 5 , which of these equations represents the relation between π‘₯ and 𝑦 .

  • A 𝑦 = 2 0 π‘₯ 2
  • B 𝑦 = 1 0 0 π‘₯ βˆ’ 6 2
  • C 𝑦 = 1 π‘₯ βˆ’ 6 2
  • D 𝑦 = 1 0 0 π‘₯ 2
  • E 𝑦 = 6 βˆ’ 1 0 0 π‘₯ 2

Q19:

The height of a right circular cylinder β„Ž varies inversely with the square of its radius π‘Ÿ . If β„Ž = 9 3 c m when π‘Ÿ = 7 . 5 c m , determine β„Ž when π‘Ÿ = 1 . 5 c m .

Q20:

Given that π‘₯ ∝ 1 √ 𝑦 2 3 , and π‘₯ = 3 when 𝑦 = 1 7 2 8 , find 𝑦 when π‘₯ = 6 .

Q21:

The speed, 𝑣 , of water spraying from the end of a garden hose varies inversely with the square of the hose nozzle radius, π‘Ÿ . If 𝑣 = 3 7 cm/s when π‘Ÿ = 6 cm/s, find 𝑣 when π‘Ÿ = 0 . 7 5 cm/s.

Q22:

If 𝑦 ∝ 1 π‘₯ , and π‘₯ = 7 8 when 𝑦 = 5 7 , find the value of 𝑦 when π‘₯ = 6 .

Q23:

Given that π‘₯ varies inversely with √ 𝑦 , and π‘₯ = 6 0 when 𝑦 = 1 6 , find the value of π‘₯ when 𝑦 = 4 .

Q24:

Given that 𝑦 = 6 + 𝑏 , where 𝑏 varies inversely with π‘₯ 2 , and 𝑦 = 1 0 6 when π‘₯ = 1 2 , find the relation between π‘₯ and 𝑦 .

  • A 𝑦 = 6 + 4 0 0 π‘₯ 2
  • B 𝑦 = 6 βˆ’ 2 5 π‘₯ 2
  • C 𝑦 = 6 βˆ’ 4 0 0 π‘₯ 2
  • D 𝑦 = 6 + 2 5 π‘₯ 2

Q25:

A company making footballs decided to cut the selling price of each football by 3 0 % . Assuming all their footballs are identical and the same price, if they wanted to get the same amount of money from sales, by what percentage will they need to increase the number of footballs sold? Round your answer to the nearest whole number.