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Worksheet: Using Dimensional Analysis to Solve Real-Life Problems

Q1:

If 1 foot is about 30 centimeters and 1 yard is about 90 centimeters, approximately how many feet are there in a yard?

Q2:

Which quantities could be measured to determine the fuel efficiency of a car?

  • Athe quantity of fuel used
  • Bthe fuel price and the distance traveled
  • Cthe distance traveled over a given period
  • Dthe distance traveled and the corresponding volume of fuel used

Which of the following is an appropriate unit for the fuel efficiency?

  • Amiles per gallon
  • Bcubic centimeters per gallon
  • Ckilometers per hour
  • Dmiles per hour
  • Ecubic centimeters per litre

Q3:

Elizabeth wants to convert 4 5 0 0 kg/m3 into meters per second

Which of the following is correct?

  • A 4 5 0 0 1 = 4 5 0 0 1 × ( 1 0 0 ) 1 × 1 1 0 0 0 k g m k g m c m m k g g 3 3 3 3
  • B 4 5 0 0 1 = 4 5 0 0 1 × 1 1 0 0 × 1 0 0 0 1 k g m k g m m c m g k g 3 3 3 3
  • C 4 5 0 0 1 = 4 5 0 0 1 × 1 0 0 1 × 1 1 0 0 0 k g m k g m c m m k g g 3 3 3 3
  • D 4 5 0 0 1 = 4 5 0 0 1 × 1 ( 1 0 0 ) × 1 0 0 0 1 k g m k g m m c m g k g 3 3 3 3
  • E 4 5 0 0 1 = 4 5 0 0 1 × 1 1 0 0 × 1 1 0 0 0 k g m k g m m c m k g g 3 3 3 3

What does she find?

Q4:

Daniel wants to convert 50 m/s into kilometers per hour.

He starts by writing the speed as a ratio of distance to time. What would this look like?

  • A 1 5 0 m s
  • B 1 5 0 s m
  • C 5 0 1 s m
  • D 5 0 1 m s
  • E 5 0 2 m s

Next, he works out how many meters would be covered in 1 hour, which is equal to 3 6 0 0 s. What does he find?

Finally, he converts his result in (2) to kilometers. What speed does he find?

Considering units as algebraic quantities, which of the following is equivalent to Daniel’s conversion?

  • A 5 0 1 = 5 0 1 × 3 6 0 0 1 × 1 1 0 0 0 m s m s s h k m m
  • B 5 0 1 = 5 0 1 × 3 6 0 0 1 × 1 1 0 0 0 m s m s s h k m m
  • C 5 0 1 = 5 0 1 × 1 3 6 0 0 × 1 0 0 0 1 m s m s s h k m m
  • D 5 0 1 = 5 0 1 × 6 0 1 × 1 1 0 0 0 m s m s s h k m m
  • E 5 0 1 = 5 0 1 × 3 6 0 0 1 × 1 1 0 0 m s m s s h k m m

Q5:

Which of the following is correct?

  • A 1 1 = 1 1 × 1 ( 1 0 0 ) × 1 0 0 0 1 g c m g c m m c m g k g 3 3 3 3
  • B 1 1 = 1 1 × 1 0 0 1 × 1 1 0 0 0 g c m g c m c m m k g g 3 3 3 3
  • C 1 1 = 1 1 × 1 1 0 0 × 1 0 0 0 1 g c m g c m m c m g k g 3 3 3 3
  • D 1 1 = 1 1 × ( 1 0 0 ) 1 × 1 1 0 0 0 g c m g c m c m m k g g 3 3 3 3
  • E 1 1 = 1 1 × 1 ( 1 0 ) × 1 0 0 0 1 g c m g c m m c m g k g 3 3 3 3

If the density of a material is 𝑑 g/cm3, what is its density expressed in kg/m3?

  • A 1 0 0 0 𝑑 kg/m3
  • B 1 0 𝑑 kg/m3
  • C 𝑑 1 0 0 0 kg/m3
  • D 𝑑 1 0 kg/m3
  • E 𝑑 kg/m3

Q6:

Which of the following is correct?

  • A 1 1 = 1 1 × 3 6 0 0 1 × 1 1 0 0 m s m s s h k m m
  • B 1 1 = 1 1 × 1 3 6 0 0 × 1 0 0 0 1 m s m s h s m k m
  • C 1 1 = 1 1 × 1 3 6 0 0 × 1 0 0 1 m s m s h s m k m
  • D 1 1 = 1 1 × 3 6 0 0 1 × 1 1 0 0 0 m s m s s h k m m
  • E 1 1 = 1 1 × 1 6 0 × 1 0 0 0 1 m s m s h s m k m

Convert 1 m/s into kilometers per hour, giving your answer as a fraction in its simplest form.

  • A 1 / = 1 8 5 / m s k m h
  • B 1 / = 1 3 6 / m s k m h
  • C 1 / = 3 6 / m s k m h
  • D 1 / = 5 1 8 / m s k m h
  • E 1 / = 5 0 3 / m s k m h

Convert 𝑎 m/s into kilometers per hour.

  • A 5 0 3 𝑎 km/h
  • B 5 1 8 𝑎 km/h
  • C 1 8 5 𝑎 km/h
  • D 3 6 𝑎 km/h
  • E 1 3 6 𝑎 km/h

Q7:

The density 𝐷 of a material can be measured in kilograms per cubic meter, often written as kg/m3. Use dimensional analysis to determine which of the following formulas is correct to calculate the density of a given object, where 𝑚 is its mass and 𝑉 its volume.

  • A 𝐷 = 𝑚 𝑉
  • B 𝐷 = 𝑉 𝑚
  • C 𝐷 = 1 𝑚 𝑉
  • D 𝐷 = 𝑚 𝑉
  • E 𝐷 = 𝑚 ( 𝑉 ) 3

Q8:

When heating up a meal, some parts of the meal can be hot before some others. For instance, chicken will heat up faster than broccoli. This is because different substances have different specific heat capacities, which is the energy needed to raise the temperature of one kilogram of a substance by one degree Celsius. The specific heat capacity is often expressed in joules per kilogram and per degree Celsius (or J/(kg⋅°C)). Using dimensional analysis, find the formula for the amount of energy 𝑄 needed to raise the temperature of a mass 𝑚 of a substance of specific heat capacity 𝑐 by Δ 𝑇 degrees.

  • A 𝑄 = 𝑚 Δ 𝑇 𝑐
  • B 𝑄 = 𝑚 𝑐 Δ 𝑇
  • C 𝑄 = 𝑐 𝑚 Δ 𝑇
  • D 𝑄 = 𝑚 𝑐 Δ 𝑇
  • E 𝑄 = 𝑚 𝑐 Δ 𝑇

Q9:

The amount of pressure an object is exerting is given by 𝑃 = 𝐹 𝐴 , where 𝐹 is the force exerted by the object and 𝐴 is the surface area on which the force is spread. When 𝐹 is measured in newtons, which can be expressed as kg⋅m/s2, and 𝐴 is measured in square meters, the pressure is measured in pascals. Which of the following is the expression of pascals in terms of the base units kilograms, meters, and seconds?

  • A k g s m / 2 2
  • B k g m s / ( ) 2
  • C k g s m / 2
  • D k g m s / 2
  • E k g s m /

Q10:

Which of the following is correct?

  • A 1 1 = 1 1 × 1 3 6 0 0 × 1 0 0 1 k m h k m h h s m k m
  • B 1 1 = 1 1 × 3 6 0 0 1 × 1 1 0 0 0 k m h k m h s h m m
  • C 1 1 = 1 1 × 3 6 0 0 1 × 1 1 0 0 k m h k m h s h m m
  • D 1 1 = 1 1 × 1 3 6 0 0 × 1 0 0 0 1 k m h k m h h s m k m
  • E 1 1 = 1 1 × 6 0 1 × 1 1 0 0 0 k m h k m h s h k m m

Convert 1 km/h into meters per second, giving your answer as a fraction in its simplest form.

  • A 1 / = 5 1 8 / k m h m s
  • B 1 / = 3 6 / k m h m s
  • C 1 / = 1 3 6 / k m h m s
  • D 1 / = 1 8 5 / k m h m s
  • E 1 / = 3 5 0 / k m h m s

Convert 𝑎 km/h into meters per second.

  • A 3 5 0 𝑎 m/s
  • B 1 8 5 𝑎 m/s
  • C 5 1 8 𝑎 m/s
  • D 1 3 6 𝑎 m/s
  • E 3 6 𝑎 m/s

Q11:

Which of the following is correct?

  • A 1 1 = 1 1 × 1 0 0 1 × 1 1 0 0 0 k g m k g m c m m k g g 3 3 3 3
  • B 1 1 = 1 1 × 1 1 0 0 × 1 0 0 0 1 k g m k g m m c m g k g 3 3 3 3
  • C 1 1 = 1 1 × 1 1 0 0 × 1 1 0 0 0 k g m k g m m c m k g g 3 3 3 3
  • D 1 1 = 1 1 × 1 ( 1 0 0 ) × 1 0 0 0 1 k g m k g m m c m g k g 3 3 3 3
  • E 1 1 = 1 1 × ( 1 0 0 ) 1 × 1 1 0 0 0 k g m k g m c m m k g g 3 3 3 3

If the density of a material is 𝑑 kg/m3, what is its density expressed in g/cm3?

  • A 𝑑 1 0 0 0 g/cm3
  • B 𝑑 1 0 0 0 0 0 g/cm3
  • C 𝑑 1 0 g/cm3
  • D 1 0 𝑑 g/cm3
  • E 1 0 0 0 𝑑 g/cm3

Q12:

Natalie wants to convert 30 km/h into meters per second

Which of the following is correct?

  • A 3 0 1 = 3 0 1 × 3 6 0 0 1 × 1 1 0 0 0 k m h k m h s h k m m
  • B 3 0 1 = 3 0 1 × 1 6 0 × 1 0 0 0 1 k m h k m h h s m k m
  • C 3 0 1 = 3 0 1 × 6 0 1 × 1 1 0 0 0 k m h k m h s h k m m
  • D 3 0 1 = 3 0 1 × 1 3 6 0 0 × 1 0 0 0 1 k m h k m h h s m k m
  • E 3 0 1 = 3 0 1 × 3 6 0 0 1 × 1 1 0 0 0 k m h k m h s h k m m

What does she find?