## Solutions to Problems on Unit Analysis

#### Unit Analysis and Converstion Between Units

1)

30 miles to the gallon = 30 (miles/gallon) = 30 (miles/gallon)(1 km/0.6214 miles)

= 30 (1/0.6214) (miles/gallon)(km/miles) = 30 (1/0.6214) (miles/gallon)(km/miles)

= 30 (1/0.6214) (km/gallon) = 30 (1/0.6214) (km/gallon)(1 gallon/3.785 liters)

= 30 (1/0.6214)(1/3.785) (km/gallon)(gallon/liters)

= 30 (1/0.6214)(1/3.785) (km/gallon)(gallon/liters) = 30 (1/0.6214)(1/3.785) (km/liter)

= 13 (km/liter)

2)
a.

10 km = 10 (km/1)(0.6214 miles/km) = 10(0.6214) (km/1)(miles/km) = 10(0.6214) (km/1)(miles/km)

= 10(0.6214) (miles/1) = 10(0.6214) miles = 6.214 miles

b.

1 light year = 3 x 108 meters so...

10 km = 10*1000 meters = 10,000 meters = 10,000 (meters/1)(light years / 3 x 108 meters)

= 10,000(1/ 3 x 108) (meters/1)(light years/meters)

= 10,000(1/ 3 x 108) (meters/1)(light years/meters)

= 10,000(1/ 3 x 108) (light years/1) = 10,000(1/ 3 x 108) light years

= 3 x 10-5 light years

3)

4 years = 4 (years/1) = 4 (years/1)(365 days/year) = 4(365) (years/1)(days/year)

= 4(365) (years/1)(days/year) = 4(365) (days/1) = 4(365) (days/1)(24 hours/day)

= 4(365)(24) (days/1)(hours/day) = 4(365)(24) (days/1)(hours/day) = 4(365)(24) (hours/1)

= 4(365)(24) (hours/1)(60 minutes/hour) = 4(365)(24)(60) (hours/1)(minutes/hour)

= 4(365)(24)(60) (hours/1)(minutes/hour) = 4(365)(24)(60) (minutes/1)

= 4(365)(24)(60) (minutes/1)(60 seconds/minute) = 4(365)(24)(60)(60) (minutes/1)(seconds/minute)

= 4(365)(24)(60)(60) (minutes/1)(seconds/minute) = 4(365)(24)(60)(60) (seconds/1)

= 4(365)(24)(60)(60) seconds = 1.26 x 108 seconds

4)

x=2 ft       y=3.5 ft       z=2 ft so...

1 ft = 1 (ft/1) = 1 (ft/1)(12 inches/ft) = 1(12) (ft/1)(inches/ft) = 1(12) (ft/1)(inches/ft)

= 1(12) (inches/1) = 1(12) (inches/1)(2.54 cm/inch) = 1(12)(2.54) (inches/1)(cm/inch)

= 1(12)(2.54) (inches/1)(cm/inch) = 1(12)(2.54) (cm/1) = 1(12)(2.54) cm

= 30.48 cm = 1ft so...

= x=60.96 cm       y=106.68 cm       z=60.96 cm

5)

30,000 ft = 30,000 (ft/1) = 30,000 (ft/1)(1 mile/5280 ft) = 30,000(1/5280) (ft/1)(mile/ft)

= 30,000(1/5280) (ft/1)(mile/ft) = 30,000(1/5280) (mile/1)

= 30,000(1/5280) (mile/1)(km/0.6214 mile) = 30,000(1/5280)(1/0.6214) (mile/1)(km/mile)

= 30,000(1/5280)(1/0.6214) (mile/1)(km/mile) = 30,000(1/5280)(1/0.6214) (km/1)

= 30,000(1/5280)(1/0.6214) km = 9.14 km

6)

60 joules/second = 60 (joules/second) = 60 (joules/second)(1 x 108 ergs/joule)

= 60(1 x 108) (joules/second)(ergs/joule)

= 60(1 x 108) (joules/second)(ergs/joule) = 60(1 x 108) (ergs/second)

= 6 x 109 (ergs/second)

7)

1 liter of water occupies a volume of 1,000 cubic centimeters (based off the definitions of these metric units). To get a box that would contain 1,000 cubic centimeters, the box would need to be 10 centimeters to a side (10cm * 10cm * 10cm = 1,000 cm3)

10 cm to a side. So (10cm)(1in/2.54cm) = (10cm)(1in/2.54cm) = (10/2.54) in = 3.94 in per side

8)

(1/6)(mile/min) = (1/6)(mile/min)(1 km/0.6214 mile) = (1/6)(mile/min)(1 km/0.6214 mile) =

(1/6)(1/0.6214)(km/min) = (1/6)(1/0.6214)(km/min)(1min/60sec) = (1/6)(1/0.6214)(km/min)(1min/60sec) =

(1/6)(1/0.6214)(1/60)(km/sec) = 4.47 x 10-3 km/s

9)

(3)(kg/m3) = (3)(kg/m3)(2.205lbs/kg) = (3)(kg/m3)(2.205lbs/kg) = (3)(2.205)(lbs/m3) =

1 inch = 2.54 cm = 0.0254 m so...

(3)(2.205)(lbs/m3)(0.0254m/in)(0.0254m/in)(0.0254m/in) = (3)(2.205)(1.64 x 10-5)(lbs/m3)(m3/in3)

(3)(2.205)(1.64 x 10-5)(lbs/m3)(m3/in3) = (3)(2.205)(1.64 x 10-5)(lbs/in3) = 1.08 x 10-4 lbs/in3