Problems #1 - 10

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Metric conversion where two units are converted

**Problem #1:** A sign gives the speed limit at 50.0 km/hr. What is this speed in centimeters per second?

**Solution:**

1) Convert km to cm:

50.0 km/hr times (10^{5}cm / km) = 5.00 x 10^{6}cm / hr

Convert hour to second:

5.00 x 10^{6}cm / hr times (1 hr / 3600 s) = 1388.9 cm/sThree sig figs gives 1390 cm/s as the answer.

**Problem #2:** An object is traveling at a speed of 7500. centimeters per second. Convert the value to kilometers per minute.

**Solution:**

Convert cm to km. Convert seconds to minutes. Here it is set up in one line:7500. cm/s times (1 km / 10

^{5}cm) times (60 s / min) = 4.500 km/min

**Problem #3:** Convert 10.6 kg/m^{3} to mg/μL

**Solution:**

1) Convert kg to mg:

10.6 kg/m^{3}times (10^{6}mg/kg) = 1.06 x 10^{7}mg/m^{3}

2) Convert m^{3} to L:

Consider that 1 m^{3}is a cube 1 meter on each side. Change each meter to its equivalent in dm:1 m^{3}= 10 dm x 10 dm x 10 dm = 1000 dm^{3}Since 1 dm

^{3}= 1 L, we arrive at this many liters for 1 m^{3}:1000 LTherefore:

1.06 x 10^{7}mg/m^{3}times (1 m^{3}/ 1000 L) = 1.06 x 10^{4}mg/L

3) Convert L to μL:

1.06 x 10^{4}mg/L x (1 L / 10^{6}μL) = 1.06 x 10^{-2}mg/μL = 0.0106 mg/μL

Suppose the problem had been to convert kg/m^{3} to mg/mL? Then, the last part of the solution would look like this:

1.06 x 10^{4}mg/L x (1 L / 10^{3}mL) = 1.06 x 10^{1}mg/mL = 10.6 mg/mL10.6 kg/m

^{3}= 10.6 mg/mL

Comment: notice that I converted from cubic meter to L, then from L to μL. The direct conversion is this:

(1 m^{3}/ 1000 L) times (1 L / 10^{6}μL) = 1 m^{3}/ 10^{9}μL

and a one-line solution looks like this:

10.6 kg/m^{3}times (10^{6}mg/kg) times (1 m^{3}/ 10^{9}μL) = 10.6 x 10^{-3}mg/μL = 0.0106 mg/μL

**Problem #4:** Convert 9.98 x 10^{12} g/mL to kg/m^{3}

**Solution:**

1) I propose to create a one-line solution in a step-by-step way. First, change g/mL to kg/mL:

9.98 x 10^{12}g/mL x (1 kg / 1000 g)

2) Next, I want to change mL to cubic meters, but suppose I really don't know (off the top of my head) what the conversion is. However, I do know this:

9.98 x 10^{12}g/mL x (1 kg / 1000 g) x (1000 mL / 1 dm^{3})That cancels the mL and replaces it with dm

^{3}.

3) I have memorized the conversion from dm^{3} to m^{3} (see the discussion in the solution to Problem #3, just above):

9.98 x 10^{12}g/mL x (1 kg / 1000 g) x (1000 mL / 1 dm^{3}) x (1000 dm^{3}/ 1 m^{3})

4) The above calculation gives this answer:

9.98 x 10^{15}kg/m^{3}

Suppose I did know the direct conversion from mL to m^{3}. My conversion might then look like this:

9.98 x 10^{12}g/mL x (1 kg / 1000 g) x (10^{6}mL / 1 m^{3})

The direct mL to m^{3} conversion depends on you knowing that 1 mL equals 1 cm^{3} and that 1 m^{3} is a cube 100 cm on each of its three sides (100 x 100 x 100 = 10^{6})

**Problem #5:** Convert 1.17 x 10^{-8} kg/cm^{3} to g/mL

**Solution:**

**Problem #6:** A light-year is the distance that light can travel in one year. If the sun is 150,000,000 km away, how many light years is the Sun from Earth? Assume light travels at a speed of 3.0 x 10^{10} cm/s

**Solution:**

1) Determine the kilometers in one light-year:

(3.0 x 10^{10}cm/s) (1 km / 10^{5}cm) (3600 s / 1 hr) (24 hr / 1 day) (365.25 day) = 9.46728 x 10^{12}km <--- this is one light-yearIn order, these are the above conversions:

cm to km

per second to per hour

per hour to per day

per day to one yearNote that 365.25 days was used. Some teachers will use 365 days. Note also that a rounded-off value for the speed of light was used. It is actually slight less than the value used.

Note that I did not use the unit year in the denominator at the end. This is to emphasize that a light-year is a measure of distance. There are 9.46728 x 10

^{12}km in one light-year.You could write the unit as km/ly.

2) Determine light-years that the Sun is from Earth:

150,000,000 km / 9.46728 x 10^{12}km/ly = 0.000015844 ly

**Problem #7:** Convert 0.153 mol/L to micromole/nm^{3}.

**Solution:**

1) Convert the numerator:

0.153 mol/L times (10^{6}μmol / mol) = 1.53 x 10^{5}μmol/L

2) Convert the denominator:

Replace L with dm^{3}to obtain 1.53 x 10^{5}μmol/dm^{3}(remember, 1 L = 1 dm^{3})deci- is 10

^{-1}and nano- is 10^{-9}, so we have an absolute exponential distance of 10^{8}1.53 x 10

^{5}μmol/dm^{3}times (1 dm / 10^{8}nm)^{3}= 1.53 x 10^{-19}μmol / nm^{3}

**Problem #8:** Convert 3.75 μm/s to km/hr

**Solution:**

micro- = 10^{-6}and kilo- = 10^{3}. The absolute exponential distance is 10^{9}.3.75 μm/s times (1 km / 10

^{9}μm) x (_______)In the empty spot, put a conversion that changes seconds to hours. There are 3600 second in one hour.

The rest is left to the reader.

**Problem #9:** 4.18 x 10^{4} kg/L to dg/mL

**Solution:**

Notice that, in there numerator, there is a total of 10

^{8}and, in the denominator is 10^{3}. That makes 10^{5}.The answer is 4.18 x 10

^{5}dg/mL.

**Problem #10:** Convert 5.00 x 10^{5} cm/s to m/day

**Solution:**

1) Convert the numerator:

5.00 x 10^{5}cm/s times (1 m / 1000 cm) <--- cancels cm, replaces it with mBased on this one conversion alone, I will use 5.00 x 10

^{2}m/s in the next step.

2) Convert the denominator:

The starting unit is now m/s and we have to take seconds up to day. The first conversion is seconds to minutes:5.00 x 10^{2}m/s times (60 s / min) <--- the seconds will cancelThe next conversion will replace minutes with the next time unit up, which is hours:

5.00 x 10^{2}m/s times (60 s / min) times (60 min / hr) <--- notice how minutes will cancelThe last conversion replaces hours with day, which is what we want:

5.00 x 10^{2}m/s times (60 s / min) times (60 min / hr) times (24 hr / day) <--- hour cancels and day stays in the denominator

3) Everything put together in one line (I start with the original cm/s value):

(5.00 x 10^{5}cm/s) x (1 m / 1000 cm) x (60 s / min) x (60 min / hr) x (24 hr / day) = 4.32 x 10^{7}m/day