Solutions

1. \(\frac{210\: \text{cm}^{3}}{1} \times \frac{1\:\text {in}^{3}}{16.387\:\text {cm}^{3}} = 12.8\:\text {in}^{3}\)


2. \(\frac{91,227,778\:\text{ft}^{3}}{1} \times \frac{0.02832\:\text{m}^{3}}{1\:\text {ft}^{3}} = 2,583,283\:\text{m}^{3}\)
   (Slight differences in rounding can cause the final answer to be off by a few numbers.)


3. \(\frac{350\:\text {cm}^{3}}{1} \times \frac{1\:\text {ml}}{1\:\text {cm}^{3}} = 350\:\text {ml}\)


4. \(\frac{30\:\text {m}^{3}}{1} \times \frac{1\:\text {yd}^{3}}{0.7646\:\text {m}^{3}} = 39.24\text {yd}^{3}\) (Written Solution)

   Step 1: Start with what you know

   We need to fill an area with dirt that measures 2 m × 3 m × 5 m. To find the volume, we multiply the three dimensions together:

   \(2\:\text{m}\times3\:\text{m}\times5\:\text{m}=30\:\text{m}^{3}\)

   So we need \(30\:\text{m}^{3}\) of dirt.

   Step 2: Determine what you want to get in the end. (Figure out what the end units should be.) = \(\text{yd}^{3}\)

   Step 3: Determine what conversion factor(s) to use. You may sometimes need more than one.

   

   We know that 1 yd = 0.9144 m and we can use this information to find \(\text{yd}^{3}\):

   1 yd = 0.9144 m

   Cube both sides of the equation:

   \(\text{(1 yard)}^{3}\) = \(\text{(0.9144 m)}^{3}\)

   Which gives us:

   \(1\:\text{yard}^{3}= 0.7646\:\text{m}^{3}\)

   Step 4: Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

   We have \(30\:\text{m}^{3}\) so we need to multiply it by \( \frac{1\:\text{yd}^{3}}{0.7646\:\text{m}^{3}}\):

   \(\frac{30\:\text{m}^{3}}{1}\times\frac{1\:\text{yd}^{3}}{0.7646\:\text{m}^{3}}\)

   Note that we chose the conversion factor that will enable us to cancel out the existing units and change them to the desired units (\(\text{m}^{3}\) is in the top of our original fraction, so we chose the conversion factor with \(\text{m}^{3}\) in the bottom so they will cancel out)

   Then cancel out the \(\text{m}^{3}\).

   \(\frac{30\:\cancel{\text{m}^{3}}}{1}\times\frac{1\:\text{yd}^{3}}{0.7646\:\cancel{\text{m}^{3}}}\)

   Next, multiply straight across:

   \(\frac{30\times 1\:\text{yd}^{3}}{1\times0.7646}\)

   Which equals:

   \(\frac{30\:\text{yd}^{3}}{0.7646}\)

   Then divide 30 by 0.7646, which gives us:

   \(39.24\:\text{yd}^{3}\)

   So we need \(39.24\:\text{yards}^{3}\) of dirt to fill that volume.



5. \(\frac{340\:\text {mm}^{3}}{1} \times \frac{1\:\text{cm}^{3}}{1000\:\text{mm}^{3}} = .34\:\text{cm}^{3}\)


6. \(\frac{300,000\:\text {ft}^{3}}{1}\times\frac{0.0283\:\text{m}^{3}}{1\:\text{ft}^{3}}=8,490\:\text{m}^{3}\)
   (Slight differences in rounding can cause the final answer to be off by a few numbers.) (Written Solution)

   Step 1: Start with what you know: The room is 300,000 cubic feet

   Step 2: Determine what you want to get in the end. (Figure out what the end units should be.): Cubic meters

   Step 3: Determine what conversion factor(s) to use:

   We know:

   1 ft = 0.3048 m

   

   To determine how many cubic meters are in one cubic foot we start with the equation:

   1 ft = 0.3048 m

   Then cube both sides of the equation:

   \(\text{(1 ft)}^{3}\) = \(\text{(0.3840 m)}^{3}\)

   Which gives us:

   \(\text{1 ft}^{3}\) = \(\text{0.0283 m}^{3}\)

   Step 4: Multiply by 1 in the form of the conversion factor that cancels out the unwanted units.

   Since we have \(300,000\:\text{ft}^{3}\) we will multiply it by \(\frac{0.0283\:\text{m}^{3}}{1\:\text{ft}^{3}}\):

   \(\frac{300,000\:\text{ft}^{3}}{1}\times\frac{0.0283\:\text{m}^{3}}{1\:\text{ft}^{3}}\)

   Note that we chose the conversion factor that will enable us to cancel out the existing units and change them to the desired units (\(\text{ft}^{3}\) is on the top of our original fraction, so we chose the conversion factor with \(\text{ft}^{3}\) in the bottom so they will cancel out)

   Then cancel out \(\text{ft}^{3}\):

   \(\frac{300,000\:\cancel{\text{ft}^{3}}}{1}\times\frac{0.0283\:\text{m}^{3}}{1\:\cancel{\text{ft}^{3}}}\)

   Then multiply straight across:

   \(\frac{300,000\times0.0283\:\text{m}^{3}}{1\times1}\)

   Which equals:

   \(8490\:\text{m}^{3}\)