### Tutorial 1.6: Part 4

Dimensional Analysis: Applications

For the box shown the length is 21.5 cm, the width is 12.3 cm, and the height is 15.4 cm. What is the volume in liters?

\begin{align*} v & = l \times w \times h\\ v & = 21.5 \, cm \times 12.3 \,cm \times 15.4 \,cm\\ v & = 4072.53 \, cm^3 \\ & = 4.07 \times 10^3 \, cm^3 \end{align*}

We want the volume in liters not cm3. By definition 1 liter = 1 dm3. We note that since 1 dm = 10 cm, 1 dm3 = 103 cm3.

\begin{align*} & 4.07 \times 10^3 \, cm^3 \times \frac{1\, dm^3}{ 10^3 \, cm^3} \times \frac{1\, liter}{1\, dm^3} \\ & = 4.07\, L \end{align*}

For the box shown the length is 24.5 cm, the width is 14.0 cm, and the height is 17.5 cm. What is the surface area in mm2.

\begin{align*} area & = 2 [ (l \times w) + (l \times h)+ (w \times h)] \\ area & = 2 [(24.5 \,cm \times 14.0 \,cm)\\ & +(24.5 \,cm \times 17.5 \,cm)+(14.0 \,cm \times 17.5 \,cm)]\\ area & = 2033.5 \, cm^2 \\ & = 2.03 \times 10^3 \, cm^2 \end{align*}

We want the area in mm2 not cm2. We note that since 1 cm = 10 mm, so 1 cm2 = 102 mm2.

\begin{align*} & 2.03 \times 10^3 \, cm^2 \times \frac{10^2\, mm^2}{ 1 \, cm^2} \\ & = 2.03 \times 10^5 \, mm^2 \end{align*}

For the cylinder shown the diameter is 960 cm and the height is 1440 cm. What is the volume in dm3 ?

\begin{align*} v &= \pi r^2h\\ v &= \pi (480 \, cm)^2(1440 \,cm)\\ v &= 1.04 \times 10^9 cm^3\\ \end{align*}

We want the volume in dm3 not cm3. 1 dm = 10 cm, so 1 dm3 = 103 cm3.

\begin{align*} & 1.04 \times 10^9 cm^3 \times \frac{1\, dm^3}{ 10^3 \, cm^3} \\ & = 1.04 \times 10^6 dm^3 \end{align*}

For the cylinder shown the diameter is 448 cm and the height is 672 cm. What is the surface area in dm2 ?

\begin{align*} area &= 2 \pi r^2 + 2 \pi rh\\ area &= 2 \pi (224 \, cm)^2 + 2 \pi (224 \, cm)(672 \, cm)\\ area &= 1.26 \times 10^6 cm^2\\ \end{align*}

We want the area in dm2 not cm2. 1 dm = 10 cm, so 1 dm2 = 102 cm2.

\begin{align*} & 1.26 \times 10^6 cm^2 \times \frac{1\, dm^2}{ 10^2 \, cm^2} \\ & = 1.26 \times 10^4 dm^2 \end{align*}