Question

Problem 2. Using dimesional analysis and units: a. For two objects with the same shape but different lengths, how does the area scale with the dimension Length? How does the volume scale with Length]? Explain. (Hint: consider the area of a square and the volume of a cube.) b. Lets apply this: Suppose you are a parent who's child needs to take medicine. You are 6 feet tall and your child is 3 feet tall (assume that you and your child have the same proportions). Knowing your own dose, estimate the dose for your child in the following cases. Explain your reasoning! . The medicine is absorbed throughout the body's volume. Your dose is 400 mg. » The medicine only coats the surface of your child's throat. Your dose is 30 ml

Answer 1

(a)

$A=a^2,V=a^3$ (Area and Volume of a Cube)

$A=[L]^2$

$V=[L]^3$

(b)

(i)

$\frac {V_c}{V_p}=\left (\frac {L_c}{L_p} \right )^3$

$\frac {V_c}{400}=\left (\frac {3}{6} \right )^3$

$V_c=50\: mg$

So the dose of the child is 50 mg.

(ii)

$\frac {A_c}{A_p}=\left (\frac {L_c}{L_p} \right )^2$

$\frac {A_c}{A_p}=\left (\frac {3}{6} \right )^2$

$\frac {A_c}{30}=\left (\frac {3}{6} \right )^2$

$A_c=\left (\frac {3}{6} \right )^2\cdot 30\: ml= 7.5\: ml$