Question

1. the volume of an object as a function of time is calculated by ( v = at^{3}+b/t ), where ( t ) is time measured in seconds and ( v ) is in cubic - meters. determine the dimension of the constants ( a ) and ( b ).

Explanation:

Step1: Analyze the units - volume

The volume $V$ is in cubic - meters ($m^{3}$). The formula for volume is $V = At^{3}+\frac{B}{t}$.

Step2: Analyze the units of each term

For the term $At^{3}$, since $t$ is in seconds ($s$), let the unit of $A$ be $x$. Then the unit of $At^{3}$ is $x\times s^{3}$. Since $At^{3}$ is part of the volume with unit $m^{3}$, we have $x\times s^{3}=m^{3}$, so $x = \frac{m^{3}}{s^{3}}$.

Step3: Analyze the second term

For the term $\frac{B}{t}$, let the unit of $B$ be $y$. Then the unit of $\frac{B}{t}$ is $\frac{y}{s}$. Since $\frac{B}{t}$ is part of the volume with unit $m^{3}$, we have $\frac{y}{s}=m^{3}$, so $y = m^{3}\times s$.

Answer:

The unit of $A$ is $\frac{m^{3}}{s^{3}}$ and the unit of $B$ is $m^{3}\cdot s$.