Chapter 3.2, Problem 14E

Solution

To determine: the exponential function that satisfies the given condition initial mass $=15\text{g}$ and decreasing at a rate of $4.6\%$ per day.

The exponential function is $15{\left(0.954\right)}^x$ .

Given information:

The initial mass $m_0$ is $15\text{g}$ .

The mass is decreasing at a rate of $4.6\%$ per day.

Concept Used:

If a population $P$ is changing at a constant percentage rate $r$ each year, then $P\left(t\right)=P_0{\left(1+r\right)}^t$ where $P_0$ is the initial population, $r$ is expressed as a decimal, and $t$ is time in years.

If $r>0$ , then $P\left(t\right)$ is an exponential growth function, and its growth factor is the base of the exponential function, $1+r$ .

Calculation:

Let $m\left(x\right)$ be the mass that changes at a constant growth rate $r$ per week, then the mass growth function is $m\left(x\right)=m_0{\left(1+r\right)}^x$ .

Here, $m_0=15$

  $\begin{array}{c}r=-4.6\%\\ =-0.046\end{array}$

Substitute $15$ for $h_0$ and $-0.046$ for $r$ in $h\left(t\right)=h_0{\left(1+r\right)}^x$ .

  $\begin{array}{c}m\left(x\right)=15{\left(1+\left(-0.046\right)\right)}^x\\ =15{\left(0.954\right)}^x\end{array}$

Hence, the exponential function is $15{\left(0.954\right)}^x$ .