Chapter 3.2, Problem 16E

Solution

To determine: the exponential function that satisfies the given condition initial population $=250$ and doubling every $7.5$ hr.

The exponential function is $250{\left(2\right)}^{\frac{x}{7.5}}$ .

Given information:

The initial population $p_0$ is $250$ .

The population is doubling every $7.5$ hr.

Concept Used:

Let $a$ and $b$ be real number constants. An exponential function in $x$ is a function that can be written in the form $f\left(x\right)=a{b}^x$ where a is nonzero, $b$ is positive, and $b\ne 1$ . The constant $a$ is the initial value of $f$ (the value at $x=0$ ), and $b$ is the base.

Calculation:

Here, $p_0=250$

  $r=2\left(\text{doubled}\right)$

Doubling every $7.5$ days means $\frac{x}{7.5}$ because $x$ is time in days.

Substitute $250$ for $p_0$ , $2$ for $r$ and replace $x$ with $\frac{x}{7.5}$ in $m\left(x\right)=m_0{r}^x$ .

  $m\left(x\right)=250{\left(2\right)}^{\frac{x}{7.5}}$

Hence, the exponential function is $250{\left(2\right)}^{\frac{x}{7.5}}$ .