Chapter 6.1, Problem 51E

a.

To determine

To write: an exponential function that is represented by the given graph:

Answer to Problem 51E

  $y=40{\left(\frac{3}{2}\right)}^x$

Explanation of Solution

Concept used:

Exponential function:

  $y=a{b}^x$ ….. (1)

Here, the y-values changes by a factor of b as x increases by 1 and the y intercept is a.

Using the given graph to make a table of values as shown below:

  

The graph intersects the y-axis at 40. This implies that y-intercept is 40.

Also from the table it can be observed that as x increases by 1 the y-values changes by a factor of $\frac{3}{2}$ .

So, substituting $a=40$ and $b=\frac{3}{2}$ in equation (1), the required exponential function is:

  $y=40{\left(\frac{3}{2}\right)}^x$

b.

To determine

the number of visitors after 5 months.

Answer to Problem 51E

Approximately 304

Explanation of Solution

Given:

The number y of visitors to the gallery after x months is given by:

  $y=40{\left(\frac{3}{2}\right)}^x$

The number of visitors after 5 months is found by substituting $x=5$ in the above function.

  $\begin{array}{c}y\left(5\right)=40{\left(\frac{3}{2}\right)}^5\\ =40\left(7.59375\right)\\ =303.75\end{array}$

So, number of visitors after 5 months is about 304.