Question

1. the amount of money in an investment account is growing exponentially. in 2010, the account had $750, and 8 years later, in 2018, the amount in the account had grown to $1,368. the exponential function modeling this growth is shown in the graph below. (a) what is the average rate that the amount in the investment account changed from 2010 to 2018? user proper units in your answer. 3 points (b)if an exponential function of the form $y = a(b)^x$ was used to model this function, what would be the value of $b$ to the nearest hundredth? show how you found your answer. 3 points

Explanation:

Part (a)

Step1: Recall average rate formula

The average rate of change is given by $\frac{\Delta y}{\Delta x}$, where $\Delta y$ is the change in amount and $\Delta x$ is the change in time.

Step2: Identify values

In 2010 (x = 0), amount $y_1 = 750$ dollars. In 2018 (x = 8), amount $y_2 = 1368$ dollars. So $\Delta y = 1368 - 750 = 618$ dollars, $\Delta x = 8 - 0 = 8$ years.

Step3: Calculate average rate

Average rate = $\frac{618}{8} = 77.25$ dollars per year.

Step1: Recall exponential function form

The exponential function is $y = a(b)^x$. From the graph, when $x = 0$, $y = 750$. Substituting $x = 0$, $y = 750$ into $y = a(b)^x$, we get $750 = a(b)^0$. Since $b^0 = 1$, $a = 750$.

Step2: Substitute another point

We know when $x = 8$, $y = 1368$. Substitute $a = 750$, $x = 8$, $y = 1368$ into $y = 750(b)^8$. So $1368 = 750(b)^8$.

Step3: Solve for $b$

First, divide both sides by 750: $\frac{1368}{750} = b^8$. Simplify $\frac{1368}{750} = 1.824$. Then take the 8th root of both sides: $b = \sqrt[8]{1.824}$. Calculate $\sqrt[8]{1.824} \approx 1.07$ (using a calculator to find the 8th root of 1.824).

Answer:

The average rate is $77.25$ dollars per year.

Part (b)