4.1: Exponential Models | Algebra and Trigonometry

How it works

An exponential model multiplies by a constant factor over equal input intervals. The initial value is the output at input zero. A growth factor above 1 increases values, while a factor between 0 and 1 models decay. Identify the time unit before interpreting a percentage rate.

A(t)=A_0b^t,\qquad b=1+r\text{ for growth}

Worked examples

Example 1

A quantity starts at $\displaystyle 200$ and grows $\displaystyle 10\%$ per year. Find its value after $\displaystyle 2$ years.

The factor is $\displaystyle 1.10$ . Thus $\displaystyle A(2)=200(1.10)^2=242$ .

Example 2

Write a model for $\displaystyle 80$ units decreasing $\displaystyle 25\%$ per hour.

The hourly factor is $\displaystyle 0.75$ , so $\displaystyle A(t)=80(0.75)^t$ .

Problems

Problem 4.1.1

Find the value of $\displaystyle P(t)=500(1.04)^t$ at $\displaystyle t=2$ .

Show worked solution

$\displaystyle P(2)=500(1.04)^2=540.8$ .

Problem 4.1.2

A quantity halves every $\displaystyle 3$ hours. Write its model when it starts at $\displaystyle 120$ .

Show worked solution

The number of half-life intervals is $\displaystyle t/3$ , so $\displaystyle A(t)=120(1/2)^{t/3}$ .

Problem 4.1.3

From the graph of $\displaystyle A(t)=100(1.2)^t$ , identify the initial value and percentage growth per time unit.

The graph of 100 times 1.2 to the t starts at 100 and increases. — Read the initial value and compare equal time intervals.

Show worked solution

$\displaystyle A(0)=100$ , so the initial value is $\displaystyle 100$ . The factor $\displaystyle 1.2=1+0.2$ gives $\displaystyle 20\%$ growth per time unit.

The graph of 100 times 1.2 to the t starts at 100 and increases. The model starts at 100 and grows by 20% per time unit. — The model starts at 100 and grows by 20% per time unit.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 4.1.1

Problem 4.1.2

Problem 4.1.3

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