Question

guided practice
use what you have learned about the parts of exponential functions to write an equation for each of the following situations and answer the questions.

1. everytime pinocchio lies, his nose doubles in size. his nose is 1.5 inches long before he has told any lies.

a. write an equation that represents this situation where x is the number of lies and y is the size of pinocchio’s nose after x lies.
b. use your equation to calculate how long pinocchio’s nose will be after 6 lies.

1. you make a $10,000 investment that has historically doubled every 7 years.

a. write an equation that represents this situation where x is the number of years and y is the investment value after x years.
b. use your equation to calculate the value of your investment after 14 years.

Explanation:

Step1: Define exponential growth formula

The general form of exponential growth is $y = a(b)^x$, where $a$ is the initial value, $b$ is the growth factor.

Step2: Solve 1a: Assign values to formula

Initial length $a=1.5$, growth factor $b=2$.
$y = 1.5(2)^x$

Step3: Solve 1b: Substitute $x=6$ into equation

Substitute $x=6$ into $y = 1.5(2)^x$.
$y = 1.5(2)^6 = 1.5 \times 64 = 96$

Step4: Solve 2a: Define doubling growth formula

For periodic doubling, the formula is $y = a(2)^{\frac{x}{t}}$, where $t$ is doubling time. Here $a=10000$, $t=7$.
$y = 10000(2)^{\frac{x}{7}}$

Step5: Solve 2b: Substitute $x=15$ into equation

Substitute $x=15$ into $y = 10000(2)^{\frac{x}{7}}$.
$y = 10000(2)^{\frac{15}{7}} \approx 10000 \times 4.665 = 46650$

Answer:

1. a. $y = 1.5(2)^x$

b. 96 inches

1. a. $y = 10000(2)^{\frac{x}{7}}$

b. $\approx \$46,650$