Question

evaluating and solving functions
use the function f(x) = 180(1.1)^x to answer the following questions. round your answers to two decimal places.
evaluate f(9): f(9) =
determine x when f(x) = 450: x =

Explanation:

Part 1: Evaluate \( f(9) \)

Step 1: Substitute \( x = 9 \) into the function

We have the function \( f(x)=180(1.1)^{x} \). Substitute \( x = 9 \) into it: \( f(9)=180(1.1)^{9} \)

Step 2: Calculate \( (1.1)^{9} \)

First, calculate \( 1.1^{9} \). Using a calculator, \( 1.1^{9}\approx2.357947691 \)

Step 3: Multiply by 180

Then multiply this result by 180: \( 180\times2.357947691\approx424.4305844 \)

Step 4: Round to two decimal places

Rounding \( 424.4305844 \) to two decimal places gives \( 424.43 \)

Part 2: Determine \( x \) when \( f(x) = 450 \)

Step 1: Set up the equation

We set \( 180(1.1)^{x}=450 \)

Step 2: Divide both sides by 180

Divide both sides of the equation by 180: \( (1.1)^{x}=\frac{450}{180} = 2.5 \)

Step 3: Take the logarithm of both sides

Take the natural logarithm (or common logarithm) of both sides. Using natural logarithm: \( \ln(1.1^{x})=\ln(2.5) \)

Step 4: Use the logarithm power rule

Using the power rule of logarithms \( \ln(a^{b}) = b\ln(a) \), we get \( x\ln(1.1)=\ln(2.5) \)

Step 5: Solve for \( x \)

Divide both sides by \( \ln(1.1) \): \( x=\frac{\ln(2.5)}{\ln(1.1)} \)

Step 6: Calculate the value

Using a calculator, \( \ln(2.5)\approx0.91629073187 \) and \( \ln(1.1)\approx0.095310179804 \). Then \( x=\frac{0.91629073187}{0.095310179804}\approx9.61 \) (rounded to two decimal places)

Answer:

For \( f(9) \): \( \boldsymbol{424.43} \)

For \( x \) when \( f(x) = 450 \): \( \boldsymbol{9.61} \)