Questions: Find C and a so that f(x)=Cax satisfies the given conditions. f(0)=16 and for each unit increase in x, the output is multiplied by 1.5. C= a= (Type an integer or a decimal.)

Find C and a so that f(x)=Cax satisfies the given conditions. f(0)=16 and for each unit increase in x, the output is multiplied by 1.5.

C=
a=

(Type an integer or a decimal.)

Transcript text: Find C and a so that $\mathrm{f}(\mathrm{x})=\mathrm{Ca}^{\mathrm{x}}$ satisfies the given conditions. $f(0)=16$ and for each unit increase in $x$, the output is multiplied by 1.5. \[ \begin{array}{l} \mathrm{C}=\square \\ \mathrm{a}=\square \end{array} \] (Type an integer or a decimal.)

Solution

Solution Steps

Step 1: Use the condition $ f(0) = 16 $ to find $ C $

The function is given by $ f(x) = Ca^x $. When $ x = 0 $, we have: \[ f(0) = Ca^0 = C \cdot 1 = C \] Since $ f(0) = 16 $, it follows that: \[ C = 16 \]

Step 2: Use the condition that the output is multiplied by 1.5 for each unit increase in $ x $ to find $ a $

For each unit increase in $ x $, the output is multiplied by 1.5. This means: \[ f(x+1) = 1.5 \cdot f(x) \] Substitute $ f(x) = Ca^x $ into the equation: \[ Ca^{x+1} = 1.5 \cdot Ca^x \] Divide both sides by $ Ca^x $: \[ a = 1.5 \]

Step 3: Write the final values of $ C $ and $ a $

From Step 1 and Step 2, we have: \[ C = 16 \quad \text{and} \quad a = 1.5 \]