We are given that the function $f(x)$ is an exponential function of the form $f(x) = ab^x$, and we know two values of the function:

$f(3.5) = 16 \quad \text{and} \quad f(7) = 62.$

We are asked to find $f(10.5)$ to the nearest hundredth.

Step 1: Write the general form

The exponential function is given by $f(x) = ab^x$, where $a$ and $b$ are constants to be determined.

Step 2: Set up equations using known values

We know:

1. $f(3.5) = ab^{3.5} = 16$
2. $f(7) = ab^7 = 62$

These give us the system of equations:

$ab^{3.5} = 16 \quad \text{(Equation 1)}$ $ab^7 = 62 \quad \text{(Equation 2)}$

Step 3: Solve for $b$

We divide Equation 2 by Equation 1 to eliminate $a$:

$\frac{ab^7}{ab^{3.5}} = \frac{62}{16}$

Simplifying:

$b^{7 - 3.5} = \frac{62}{16}$

$b^{3.5} = \frac{62}{16} = 3.875$

Now, solve for $b$ by taking the cube root:

$b = (3.875)^{\frac{1}{3.5}}$

Step 4: Solve for $a$

Once we have $b$, we can substitute it back into one of the original equations (say, Equation 1) to find $a$.

Step 5: Find $f(10.5)$

Finally, using the values of $a$ and $b$, we can find $f(10.5)$ by substituting $x = 10.5$ into the equation $f(x) = ab^x$.

I'll calculate this step by step.The value of $f(10.5)$ is approximately 240.25, to the nearest hundredth.

Would you like further details or have any questions about the steps? Here are some related questions to explore:

1. What happens to an exponential function as $x$ increases indefinitely?
2. How does the base $b$ of an exponential function affect the function's growth rate?
3. Can exponential functions ever have negative values for $f(x)$?
4. How would you find the inverse of an exponential function?
5. What are some real-life applications of exponential growth or decay?

Tip: For exponential functions, the base $b$ determines how quickly the function grows (if $b > 1$) or decays (if $0 < b < 1$).