To determine the exponential function $f(t) = ab^t$ that passes through the points $(0,20)$ and $(5,110)$, we can use the following steps:

Step 1: Identify $a$ from the point $(0,20)$

For an exponential function $f(t) = ab^t$, the point $(0,20)$ means: $f(0) = ab^0 = a = 20$ So, we have $a = 20$.

Step 2: Substitute $a$ and use the point $(5,110)$ to find $b$

Now that we know $a = 20$, we can rewrite the function as: $f(t) = 20b^t$

Using the point $(5,110)$, we substitute $t = 5$ and $f(5) = 110$: $110 = 20b^5$

Step 3: Solve for $b$

Divide both sides by 20 to isolate $b^5$: $\frac{110}{20} = b^5$ $5.5 = b^5$

Now, take the fifth root of both sides to solve for $b$: $b = \sqrt[5]{5.5}$ Calculating $\sqrt[5]{5.5}$ gives approximately: $b \approx 1.3797$

Step 4: Write the final function

Now we have both $a$ and $b$, so the function is: $f(t) = 20 \cdot (1.3797)^t$

Final Answer

The exponential function in standard form is: $f(t) = 20 \cdot 1.3797^t$

Would you like further details or have any questions?

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Here are five related questions to deepen your understanding:

1. How would the function change if the point $(0,20)$ was replaced with a different y-intercept?
2. What is the general method to find an exponential function given any two points?
3. How can you verify that $f(t) = 20 \cdot 1.3797^t$ fits both points?
4. What would happen to the function if $t$ represented time in years, and $f(t)$ represented population growth?
5. How do changes in the value of $b$ affect the growth rate of the function?

Tip: In exponential functions, $a$ represents the initial value (or y-intercept), while $b$ represents the growth (if $b > 1$) or decay (if $0 < b < 1$) factor.