Question

(0,4) (1,7.6) (2,14.44)
fill in the missing elements of the exponential growth equation.

Answer 1

To deduce the exponential growth equation from the given points \((0,4)\), \((1,7.6)\), and \((2,14.44)\), we can assume the general form of an exponential growth equation as:

\[
y = a \cdot e^{(bt)}
\]

where \(a\) is the initial value, \(b\) is the growth rate, and \(t\) is time.

From the first point \((0, 4)\):
\[
y(0) = a \cdot e^{(b \cdot 0)} = a \cdot 1 = a = 4
\]

So, we have:
\[
a = 4
\]

Now we can rewrite the equation as:
\[
y = 4 \cdot e^{(bt)}
\]

Using the second point \((1, 7.6)\):
\[
7.6 = 4 \cdot e^{b \cdot 1}
\]
\[
e^{b} = \frac{7.6}{4} = 1.9
\]

To solve for \(b\):
\[
b = \ln(1.9)
\]

Now, we can also confirm this using the third point \((2, 14.44)\):
\[
14.44 = 4 \cdot e^{(b \cdot 2)}
\]
\[
e^{2b} = \frac{14.44}{4} = 3.61
\]

We can express \(e^{2b}\) as \((e^b)^2\):
\[
(e^b)^2 = 3.61
\]
\[
(1.9)^2 = 3.61
\]

This confirms that our value of \(b = \ln(1.9)\) is consistent.

Now that we have \(a\) and \(b\):
1. \(a = 4\)
2. \(b = \ln(1.9)\)

Thus, the complete exponential growth equation can be written as:
\[
y = 4 \cdot e^{(\ln(1.9) \cdot t)}
\]

This can also be rewritten with \(b \approx 0.624\) for simplicity, so:
\[
y \approx 4 \cdot e^{0.624t}
\]

This function should fit the given data points well.