Question

3. An exponential equation of the form, =⋅, passes through the points (12,147) and (18, 1029). A. Find this equation. Be sure to show your work.B. Using this equation, find y when x=8. Round to 3 decimal places.

Answer 1

We have an exponential function and 2 points that belong to that function.

We have to find the parameters "a" and "b" of the function.

We can start wit parameter b using both points. We divide the value of y for both points as:

$\begin{gathered} \frac{y_2}{y_1}=\frac{a\cdot b^{x_2}}{a\cdot b^{x_1}}=\frac{b^{x_2}}{b^{x_1}}=b^{x_2-x_1} \\ b=\sqrt[x_2-x_1]{\frac{y_2}{y_1}} \end{gathered}$

Replacing with the values (x1, y1) = (12, 147) and (x2, y2) = (18, 1029) we get the value of b:

$\begin{gathered} b=\sqrt[18-12]{\frac{1029}{147}} \\ b=\sqrt[6]{7} \\ b\approx1.383 \end{gathered}$

Now, we can use this result and one of the points to find the parameter a as:

$\begin{gathered} y_1=a\cdot1.383^{x_1} \\ 147=a\cdot1.383^{12} \\ a=\frac{1.383^{12}}{147} \\ a=\frac{49}{147} \\ a=\frac{1}{3} \\ a\approx0.333 \end{gathered}$

Now we can write the equation as:

$y=0.333\cdot1.383^x$

We can find the value of y when x=8 replacing x with 8 in the exponential function and calculating y:

$\begin{gathered} y(8)=0.333\cdot1.383^8 \\ y(8)=0.333\cdot13.387 \\ y(8)=4.457 \end{gathered}$

Answer:

A. The equation is y=0.333 * 1.383^x.

B. The value of y(8) is 4.457