Question

directions: use this information to answer parts a and b. solve the equation $6e^{0.11x} - 18 = 72$ by graphing. round to the nearest tenth.

Explanation:

Step1: Rewrite the equation

First, rewrite the equation \(6e^{0.11x}-18 = 72\) as two separate functions: \(y = 6e^{0.11x}-18\) and \(y = 72\).

Step2: Analyze the functions

The function \(y = 6e^{0.11x}-18\) is an exponential function. The exponential function \(y = e^{kx}\) (here \(k = 0.11\)) has a horizontal asymptote. For \(y = 6e^{0.11x}-18\), as \(x\to-\infty\), \(e^{0.11x}\to0\), so \(y\to - 18\). As \(x\) increases, \(e^{0.11x}\) increases exponentially, so \(y = 6e^{0.11x}-18\) increases exponentially. The function \(y = 72\) is a horizontal line.

Step3: Find the intersection point

To solve the equation, we need to find the \(x\)-value where the two graphs intersect. We can use a graphing calculator or software to graph both functions.
First, simplify the original equation:

$$
LATEXBLOCK0
$$

Now, take the natural logarithm of both sides:

$$
LATEXBLOCK1
$$

Calculate \(\ln(15)\approx2.70805\), then \(x=\frac{2.70805}{0.11}\approx24.6\) (when rounded to the nearest tenth). When we graph \(y = 6e^{0.11x}-18\) and \(y = 72\), the \(x\)-coordinate of their intersection point is approximately \(24.6\).

Answer:

\(x\approx24.6\)