The population of a fast-growing city in Texas can be modeled with the equation \(p(t) = 82 \boldcdot e^{(0.078t)}\). The population of a fast-growing city in Tennessee can be modeled with \(q(t) = 132 \boldcdot e^{(0.047t)}\). In both equations, \(t\) represents years since 2016, and the population is measured in thousands. The graphs representing the two functions are shown. The point where the two graphs intersect has a \(y\)-coordinate of about 271.7.

1. What does the intersection mean in this situation?
2. Find the \(t\)-coordinate of the intersection point by solving each equation. Show your reasoning.

   \(p(t) = 271.7\)

   \(q(t)=271.7\)
3. Explain why we can find out the \(t\) value at the intersection of the two graphs by solving \(p(t) = q(t)\).

The function \(f\) is given by \(f(x) = 100 \boldcdot 3^x\). Select all equations whose graph meets the graph of \(f\) for a positive value of \(x\).

The half-life of nickel-63 is 100 years. A student says, “An artifact with nickel-63 in it will lose a quarter of that substance in 50 years.”

Do you agree with this statement? Explain your reasoning.

Technology required. Estimate the value of each expression and record it. Then use a calculator to find its value and record it.

Here is a graph that represents \(f(x) = e^x\). 

Explain how we can use the graph to estimate:

1. the solution to an equation such as \(300 = e^x\).
2. the value of \(\ln 700\).