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Kylee manages a small theme park and she has been analyzing the attendance data. Kylee finds that the number of visitors increases exponentially as the temperature increases, and this situation is represented by the function f(x) = 4x. Kylee also finds a linear equation that models the number of people who leave the park early depending on the change in temperature, and it is represented by g(x) = −x + 5. The graph of the two functions is below. Find the solution to the two functions and explain what the solution represents.

Kylee Manages A Small Theme Park And She Has Been Analyzing The Attendance Data Kylee Finds That The Number Of Visitors Increases Exponentially As The Temperatu class=

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Thus, the solution is (1,4) and that represents the number in which the number of visitors incoming when the temperature is increasing matches the number of visitors leaving early when the temperature is decreasing.

1) The solution to both functions is that point that is located at the intersection of the curve and the line.

2) So, let's solve this system of equations:

[tex]\begin{gathered} \begin{matrix}y=4^x\end{matrix} \\ y=-x+5 \\ \end{gathered}[/tex]

Note that we can apply the Substitution Method:

[tex]\begin{gathered} 4^x=-x+5 \\ \ln4^x=\ln_(-x+5) \\ x\ln(4)=\ln_(-x+5) \\ x=\frac{\ln(-x+5)}{\ln(4)} \\ \frac{\ln \left(-x+5\right)}{\ln \left(4\right)}=x \\ \frac{\ln \left(-x+5\right)}{\ln \left(4\right)}\ln \left(4\right)=x\ln \left(4\right) \\ \ln \left(-x+5\right)=x\ln \left(4\right) \\ \ln \left(-x+5\right)=2\ln \left(2\right)x \\ e^{\ln \left(-x+5\right)}=e^{2\ln \left(2\right)x} \\ -x+5=4^x \\ -(1)+5=4^1 \\ 4=4 \\ x=1 \end{gathered}[/tex]

With the quantity of x=1 we can plug it into the second formula:

3)

[tex]\begin{gathered} y=-x+5 \\ y=-1+5 \\ y=4 \end{gathered}[/tex]

4) Thus, the solution is (1,4) and that represents the number in which the number of visitors incoming when the temperature is increasing matches the number of visitors leaving.

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