Respuesta :
ANSWER
[tex]h(x)=52.48\cdot(0.7)^x^{}[/tex]EXPLANATION
We have to find the exponential function h(x) such that:
[tex]\begin{gathered} h(3)=18 \\ h(6)=6.174 \end{gathered}[/tex]The general form of an exponential function is:
[tex]h(x)=a\cdot b^x^{}[/tex]where a = coefficient; b = base
We have to find the values of a and b.
First, substitute x = 3 and h(x) = 18 into the general function:
[tex]18=a\cdot b^3[/tex]Next, repeat the procedure for x = 6 and h(x) = 6.174:
[tex]6.174=a\cdot b^6[/tex]Next, divide the two equations:
[tex]\begin{gathered} \Rightarrow\frac{6.174}{18}=\frac{a\cdot b^6}{a\cdot b^3} \\ \Rightarrow0.343=b^{6-3} \\ \Rightarrow b^3=0.343 \end{gathered}[/tex]Find the cube root of both sides:
[tex]\begin{gathered} b=\sqrt[3]{0.343} \\ b=0.7 \end{gathered}[/tex]Now, substitute the value of b into the first equation to find a:
[tex]\begin{gathered} 18=a\cdot(0.7)^3 \\ 18=a\cdot0.343 \\ \Rightarrow a=\frac{18}{0.343} \\ a\approx52.48 \end{gathered}[/tex]Therefore, the formula for the exponential function h(x) is:
[tex]h(x)=52.48\cdot(0.7)^x[/tex]