Find values for the constants $a, b$, and $t_{0}$ so that the quantity described is represented by the function $Q=a b^{t-t_{0}}$. A house whose value increases by $7.7 \%$ per year is worth $\$ 350,000$ in year $t=7$. \[ \begin{array}{l} a= \\ b= \\ t_{0}= \end{array} \]
Solution
Step 1 ：The problem is asking us to find the values of the constants \(a\), \(b\), and \(t_{0}\) for the function \(Q=a b^{t-t_{0}}\) that represents the value of a house that increases by \(7.7 \%\) per year and is worth \$350,000 in year \(t=7\).
Step 2 ：We know that the value of the house increases by \(7.7 \%\) per year, so \(b\) should be \(1.077\) (since \(1\) represents the initial value and \(0.077\) is the increase per year).
Step 3 ：We also know that the house is worth \$350,000 in year \(t=7\), so we can substitute these values into the function to solve for \(a\) and \(t_{0}\).
Step 4 ：Let's first solve for \(a\) by setting \(t=7\) and \(Q=350000\) in the function and assuming \(t_{0}=0\) (since we don't have any information about \(t_{0}\) yet). After finding \(a\), we can try to find a suitable \(t_{0}\) that makes the function valid for all \(t \geq 0\).
Step 5 ：We have found the value of \(a\) to be approximately \(208237.106853773*1077.0^{t0}/10.0^{3*t0}\). However, this value of \(a\) is dependent on \(t_{0}\), which we have not yet determined.
Step 6 ：Since we don't have any information about \(t_{0}\), we can assume that \(t_{0}=0\) for simplicity. This means that the house started increasing in value from year \(t=0\).
Step 7 ：Let's substitute \(t_{0}=0\) into the value of \(a\) and see what we get.
Step 8 ：After substituting \(t_{0}=0\) into the value of \(a\), we get \(a \approx 208237.106853773\).
Step 9 ：So, the constants \(a\), \(b\), and \(t_{0}\) that make the function \(Q=a b^{t-t_{0}}\) represent the value of a house that increases by \(7.7 \%\) per year and is worth \$350,000 in year \(t=7\) are approximately \(a \approx 208237.106853773\), \(b = 1.077\), and \(t_{0}=0\).
Step 10 ：Final Answer: \(a \approx \boxed{208237.106853773}\), \(b = \boxed{1.077}\), \(t_{0} = \boxed{0}\)
