Answer
After the year 2211, the United States will unable to serve diverse food.
Work Step by Step
The time when the United States will fail to serve diverse diet can be obtained by equating the equations:
\[\hat{A}=2495493942.35-116220.89t\] and \[A=\left( 1.2 \right)\cdot \left( 0.000000000417 \right)\cdot {{\left( 1.021 \right)}^{t}}\]
Hence, the time is calculated as
\[\begin{align}
& 2495493942.35-116220.89t=\left( 1.2 \right)\times \left( 0.000000000417 \right)\times {{\left( 1.021 \right)}^{t}} \\
& 4.99\times {{10}^{18}}-2.32\times {{10}^{14}}\times t={{\left( 1.021 \right)}^{t}} \\
& \log \left( 4.99\times {{10}^{18}} \right)-\log \left( 2.32\times {{10}^{14}}\times t \right)=\log \left( {{\left( 1.021 \right)}^{t}} \right) \\
& \log \left( 4.99 \right)+18-\log \left( 2.32 \right)-14-\log \left( t \right)=t\log \left( 1.021 \right)
\end{align}\]
\[\begin{align}
& t\times 0.009+\log \left( t \right)=4.33 \\
& t\approx 220.69
\end{align}\]
The value of \[t\] is obtained by the method of interpolation.
