The gross annual earnings of a certain company were $E ( t ) = 0.2 t ^ { 2 } + 9 t + 30$ thousand dollars where t is the number of years since its formation in 1990. The gross annual earnings with respect to t in 1995 are growing at 13.75%.

Question

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To find the rate of growth of the gross annual earnings in 1995, we differentiate $E(t)$ with respect to $t$ to get $E'(t) = 0.4t + 9$. Substituting $t = 5$ (since 1995 is 5 years after 1990) into $E'(t)$, we get $E'(5) = 0.4(5) + 9 = 11$ thousand dollars per year. To find the percentage growth rate, we divide this by the total earnings in 1995, $E(5) = 0.2(5)^2 + 9(5) + 30 = 2.5 + 45 + 30 = 77.5$ thousand dollars, and then multiply by 100. Thus, the growth rate is $\frac{11}{77.5} \times 100 = 14.19\%$. However, the statement claims a growth rate of 13.75%, which seems to be a slight miscalculation. Given the context, the correct calculation should yield a growth rate close to 14.19%, but the assertion of a specific growth rate of 13.75% could be considered a simplification or rounding in the question's context. Therefore, based on the precise calculation, the statement as presented would be incorrect, but if the question implies a rounded or approximate value, it could be seen as a misunderstanding of the exact calculation.

The Gross Annual Earnings of a Certain Company Were E(t)=0.2t2+9t+30E ( t ) = 0.2 t ^ { 2 } + 9 t + 30E(t)=0.2t2+9t+30

The Gross Annual Earnings of a Certain Company Were $E ( t ) = 0.2 t ^ { 2 } + 9 t + 30$