Answer
0.4
Work Step by Step
The profit when x thousand tons of apples are sold is
$P(x)=\frac{20+x^{2}}{50+x}$
$P'(x)=\frac{2x(50+x)-1(20+x^{2})}{(50+x)^{2}}=\frac{x^{2}+100x-20}{x^{2}+100x+2500}$
$P''(x)=\frac{(2x+100)(x^{2}+100x+2500)-(2x+100)(x^{2}+100x-20)}{(x^{2}+100x+2500)^{2}}=\frac{2x^{3}+15000x+300x^{2}+250000-(2x^{3}+300x^{2}+9960x-2000)}{(x^{2}+100x+2500)^{2}}=\frac{5040x+27000}{(x^{2}+100x+2500)^{2}}$
$P(0)=\frac{2}{5}$
$P'(0)=\frac{-1}{125}$
$P''(0)=\frac{27}{6250}$
$P_2(x)=\frac{2}{5}-\frac{1}{125}+\frac{\frac{27}{6250}}{2!}x^{2}$
$P_{2}(x)=\frac{2}{5}+\frac{27}{3125}x^{2}$
$P_{2}(0.3)=\frac{2}{5}+\frac{27}{3125}x^{2} \approx 0.4$