Answer to Question #204603 in Calculus for Jace

Question

In “x” years from now, one investment plan will be generating profit at the rate of R­₁(x) = 50 + x² pesos per year, while a second plan will be generating profit at the rate R₂(x) = 200 + 5x pesos per year. For how many years will the second plan be more profitable one? Compute also the net excess profit if the second plan would be used instead of the first.

Answer 1

$$R_1(x)<R_2(x)$$

$$50+x^2<200+5x, x\geq0$$

$$x^2-5x-150<0$$

$$(x+10)(x-15)<0$$

Since $x\geq0,$ we take

$$0\leq x<15.$$

The excess profit of plan 2 over the plan 1 is

$$E(t)=R_2(t)-R_1(t)=200+5x-(50+x^2)$$

$$=150+5x-x^2$$

The net excess profit over the tine period $0\leq t\leq 15$

$$Net\ Profit=\displaystyle\int_{0}^{15}(150+5x-x^2)dx$$

$$=[150x+\dfrac{5x^2}{2}-\dfrac{x^3}{3}]\begin{matrix} 15 \\ 0 \end{matrix}$$

$$=2250+562.5-1125=1687.5$$

$0\leq x<15\ years, P1687.50$