Answer to Question #159570 in Calculus for dsfd

Question

 Angela forgot to study for her APUSH exam and it is the day before the exam. She worries that she will fail if she doesn’t study at all, so she decides she will study for at least 2 hours. She estimates her potential score to be 𝑆 = (120𝑡)/( 𝑡+2 )if she studies for 𝑡 hours. But she also knows that the longer she studies, the more tired she will become, and she won’t reach her potential. She estimates that her “fatigue factor” is 𝐹 = (10)/ (𝑡+10). To find her projected grade, 𝐺, she multiplies her potential score by her fatigue factor so that 𝐺 = 𝑆 ⋅ 𝐹. How many hours should Angela study to maximize her grade, 𝐺?

Answer 1

Consider the function

$G(t)=S(t)\cdot F(t)=\frac{120𝑡}{ 𝑡+2}\cdot\frac{10}{t+10}=1200\frac{t}{t^2+12t+20}$

Let us find the derivative:

$G'(t)=1200\frac{t^2+12t+20-t(2t+12)}{(t^2+12t+20)^2}=1200\frac{-t^2+20}{(t^2+12t+20)^2}$

It follows that for $t\ge 0$ we have that $G'(t)=0$ if $t=\sqrt{20}=2\sqrt{5}$

Consequently, her projected grade $𝐺(t)$ is increasing function for all $0\le t\le 2\sqrt{5}$, and $𝐺(t)$ is decreasing function for all $t\ge 2\sqrt{5}$. Thererfore, Angela should study $2\sqrt{5}\approx 4.47$ hours to maximize her grade.

Answer: 4 hours and 28 minutes