AP Calculus BC 5.10 Optimization Overview

Review optimisation by defining a target function, finding feasible critical points and justifying the best value in context.

[Maximum number: 3]

An invasive species of plant appears in a fruit grove at time t=0 and begins to spread. The

function C defined by $C(t)=7.6 \arctan (0.2 t)$ models the number of acres in the fruit grove

affected by the species t weeks after the species appears. It can be shown that $C^{\prime}(t)=\frac{38}{25+t^{2}}$ .

(Note: Your calculator should be in radian mode.)

At time t=4 weeks after the invasive species appears in the fruit grove, measures are taken

to counter the spread of the species. The function A, defined by $A(t)=C(t)-\int_{4}^{t} 0.1 \cdot \ln (x) d x$ ,

models the number of acres affected by the species over the time interval $4 \leq t \leq 36$ . At what

time t, for $4 \leq t \leq 36, \operatorname{does} A$ attain its maximum value? Justify your answer.

D At time t=4 weeks after the invasive species appears in the fruit grove, measures are taken to counter

the spread of the species. The function A, defined by $A(t)=C(t)-\int_{4}^{t} 0.1 \cdot \ln (x) d x$ , models the number

of acres affected by the species over the time interval $4 \leq t \leq 36$ . At what time t, for $4 \leq t \leq 36$ , does

A attain its maximum value? Justify your answer.

$A^{\prime}(t)=C^{\prime}(t)-0.1 \cdot \ln t$	Considers $A^{\prime}(t)=0 \quad$ Point 7 (P7)
For $4 \leq t \leq 36$ , the maximum value of A(t) occurs when $A^{\prime}(t)=0$ or at an endpoint. $A^{\prime}(t)=C^{\prime}(t)-0.1 \cdot \ln t=0 \Rightarrow C^{\prime}(t)=0.1 \cdot \ln t$
$\Rightarrow t=11.441700$ t	A(t)<br>4	5.128031<br>11.441700	7.316978<br>36	1.743056	Justification Point 8 (P8)
Therefore, the number of acres affected by the species is a maximum at time t=11.442 (or 11.441) weeks.	Answer with supporting work

Scoring Notes for

All question bank results loaded

Covered syllabus points

5.10 Introduction to Optimization Problems question 1