I sat for an exam a few days ago. I managed to answer every question except for question $1$c in the calculus paper. Provided that I got question $2$d correct (my answer was $m=0.5$), the absence of an answer for question $1$c should not in any way affect my overall grade. Out of interest, however, I am wondering if you could start me off on how to approach problem $1$c. Both the questions can be found below.
Thank you very much.
Question $1$c:
An enclosure at a reserve is divided into two rectangular cages.
One side of the enclosure is a solid wall.
There is a fence around the rest of the enclosure and another between the male and female cages.
The total length of the fences is $275$ m.
$5$ male birds are kept in the smaller cage and $8$ female birds in the larger cage.
The males have an average area of $250$ sq.m each in their cage.
The area of the female cage is to be maximised.
Find the average area for each of the female birds in their cage.
Question $2$d:
The diagram below shows part of the graph of the function $y=x^2, x>0$.
The shaded region between the curve and the $X$-axis, and between $x = m$ and $x = m + 2$, has an area of $5 \frac16$ sq.units.
Find the value of $m$.
Explain the choice of the solution.
Best Answer
Using the following labeling:
Therefore $3x+y+z=275$.
Therefore $\frac {xz} 5 = 250$. You are to maximize the area of the female cage, which is given by $A(x,y) = xy$.
You can use substitution (with regard to the constraints) to write a function for the area of the cage which is only dependent on one variable, and then maximize it using the normal approach of taking the derivative and setting it to zero. Alternatively, you can use the Lagrange multiplier method.