For an upcoming grade 11 test, there is this question i don't understand. Here are the requirements: The perimeter is a maximum of $60$ cm.
The question wants me to come up with the dimensions of this shape:
I cannot change the fact that there are two sets of equal side, so there are 2 variables that need to be found. I am stuck, i don't know what to do. I have the correct answer from my teacher but not the process. Answer $= 14.06$cm and $8.91$cm
Can someone please show me how to do this? My test starts on 2017-09-27. If you answer it after that, it is still okay, hopefully others will learn from it.
$\endgroup$1 Answer
$\begingroup$Call $a$ the base of the shape, and $b$ its height. The perimeter $P$ is
$$ P = 3a + 2b ~~~\Rightarrow~~~ b = (P - 3a)/2 \tag{1} $$
and its area is
$$ A = ab + \frac{\sqrt{3}}{4}a^2 \tag{2} $$
where the first term is the area of a rectangle, whereas the second one is the area of an equilateral triangle of side $a$.
After you replace (2) in (1) you get
$$ A = \frac{1}{4} a ((-6 + \sqrt{3}) a + 2 P) = \frac{1}{4} (\sqrt{3} - 6) \left(\frac{P}{\sqrt{3} - 6} + a\right)^2 - \frac{P^2}{4 (\sqrt{3} - 6)} \tag{3} $$
The first term is negative, the second is positive, so the maximum of the area is reached when the first term is zero, and that happens when
$$ a = \frac{P}{6 -\sqrt{3}} \tag{4} $$
replacing in (1) you can get the value of $b$
$$ b= \frac{1}{22} (5 - \sqrt{3}) P \tag{5} $$
$\endgroup$ 7
