2u q (1 Viewer)

xibu34 · Sep 24, 2023

How do I do this question ?

I've done the show part and found r as 7, although I'm unsure on how I can actually show this as the minimum perimeter.

Edit: Think I've got it but would like to see someone else's working

synthesisFR · Sep 24, 2023

xibu34 said:
View attachment 39907
How do I do this question ?

I've done the show part and found r as 7, although I'm unsure on how I can actually show this as the minimum perimeter.

Edit: Think I've got it but would like to see someone else's working

Wait this question was in my schools 2u paper
I’ll send when I get home
Idk if my school bought a paper then or made their own I think usually they do both and mix it

xibu34 · Sep 24, 2023

synthesisFR said:
Wait this question was in my schools 2u paper

I go to a 600+ ranked school or something

Luukas.2 · Sep 24, 2023

From the area:

$\begin{align*} A &= \cfrac{1}{2} r^2 \theta \\ 49 &= \cfrac{1}{2} r^2 \theta \\ \theta &= \cfrac{98}{r^2} \end{align*}$

The perimeter is then

$\begin{align*} P &= r + r + r\theta \\ &= 2r + r \times \cfrac{98}{r^2} \\ &= 2r + \cfrac{98}{r} \\ &= 2r + 98r^{-1} \end{align*}$

Calculus:

$\begin{align*} \cfrac{dP}{dr} &= 2 - 98r^{-2} \\ &= 2 - \cfrac{98}{r^2} \\ \text{Put $\cfrac{dP}{dr} = 0$}: \qquad 2 &= \cfrac{98}{r^2} \\ r^2 &= 49 \\ r &= 7 \qquad \text{as $r$ is a distance and so $r > 0$} \end{align^}$

Nature of the Stationary Point at $r = 7 \text{ cm}$ :

$\begin{align*} \cfrac{d^2P}{dr^2} &= \cfrac{d}{dr}\left( 2 - 98r^{-2}\right) \\ &= 0 - 98 \times -2r^{-3} \\ &= \cfrac{196}{r^3} \\ \text{At $r = 7$ cm}: \qquad \cfrac{d^2P}{dr^2} &= \cfrac{4}{7} > 0 \end{align*}$

So, the stationary point for the perimeter is a minimum, and thus the minimum perimeter occurs when $r = 7 \text{ cm}$ and $\theta = 2 \text{ radians}$ . Under these conditions:

$\begin{align*} \text{Arc length } &= r\theta = 7 \times 2 = 14 \text{ cm} \\ \text{Diameter } &= 2r = 2 \times 7 = 14 \text{ cm} \end{align*}$

and so the minimum perimeter length occurs when the diameter of the circle and the arc length are equal, as required.

Luukas.2 · Sep 24, 2023

synthesisFR said:
Wait this question was in my schools 2u paper
I’ll send when I get home
Idk if my school bought a paper then or made their own I think usually they do both and mix it

It's from the 2023 Western Maths paper

scaryshark09 · Sep 24, 2023

xibu34 said:
I go to a 600+ ranked school or something

some schools share questions for their trials

xibu34 · Sep 24, 2023

synthesisFR said:
Wait this question was in my schools 2u paper
I’ll send when I get home
Idk if my school bought a paper then or made their own I think usually they do both and mix it

what was your schools average ? was this an easy paper this year ?

synthesisFR · Sep 24, 2023

xibu34 said:
what was your schools average ? was this an easy paper this year ?

Idk tbh I think it was alr some ppl did rlly bad some did rlly good

xibu34 · Sep 24, 2023

synthesisFR said:
Wait this question was in my schools 2u paper
I’ll send when I get home
Idk if my school bought a paper then or made their own I think usually they do both and mix it

Did you also have the lightbulb PDF question in it ?

synthesisFR · Sep 24, 2023

xibu34 said:
Did you also have the lightbulb PDF question in it ?

i think so i dont remember

2u q (1 Viewer)

xibu34

Active Member

synthesisFR

School Spirit Skit 1

xibu34

Active Member

Luukas.2

Well-Known Member

Luukas.2

Well-Known Member

scaryshark09

∞∆ who let 'em cook dis long ∆∞

xibu34

Active Member

synthesisFR

School Spirit Skit 1

xibu34

Active Member

synthesisFR

School Spirit Skit 1

Users Who Are Viewing This Thread (Users: 0, Guests: 1)