2u maxima and minima help (1 Viewer)

mandemindiguise · Jan 15, 2024

How do i do q.i here

Luukas.2 · Jan 15, 2024

$\begin{align*} A_{\text{total}} &= A_{\Delta OKJ} + A_{\Delta OMP} \\ &= \frac{1}{2} \times JK \times JO \times \sin{\angle{OJK}} + \frac{1}{2} \times MO \times MP \times \sin{\angle{OMP}} \\ &= \frac{1}{2}sx\sin{\alpha} + \frac{1}{2}(l - x) \times MP \times \sin{\alpha} \qquad \text{as $\angle{OMP} = \angle{OJK} = \alpha$ (alternate angles on parallel lines)} \\ &= \frac{1}{2}\left[sx + (l - x)MP\right]\sin{\alpha} \qquad \text{. . . (1)} \end{align*}$

Now, we need to find the length of the side $MP$ .

This can be done once we recognise that we have a pair of similar triangles: $\Delta OJK ||| \Delta OMP\ \text{(AA)}$

$\begin{align*} \frac{OJ}{OM} = &\frac{OK}{OP} = \frac{JK}{MP} \qquad \text{(corresponding sides in similar triangles)} \\ \frac{x}{l - x} = &\frac{OK}{OP} = \frac{s}{MP} \\ MP &= \frac{s(l - x)}{x} \qquad \text{. . . (*)} \end{align*}$

$\begin{align*} \text{Put (*) into (1):} \qquad A_{\text{total}} &= \frac{1}{2}\left[sx + (l - x) \times \frac{s(l - x)}{x}\right]\sin{\alpha} \\ &= \frac{s}{2}\left[x + \frac{(l - x)^2}{x}\right]\sin{\alpha} \\ &= \frac{s}{2}\left(x + \frac{l^2 - 2xl + x^2}{x}\right)\sin{\alpha} \\ &= \frac{s}{2}\left(x + \frac{l^2}{x} - 2l + x\right)\sin{\alpha} \\ &= \frac{s}{2}\left(2x - 2l + \frac{l^2}{x}\right)\sin{\alpha} \\ &= s\left(\frac{2x}{2} - \frac{2l}{2} + \frac{l^2}{2x}\right)\sin{\alpha} \\ &= s\left(x - l + \frac{l^2}{2x}\right)\sin{\alpha} \qquad \text{as required} \end{align*}$

mandemindiguise · Jan 15, 2024

THANK U SM LUUKAS, UR GOATED

johnnnCENAAAA · Jan 15, 2024

mandemindiguise said:
THANK U SM LUUKAS, UR GOATED

shut up

johnnnCENAAAA · Jan 15, 2024

mandemindiguise said:
THANK U SM LUUKAS, UR GOATED

My presence sweet and my aura bright
Diamonds good for my appetite
Guess it fuckin' just clicked one night
All them demons helped me see shit differently
So don't be sad for me

How you been spending your time? (How?)
How you be using your tongue? (How?)
You be so worried 'bout mine (Mine)
Can't even get yourself none
You know you sound so dumb (So dumb, so dumb)
You know you sound so dumb (So dumb)

So maybe you should shut up, mmm
Yeah, maybe you should shut up
Said if only you would shut up (Mmm)
Yeah, that's right, you should shut up (Ah)

2u maxima and minima help (1 Viewer)

mandemindiguise

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Luukas.2

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mandemindiguise

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johnnnCENAAAA

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johnnnCENAAAA

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