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MC3 Algebra Help
Can I get a hint for 4.26 and 4.28?
Question 4.26:
In tetrahedron $P-ABC$, $\angle APB=\angle BPC=\angle CPA=90^\circ$, and the sum of the lengths of the six edges is $S$. Find the maximum possible value for its volume.This is like a corner of a rectangular block. To begin, you can let $a=PA$, $b=PB$, and $c=PC$. Then $AB=\sqrt{a^2+b^2}$, $BC=\sqrt{b^2+c^2}$, $CA=\sqrt{c^2+a^2}$, and \[ S = a+b+c+\sqrt{a^2+b^2} + \sqrt{b^2+c^2} + \sqrt{c^2+a^2}, \] and also the volume \[ V=\frac{abc}{6}. \]
Using AM-GM we can have $a+b+c\geq 3\sqrt[3]{abc}$, and $a^2+b^2\geq 2ab$, etc. (note that equality holds for all those if $a=b=c$), and from here try to find an inequality between $S$ and $V$.Question 4.28:
Find the maximum and minimum values of: $\sqrt{5-x^2} +\sqrt{3}x$When there is a term like $\sqrt{5-x^2}$, a trig substitution is usually helpful (this is a very handy technique when you study calculus, either AP Calc or college level Calc). Let $x=\sqrt{5}\sin\alpha$ where $-90^\circ\leq \alpha\leq 90^\circ$. Then \[ y = \sqrt{5-x^2} +\sqrt{3}x = \sqrt{5}\cos\alpha + \sqrt{15}\sin\alpha = 2\sqrt{5}\sin(\alpha+30^\circ). \] Note that in the chosen interval for $\alpha$, this function does reach its maximum of $2\sqrt{5}$, but it does not reach the minimum $-2\sqrt{5}$. So what is the actual minimum?
For 4.28, why is there the restriction \( -\pi/2 \leq \alpha \leq \pi /2 \)?
I also need help with some other problems.
4.30. I tried completing the square or squaring the equation but that didn't work.
4.31. I applied Cauchy-schwarz with \( sin^2( \theta) + cos^2( \theta) = 1 \) and got \( 256=(sin^2 \theta + cos^2 \theta)(1/sin^6 \theta + 81/ cos^6 \theta) \geq(1/sin^2 \theta + 9/cos^2 \theta)^2 \), but I don't know where to go from there.
For 4.34, I tried AM-GM with \( a+b+c+d=8 a^2+b^2+c^2+d^2=8 \) but that didn't get me anywhere.
For 4.37, I tried factoring in the denominator and substituting \( 1-x^4=y^4+z^4 \) etc. but that didn't work.
For 4.38, no clue
For 4.28, it is not a restriction. In this interval, all possible $x$ is covered, and also $\cos\alpha \geq 0$, so you can express the square root without worrying about signs.
For 4.30, see each radical as the distance between two points.
For 4.31, you are on the right track, just do it again.
For 4.34, write $a+b+c=4-d$ and $a^2+b^2+c^2=8-d^2$, then apply Cauchy-Schwarz.
For 4.37, handle each term separately and try something similar to Problem 4.9.
For 4.38, try to use the basic inequality $y^2+z^2\geq 2yz$, and also a weighted AM-GM like $kx^2 + \dfrac{y^2}{4k} \geq xy$.
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