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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematical Methods CAS => Topic started by: Tyleralp1 on February 09, 2014, 04:01:29 pm

Title: Need Help with a few questions. Simultaneous and Transposing
Post by: Tyleralp1 on February 09, 2014, 04:01:29 pm: I'm having some slight difficulty in working out the following two problems.

For the volume question, I've figured out the volume for each respective shape with t being the radius. Not sure how to get the ratio of each, and then find the values of each letter and thus total area.

For wine and water questions, I've tried making several simultaneous equations, but none seem to work.

All help is highly appreciated :)

(http://s21.postimg.org/9mzagbp07/IMAG0481.jpg)(http://s8.postimg.org/dlhra0qmt/IMAG0480.jpg)

Mod edit: resizing images - Phy124
Title: Re: Need Help with a few questions. Simultaneous and Transposing
Post by: b^3 on February 09, 2014, 04:41:39 pm: I have an odd feeling I stuffed something up somewhere but the numbers seemed to work out.

You just got to try and work with what you've got, when you equate the volumes you can get each variable in terms of $t$ , then from that you know their ratio with respect to $t$ from the coefficients when the coefficient on $t$ is $1$ . Once you have that you can substitute back in the other two variables in terms of $t$ to expressed everything you have in terms of only one variable, and use the piece of information given in the last part to solve for $t$ , then substituting back into your total volume.
$\begin{alignedat}{1}V_{hemisphere} & =\frac{1}{2}\times\frac{4}{3}\pi r^{3} \\ & =\frac{2}{3}\pi t^{3} \\ V_{cylinder} & =\pi r^{2}h \\ & =\pi t^{2}s \\ V_{cone} & =\frac{1}{3}\pi r^{2}h \\ & =\frac{1}{3}\pi t^{2}w \\ V_{hemi}= & V_{cyl} \\ \frac{2}{3}\pi t^{3} & =\pi t^{2}s \\ \frac{2}{3}t & =s \\ s & =\frac{2}{3}t \\ V_{hemi} & =V_{cone} \\ \frac{2}{3}\pi t^{3} & =\frac{1}{3}\pi t^{2}w \\ 2t & =w \\ w & =2t \\ \text{Taking each ratio with respect to }t \\ w:s:t & =2:\frac{2}{3}:1 \\ w:s:t & =6:2:3 \\ V_{total} & =V_{hemi}+V_{cyl}+V_{cone} \\ & =\frac{2}{3}\pi t^{3}+\pi t^{2}s+\frac{1}{3}\pi t^{2}w \\ & =\frac{1}{3}\pi t^{2}\left(2t+3s+w\right) \\ \text{But }s=\frac{2}{3}t,\: w=2t \\ V_{total} & =\frac{1}{3}\pi t^{2}\left(2t+3\left(\frac{2}{3}\right)t+2t\right) \\ & =\frac{1}{3}\pi t^{2}\left(6t\right) \\ & =2\pi t^{3} \\ w+s+t & =11 \\ 2t+\frac{2}{3}t+t & =11 \\ 6t+2t+3t & =33 \\ 11t & =33 \\ t & =3 \\ V_{total} & =2\pi\left(3\right)^{3} \\ & =54\pi\: u^{3} \end{alignedat}$
Title: Re: Need Help with a few questions. Simultaneous and Transposing
Post by: Tyleralp1 on February 09, 2014, 05:41:55 pm: Thank you sooo much :D :D :D