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v1 Algebra, STD2 A1 2012 HSC 21 MC

Which of the following correctly expresses  \(r\)  as the subject of  \(V=\pi r^2+x\) ?

  1. \(r=\pm\sqrt{\dfrac{V}{\pi}}-x\)
  2. \(r=\pm\sqrt{\dfrac{V}{\pi}-x}\)
  3. \(r=\pm\sqrt{\dfrac{V-x}{\pi}}\)
  4. \(r=\pm\dfrac{\sqrt{V-x}}{\pi}\)
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\(C\)

Show Worked Solution
\(V\) \(=\pi r^2+x\)
\(\pi r^2\) \(=V-x\)
\(r^2\) \(=\dfrac{V-x}{\pi}\)
\(\therefore\ r\) \(=\pm\sqrt{\dfrac{V-x}{\pi}}\)

\(\Rightarrow C\)

v1 Algebra, STD2 A1 2011 HSC 18 MC

Which of the following correctly expresses  \(b\)  as the subject of  \(y= ax+\dfrac{1}{4}bx^2\)?

  1. \(b=\dfrac{4y-ax}{x^2}\)
  2. \(b=\dfrac{4(y-ax)}{x^2}\)
  3. \(b=\dfrac{\dfrac{1}{4}y-ax}{x^2}\)
  4. \(b=\dfrac{\dfrac{1}{4}(y-ax)}{x^2}\)
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\(B\)

Show Worked Solution
\(y\) \(= ax+\dfrac{1}{4}bx^2\)
\(\dfrac{1}{4}bx^2\) \(=y-ax\)
\(bx^2\)  \(=4(y-ax)\)
\(b\) \(=\dfrac{4(y-ax)}{x^2}\)

 
\(\Rightarrow B\)

v1 Algebra, STD2 A1 2004 HSC 11 MC

If  \(m = 8n^2\), what is a possible value of  \(n\)  when  \(m=7200\)?

  1. \(0.0\dot{3}\)
  2. \(30\)
  3. \(240\)
  4. \(900\)
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\(B\)

Show Worked Solution
\(m\) \(=8n^2\)
\(n^2\) \(=\dfrac{m}{8}\)
\(n\) \(=\pm\sqrt{\dfrac{m}{8}}\)

 
\(\text{When}\ m=7200:\)

\(n\) \(=\pm\sqrt{\dfrac{7200}{8}}\)
  \(=\pm 30\)

 
\(\Rightarrow B\)

v1 Algebra, STD2 A1 2006 HSC 18 MC

What is the formula for \(g\) as the subject of \(7d=8e+5g^2\)?

  1. \(g =\pm\sqrt{\dfrac{8e-7d}{5}}\)
  2. \(g =\pm\sqrt{\dfrac{7d-8e}{5}}\)
  3. \(g =\pm\dfrac{\sqrt{7d+8e}}{5}\)
  4. \(g =\pm\dfrac{\sqrt{8e-7d}}{5}\)
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\(B\)

Show Worked Solution
\(7d\) \(=8e+5g^2\)
\(5g^2\) \(=7d-8e\)
\(g^2\) \(=\dfrac{7d-8e}{5}\)
\(g\) \(=\pm\sqrt{\dfrac{7d-8e}{5}}\)

\(\Rightarrow B\)

v1 Algebra, STD2 A1 SM-Bank 11

Make  \(V\)  the subject of the equation  \(E=\dfrac{3}{2}mV^3\).  (3 marks)

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\(v=\sqrt[3]{\dfrac{2E}{3m}}\)

Show Worked Solution
\(E\) \(=\dfrac{3}{2}mV^3\)
\(2E\) \(=3mV^3\)
\(\dfrac{2E}{3}\) \(=mV^3\)
\(V^3\) \(=\dfrac{2E}{3m}\)
\(V\) \(=\sqrt[3]{\dfrac{2E}{3m}}\)

v1 Algebra, STD2 A1 SM-Bank 9

The volume of a sphere is given by  \(V=\dfrac{4}{3}\pi r^3\)  where  \(r\)  is the radius of the sphere.

If the volume of a sphere is  \(385\ \text{cm}^3\), find the radius, to 1 decimal place.  (3 marks)

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\(4.5\ \text{cm  (to 1 d.p.)}\)

Show Worked Solution
\(V\) \(=\dfrac{4}{3}\pi r^3\)
\(3V\) \(= 4\pi r^3\)
\(r^3\) \(=\dfrac{3V}{4\pi}\)

 

\(\text{When}\ \ V =385\)

\(r^3\) \(=\dfrac{3\times 385}{4\pi}\)
  \(=91.911\dots\)
\(\therefore\ r\) \(=\sqrt[3]{91.911\dots}\)
  \(=4.512\dots\ \ \text{(by calc)}\)
  \(=4.5\ \text{cm   (to 1 d.p.)}\)