Aussie Maths & Science Teachers: Save your time with SmarterEd
The masses of avocados in a crop may be assumed to be normally distributed, with a mean of 200 grams and a standard deviation of 7.5 grams.
After an avocado of mass \(M\) grams is peeled and the stone is removed, the mass of edible flesh \(F\) grams is given by \(F=0.70 M\). Four avocados are randomly selected from the crop.
What is the probability, correct to four decimal places, that a total of more than 570 grams of edible flesh is obtained?
Particle 1 has position vector \({\underset{\sim}{r}}_1(t)=\cos (t) \underset{\sim}{ i }+\sin (t) \underset{\sim}{ j }+\sqrt{\sin (2 t)} \underset{\sim}{ k }\) and Particle 2 has position vector \({\underset{\sim}{r}}_2(t)=\sin (t) \underset{\sim}{ i }+\cos (t) \underset{\sim}{ j }+\sqrt{\sin (2 t)} \underset{\sim}{ k }\), where \(t\) is measured in seconds and \(t \in\left(0, \dfrac{\pi}{2}\right)\).
The number of times the velocity of Particle 1 is perpendicular to the position vector \({\underset{\sim}{r}}_2(t)\) during the first \(\dfrac{\pi}{2}\) seconds is
The position of a moving body is given by \(\underset{\sim}{ r }(t)=\sin (t) \underset{\sim}{ i }+\cos (2 t) \underset{\sim}{ j }\), where \(t\) is measured in seconds, for \(t \geq 0\).
The motion of the body can be described as moving along a parabolic path given by
Analyse how the different components of the FITT principle could be manipulated to create a periodised anaerobic training program for a 400 m runner. (8 marks)
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Compare and contrast how the FITT principle would be implemented for a basketball player focusing on anaerobic training versus aerobic training. (8 marks)
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Consider the vectors \(\underset{\sim}{r}\) and \(\underset{\sim}{s}\) where \(\abs{\underset{\sim}{ r }}=9\) and \(\underset{\sim}{ s }=2 \underset{\sim}{ i }-2 \underset{\sim}{ j }+\underset{\sim}{k}\).
If the vector resolute of \(\underset{\sim}{r}\) in the direction of \(\underset{\sim}{s}\) is equal to \(-4 \underset{\sim}{i}+4 \underset{\sim}{j}-2 \underset{\sim}{k}\), then the scalar resolute of \(\underset{\sim}{s}\) in the direction of \(\underset{\sim}{r}\) is equal to
The position, \(x\) metres, of a particle moving in a straight line from a fixed origin \(O\) at time, \(t\) seconds, is given by \(x=e^{(k-1) t}\), where \(k>1\).
The acceleration of the particle, in m s\(^{-2}\), when \(x=k+1\) is
The length of the curve specified by \(x=1-\cos (t)\) and \(y=t-\sin (t)\), where \(t \in[0,2 \pi]\), is given by
Let the lines \(l_1\) and \(l_2\) be defined by \(l_1:{\underset{\sim}{r}}_1(\lambda)=\underset{\sim}{i}+m \underset{\sim}{k}+\lambda(\underset{\sim}{i}+2\underset{\sim}{j}+\underset{\sim}{k})\) and \(l_2:{\underset{\sim}{r}}_2(\mu)=2 \underset{\sim}{ i }-\underset{\sim}{ k }+\mu(-\underset{\sim}{ i }+3 \underset{\sim}{ j }+2 \underset{\sim}{ k })\), where \(m \in R \backslash\left\{-\dfrac{4}{5}\right\}\) and \(\lambda, \mu \in R\). If the shortest distance between the two skew lines \(l_1\) and \(l_2\) is \(\dfrac{14}{\sqrt{35}}\), find the values of \(m\). (3 marks) --- 10 WORK AREA LINES (style=lined) ---
Evaluate the effectiveness of different aerobic training methods for developing a marathon runner's conditioning program based on the FITT principle. (10 marks)
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A triathlete is designing an aerobic training program based on the FITT principle. Which progression model would be most appropriate for the "Time" component over an 8-week period?
A function is defined as \(f(x)=\dfrac{x-h}{(x+1)(x-4)}\)
Determine the range of \(h\) where the graph of \(f(x)\) have no turning points. (3 marks)
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Consider the function \(f\) with rule \(f(x)=\left\{\begin{array}{cl}\dfrac{x^2+3 x-10}{x-2} & , x \in R \backslash\{2\} \\ 7 & , x=2\end{array}\right.\)
Which of the following statements is correct?
A car is travelling along a straight, flat road. The velocity, \(v\) km h\(^{-1}\), of the car and its position, \(x\) kilometres, are measured from the position on the road where \(x=0\). The velocity \(v\) and the position \(x\) of the car are related by \(v^2=1600+\dfrac{672}{\pi} \arccos \left(\dfrac{x}{20}\right)\), where \(-15 \leq x \leq 15\) and \(v \geq 0\). A speed detection device is positioned to detect the speed of a car as it passes the position \(x=0\). The speed limit on the road is 40 km h\(^{-1}\). The speed detection device will be activated if the car is travelling at 10% or more above the speed limit. --- 5 WORK AREA LINES (style=lined) --- --- 8 WORK AREA LINES (style=lined) ---
Consider the relation \(x^2 y^2+x y=2\), where \(x, y \in R\). --- 6 WORK AREA LINES (style=lined) --- --- 6 WORK AREA LINES (style=lined) ---
The production of a brand of weed trimmer involves three stages, Stage 1, Stage 2 and Stage 3, which take \(W_1\) hours, \(W_2\) hours and \(W_3\) hours, respectively. Here \(W_1, W_2\) and \(W_3\) are independent random variables, which may be assumed to be normally distributed. Assume that Stage 2 starts immediately after Stage 1 ends and that Stage 3 starts immediately after Stage 2 ends. The mean, standard deviation and cost at each stage are shown in the table below. \begin{array}{|c|c|c|c|c|} --- 5 WORK AREA LINES (style=lined) --- --- 8 WORK AREA LINES (style=lined) --- --- 10 WORK AREA LINES (style=lined) ---
\hline
\rule{0pt}{2.5ex} \textbf{Stage} & \textbf{Time (h) } & \textbf{Mean(h)} &\textbf{Standard } & \textbf{Cost (\$/h)} \\
& & & \rule[-1ex]{0pt}{0pt} \textbf{deviation (h)}\\
\hline \rule{0pt}{2.5ex} 1 \rule[-1ex]{0pt}{0pt}& W_1 & 1.0 & 0.3 & 10 \\
\hline \rule{0pt}{2.5ex} 2 \rule[-1ex]{0pt}{0pt}& W_2 & 1.5 & 0.4 & 20 \\
\hline \rule{0pt}{2.5ex} 3 \rule[-1ex]{0pt}{0pt}& W_3 & 2.0 & 0.5 & 15 \\
\hline
\end{array}
Design a 6-week aerobic training program for a netball center court player using the FITT principle and explain how you would progress the program to ensure continuous improvement. (12 marks)
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Analyse the differences between High Intensity Interval Training (HIIT) and Sprint Interval Training (SIT), and explain how each could be effectively incorporated into a training program for track cyclists. Provide specific examples to support your response. (8 marks)
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The table below compares two different interval training methods.
\begin{array} {|l|l|l|l|l|}
\hline \textbf{Training} & \textbf{Work} & \textbf{Work Intensity} & \textbf{Rest} & \textbf{Total Session} \\
\textbf{Method} & \textbf{Duration} & & \textbf{Duration} & \textbf{Duration}\\
\hline \text{Method X} & \text{30 seconds} & \text{All-out maximal effort} & \text{4 minutes} & \text{20 - 25 minutes} \\
\hline \text{Method Y} & \text{4 minutes} & \text{90% of MHR} & \text{3 minutes} & \text{30 - 35 minutes} \\
\hline \end{array}
Which of the following correctly identifies Methods X and Y and their primary training effects?
Let \(f: R \backslash\{-1\} \rightarrow R, f(x)=\dfrac{(x-1)^2}{(x+1)^2}\) The rule \(f(x)\) can be written in the form \(f(x)=A+\dfrac{B}{x+1}+\dfrac{C}{(x+1)^2}\), where \(A, B, C \in Z\). --- 5 WORK AREA LINES (style=lined) --- --- 6 WORK AREA LINES (style=lined) --- --- 3 WORK AREA LINES (style=lined) --- a. \(\text{See worked solutions.}\) b. \(\text{Turning point at}\ (1,0)\) c. b. \(f^{′}(x)=\dfrac{4}{(x+1)^2}-\dfrac{8}{(x+1)^3}\) \(\text{SP’s when}\ \ f^{′}(x)=0:\) \(\therefore \text{Turning point at}\ (1,0)\) c. \(\text{Vertical asymptote at}\ \ x=-1\) \(f(0)=\dfrac{(-1)^2}{(1)^2}=1\ \ \Rightarrow \ \ y\text{-intercept at}\ (0,1)\) \(\text{As}\ x\rightarrow \infty, \ 1-\dfrac{4}{(x+1)}+\dfrac{4}{(x+1)^2} \rightarrow 1^{-} \) \(\text{As}\ x\rightarrow -\infty, \ 1-\dfrac{4}{(x+1)}+\dfrac{4}{(x+1)^2} \rightarrow 1^{+} \) \(\text{Horizontal asymptote at}\ \ y=1 \)
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Show Worked Solution
a.
\(\dfrac{(x-1)^2}{(x+1)^2}\)
\(=\dfrac{x^2-2x+1}{x^2+2x+1}\)
\(=\dfrac{x^2+2x+1}{x^2+2x+1}-\dfrac{4x}{x^2+2x+1}\)
\(=1-\dfrac{4(x+1)}{(x+1)^2}+\dfrac{4}{(x+1)^2}\)
\(=1-\dfrac{4}{(x+1)}+\dfrac{4}{(x+1)^2}\)
\(\dfrac{4}{(x+1)^2}\)
\(=\dfrac{8}{(x+1)^3}\)
\(4(x+1)\)
\(=8\)
\(x\)
\(=1\)
\(f(1)=0\)
♦♦ Mean mark (c) 31%.
An aircraft flies with a constant velocity of 95 ms\(^{-1}\) North. During the flight, a the plane experiences 2 different cross winds. --- 3 WORK AREA LINES (style=lined) --- --- 8 WORK AREA LINES (style=lined) ---
a. → Using vector addition as shown in the diagram above: \(R=\sqrt{40^2+95^2} = 103.1\ \text{ms}^{-1}\) → The resultant velocity is \(103.1\ \text{ms}^{-1}\) at \(337^{\circ}\)T. b. → Using the cosine rule, \(c^2=a^2+b^2-2ab\cos C\) where \(C\) is the angle opposite the unknown side. \(c=\sqrt{50^2+95^2-2 \times 50 \times 95 \times \cos 30} = 57.4261\ \text{ms}^{-1}\) To determine the angle, use the sine rule: → The resultant velocity of the plane is \(57.4\ \text{ms}^{-1}\) at \(026^{\circ}\)T.Show Worked Solution
\(\tan \theta\)
\(=\dfrac{40}{95}\)
\(\theta\)
\(=\tan^{-1}\left(\dfrac{40}{95}\right)\)
\(=23^{\circ}\)
\(\dfrac{\sin \theta}{50}\)
\(=\dfrac{\sin 30}{R}\)
\(\sin \theta\)
\(=\dfrac{50 \times \sin 30}{57.4261}\)
\(\theta\)
\(=\sin^{-1}\left(\dfrac{50 \times \sin 30}{57.4261}\right)\)
\(=25.8^{\circ}\)
Consider the function with rule \(f(z)=3 z^3+2 i z^2+3 z+2 i\), where \(z \in C\). --- 3 WORK AREA LINES (style=lined) --- --- 5 WORK AREA LINES (style=lined) --- --- 0 WORK AREA LINES (style=lined) --- a. \(\text{Method 1}\) \(f(z)=3 z^3+2 i z^2+3 z+2 i = (3z+2i)(z^2+1)\) \(\text{Method 2}\) \(f\left( -\dfrac{2i}{3} \right)=\dfrac{8}{9}-\dfrac{8}{9}-2i+2i=0\) \(\text{By factor theorem:}\) \(z+ \dfrac{2}{3} i\ \ \text{is a factor of}\ f(z) \) \(\text{Hence,}\ 3\left( z+\dfrac{2i}{3} \right) = 3z+2i\ \ \text{is a factor.}\) b. \(f(z)= (3z+2i)(z^2+1)\) \(z=-\dfrac{2i}{3} , z= \pm i\) c. a. \(\text{Method 1}\) \(f(z)=3 z^3+2 i z^2+3 z+2 i = (3z+2i)(z^2+1)\) \(\text{Method 2}\) \(f\left( -\dfrac{2i}{3} \right)=\dfrac{8}{9}-\dfrac{8}{9}-2i+2i=0\) \(\text{By factor theorem:}\) \(z+ \dfrac{2}{3} i\ \ \text{is a factor of}\ f(z) \) \(\text{Hence,}\ 3\left( z+\dfrac{2i}{3} \right) = 3z+2i\ \ \text{is a factor.}\) b. \(f(z)= (3z+2i)(z^2+1)\) \(z=-\dfrac{2i}{3} , z= \pm i\) c.
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Show Worked Solution
♦ Mean mark (c) 49%.
Suppose that a function \(f(x)\) and its derivative \(f^{\prime}(x)\) satisfy \(f(4)=25\) and \(f^{\prime}(4)=15\).
Determine the gradient of the tangent line to the graph of \( {\displaystyle y=\sqrt{f(x)} } \) at \( x=4 \).
The function \(f(x)=\dfrac{x}{2}+\dfrac{2}{x}\) undergoes the following sequence of transformations to become \(g(x)\):
The function \(g\) has a local minimum at the point with the coordinates
The graph of \(y=f(x)\) is shown below.
Which of the following options best represents the graph of \(y=f(2 x+1)\) ?
Twelve students sit in a classroom, with seven students in the first row and the other five students in the second row. Three students are chosen randomly from the class.
The probability that exactly two of the three students chosen are in the first row is
At a Year 12 formal, 45% of the students travelled to the event in a hired limousine, while the remaining 55% were driven to the event by a parent.
Of the students who travelled in a hired limousine, 30% had a professional photo taken.
Of the students who were driven by a parent, 60% had a professional photo taken.
Given that a student had a professional photo taken, what is the probability that the student travelled to the event in a hired limousine? (3 marks)
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| \(\text{Pr(Limo|PP)}\) | \(=\dfrac{\text{Pr(Limo)}\ \cap\ \text{Pr(PP)}}{\text{Pr(PP)}}\) |
| \(=\dfrac{0.45\times 0.30}{(0.45\times 0.30)+(0.55\times 0.6)}\) | |
| \(=\dfrac{0.135}{0.465}\) | |
| \(=\dfrac{9}{31}\) |
Analyse the benefits and limitations of implementing either continuous aerobic training or High Intensity Interval Training (HIIT) as the primary training method for a cricket team during pre-season. (8 marks)
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Outline an experimental procedure to determine the acceleration of a falling steel ball. Your explanation should include all the measurements that must be recorded, the calculations needed to compute the acceleration, and an identification of any potential sources of error in the experiment. (6 marks)
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Evaluate the benefits and limitations of Fartlek training compared to structured interval training for a hockey team. (8 marks)
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Evaluate the effectiveness of different training methods for a 400 m track athlete throughout a competition season. (8 marks)
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Analyse how contemporary training methods like HIIT and SIT have evolved from traditional aerobic and anaerobic training approaches. Use examples from two different sports. (8 marks)
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An athlete's training program includes:
This represents:
Which training method would be MOST appropriate for a 400 m track athlete?
Compare how continuous training would be applied differently for an individual sport athlete versus a team sport athlete. (6 marks)
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Explain how a soccer player could use continuous training to improve their aerobic capacity during pre-season. (5 marks)
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Solve \(2 \log _3(x-4)+\log _3(x)=2\) for \(x\). (4 marks) --- 8 WORK AREA LINES (style=lined) ---
\(1^3-8(1)^2+16(1)-9=0\) \(\therefore\ x-1\ \text{is a factor} \) \((x-1)(x^2-7x+9)=0\) \(\text{Using quadratic formula to solve}\ \ x^2-7x+9=0:\) \( x=1, \dfrac{7- \sqrt{13}}{2}, \dfrac{7 + \sqrt{13}}{2}\) \(\therefore\ \dfrac{7 + \sqrt{13}}{2}\ \text{ is the only possible solution.}\)
Show Worked Solution
\(2\log_3(x-4)+\log_3(x)\)
\(=2\)
\(\log_3x(x-4)^2\)
\(=2\)
\(x(x-4)^2\)
\(=3^2\)
\(x(x^2-8x+16)-9\)
\(=0\)
\(x^3-8x^2+16x-9\)
\(=0\)
\(\text{Find a factor}\ \ \Rightarrow\ \ \text{Test}\ \ x=1:\)
♦♦ Mean mark 36%.
\(x\)
\(=\dfrac{-(-7)\pm\sqrt{(-7)^2-4(1)(9)}}{2(1)}\)
\(=\dfrac{7\pm \sqrt{49-36}}{2}\)
\(=\dfrac{7\pm \sqrt{13}}{2}\)
\(\text{For }\log_3(x-4)\ \text{to exist}\ x>4\)
Evaluate the importance of glycogen loading for aerobic versus anaerobic activities. (4 marks)
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Analyse how micronutrient requirements vary between aerobic and anaerobic athletes. (4 marks)
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Compare pre and post-training nutrition requirements for a gymnast. (5 marks)
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Assess how different types of dietary fat impact an athlete's performance. (5 marks)
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During a 3-hour tennis match in summer, an athlete consumes only water. What micronutrient imbalance is likely to occur first?
Compare the nutritional requirements of a sprinter versus a marathon runner in preparation for competition. (5 marks)
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Assess the importance of iron intake for female endurance athletes. (5 marks)
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A powerlifter notices their muscular strength has declined despite maintaining their training program. Which micronutrient deficiency is most likely causing this issue?
Compare the causes of fatigue between a 400m sprint and a marathon runner. (4 marks)
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Which row correctly identifies the main cause of fatigue for different duration activities?
\begin{align*}
\begin{array}{l}
\rule{0pt}{2.5ex} \ \rule[-1ex]{0pt}{0pt}& \\
\rule{0pt}{2.5ex}\textbf{A.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex}\textbf{B.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex}\textbf{C.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex}\textbf{D.}\rule[-1ex]{0pt}{0pt}\\
\end{array}
\begin{array}{|l|l|l|}
\hline
\rule{0pt}{2.5ex}\textbf{10 second sprint}\rule[-1ex]{0pt}{0pt}& \textbf{2 minute swim}& \textbf{2 hour run} \\
\hline
\rule{0pt}{2.5ex}\text{ATP depletion}\rule[-1ex]{0pt}{0pt}&\text{Lactic acid}&\text{Glycogen depletion}\\
\hline
\rule{0pt}{2.5ex}\text{Lactic acid}\rule[-1ex]{0pt}{0pt}& \text{Glycogen depletion}&\text{ATP depletion}\\
\hline
\rule{0pt}{2.5ex}\text{Glycogen depletion}\rule[-1ex]{0pt}{0pt}& \text{ATP depletion}&\text{Lactic acid} \\
\hline
\rule{0pt}{2.5ex}\text{Lactic acid}\rule[-1ex]{0pt}{0pt}& \text{ATP depletion}&\text{Glycogen depletion} \\
\hline
\end{array}
\end{align*}
Analyse how the three energy systems interact during a 3-minute boxing round. (5 marks)
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A manufacturer \((M)\) makes deliveries to the supermarket \((S)\) via a number of storage warehouses, \(L, N, O, P, Q\) and \(R\). These eight locations are represented as vertices in the network below. The numbers on the edges represent the maximum number of deliveries that can be made between these locations each day. --- 2 WORK AREA LINES (style=lined) --- --- 3 WORK AREA LINES (style=lined) ---
a. \(13+18+6+9=46\) \(\text{(Reverse flow}\ Q → O\ \text{is not counted.)}\) b. \(\text{Max deliveries (min cut)}\ =13+5+11+8=37\)
Show Worked Solution
♦ Mean mark (b) 29%.
An athlete trains using different intensities across a session. Which energy system requires the LONGEST recovery period?
Analyse how varying durations of training activities affect energy system development in soccer players. (5 marks)
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During a marathon run lasting over 3 hours, which energy system contribution would be most accurate?
Analyse how different intensities of training can affect energy system adaptations. (5 marks)
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A health rating, `R`, is calculated by dividing a person’s weight, `w`, in kilograms by the square of the person’s height, `h`, in metres.
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The cost of hiring an open space for a music festival is $120 000. The cost will be shared equally by the people attending the festival, so that `C` (in dollars) is the cost per person when `n` people attend the festival.
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i.
\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 240 & 120 & 80 & 60 & 48\ & 40 \ \\
\hline
\end{array}
| ii. |  |
iii. `C = (120\ 000)/n`
`n\ text(must be a whole number)`
iv. `text(Limitations can include:)`
`•\ n\ text(must be a whole number)`
`•\ C > 0`
v. `text(If)\ C = 94:`
| `94` | `= (120\ 000)/n` |
| `94n` | `= 120\ 000` |
| `n` | `= (120\ 000)/94` |
| `= 1276.595…` |
`:.\ text(C)text(ost cannot be $94 per person,)`
`text(because)\ n\ text(isn’t a whole number.)`
i.
\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of people} (n) \rule[-1ex]{0pt}{0pt} & \ 500\ & \ 1000 \ & 1500 \ & 2000 \ & 2500\ & 3000 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 240 & 120 & 80 & 60 & 48\ & 40 \ \\
\hline
\end{array}
| ii. | |
iii. `C = (120\ 000)/n`
iv. `text(Limitations can include:)`
`•\ n\ text(must be a whole number)`
`•\ C > 0`
v. `text(If)\ C = 94`
| `=> 94` | `= (120\ 000)/n` |
| `94n` | `= 120\ 000` |
| `n` | `= (120\ 000)/94` |
| `= 1276.595…` |
`:.\ text(C)text(ost cannot be $94 per person,)`
`text(because)\ n\ text(isn’t a whole number.)`
The number of trees that can be planted along the fence line of a paddock varies inversely with the distance between each tree.
There will be 108 trees if the distance between them is 5 metres.
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A student believes that the time it takes for an ice cube to melt (`M` minutes) varies inversely with the room temperature `(T^@ text{C})`. The student observes that at a room temperature of `15^@text{C}` it takes 12 minutes for an ice cube to melt.
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\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \ \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \ & \ \ 15\ \ \ & \ \ \ 30\ \ \ \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\end{array}
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| a. | `M` | `prop 1/T` |
| `M` | `=k/T` | |
| `12` | `=k/15` | |
| `k` | `=15 xx 12` | |
| `=180` |
`:.M=180/T`
b.
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \ \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \ & \ \ 15\ \ \ & \ \ 30\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & 36 & 12 & 6 \\
\hline
\end{array}
| a. | `M` | `prop 1/T` |
| `M` | `=k/T` | |
| `12` | `=k/15` | |
| `k` | `=15 xx 12` | |
| `=180` |
`:.M=180/T`
b.
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \ \rule[-1ex]{0pt}{0pt} & \ \ \ 5\ \ \ & \ \ 15\ \ \ & \ \ 30\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ M\ \ \rule[-1ex]{0pt}{0pt} & 36 & 12 & 6 \\
\hline
\end{array}
When people walk in snow, the depth (`D` cm) of each footprint depends on both the area (`A` cm²) of the shoe sole and the weight of the person. The graph shows the relationship between the area of the shoe sole and the depth of the footprint in snow, for a group of people of the same weight.
Find the equation relating `D` and `A`. (2 marks)
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Use your equation from part (i) to calculate the area of his shoe sole. (1 mark)
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The air pressure, `P`, in a bubble varies inversely with the volume, `V`, of the bubble.
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By finding the value of the constant, `a`, find the value of `P` when `V = 4`. (2 marks)
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Use the horizontal axis to represent volume and the vertical axis to represent air pressure. (2 marks)
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| i. | `P` | `prop 1/V` |
| `= a/V` |
| ii. | `text(When)\ P=3,\ V = 2` |
| `3` | `= a/2` |
| `a` | `=6` |
`text(Need to find)\ P\ text(when)\ V = 4`
| `P` | `=6/4` |
| `= 1 1/2` |
| iii. | |
The time taken to clean a warehouse varies inversely with the number of cleaners employed.
It takes 8 cleaners 60 hours to clean a warehouse.
Working at the same rate, how many hours would it take 10 cleaners to clean the same warehouse.