smc-4445-60-Hyperbola applications

v1 Algebra, STD2 A4 2014 HSC 29a

A golf club hires an entire course for a charity event at a total cost of `$40\ 000`. The cost will be shared equally among the players, so that `C` (in dollars) is the cost per player when `n` players attend.

Complete the table below by filling in the three missing values. (1 mark)
\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of players} (n) \rule[-1ex]{0pt}{0pt} & \ 50\ & \ 100 \ & 200 \ & 250 \ & 400\ & 500 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & & & & 160 & 100\ & 80 \ \\
\hline
\end{array}
Using the values from the table, draw the graph showing the relationship between `n` and `C`. (2 marks)
What equation represents the relationship between `n` and `C`? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
Give ONE limitation of this equation in relation to this context. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Is it possible for the cost per person to be $94? Support your answer with appropriate calculations. (1 mark)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of players} (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}

c. `C = (40\ 000)/n`

`n\ text(must be a whole number)`

d. `text(Limitations can include:)`

`•\ n\ text(must be a whole number)`

`•\ C > 0`

e. `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, because)\ n\ text(isn’t a whole number.)`

Show Worked Solution

\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Number of players} (n) \rule[-1ex]{0pt}{0pt} & \ 50\ & \ 100 \ & 200 \ & 250 \ & 400\ & 500 \ \\
\hline
\rule{0pt}{2.5ex}\text{Cost per person} (C)\rule[-1ex]{0pt}{0pt} & 800 & 400 & 200 & 160 & 100\ & 80 \ \\
\hline
\end{array}

c. `C = (40\ 000)/n`

TIP: Limitations require looking at possible restrictions of both the domain and range.

d. `text(Limitations can include:)`

`•\ n\ text(must be a whole number)`

`•\ C > 0`

e. `text(If)\ C = 120`

`120`	`= (40\ 000)/n`
`120n`	`= 40\ 000`
`n`	`= (40\ 000)/120`
	`= 333.33..`

`:.\ text{Cost cannot be $120 per person because the required}\ n\ \text{is not a whole number.}`

v1 Algebra, STD2 A4 2022 HSC 24

A chef believes that the time it takes to defrost a turkey (`D` hours) varies inversely with the room temperature (`T^\circ \text{C}`). The chef observes that at a room temperature of `20^\circ \text{C}`, it takes 15 hours for the turkey to fully defrost.

Find the equation relating `D` and `T`. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
By first completing this table of values, graph the relationship between temperature and time. (2 marks)

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \ \rule[-1ex]{0pt}{0pt} & \ \ \ 10\ \ \ & \ \ 15\ \ \ & \ \ \ 20\ \ \ & \ \ \ 25\ \ \ & \ \ \ 30\ \ \ \\
\hline
\rule{0pt}{2.5ex} \ \ D\ \ \rule[-1ex]{0pt}{0pt} & \ \ \ \ & \ \ \ & \ \ \ \ & \ \ \ \ & \ \ \ \\
\hline
\end{array}

--- 0 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `D \prop 1/T\ \ =>\ \ D=k/T`

	`15`	`=k/20`
	`k`	`=15 xx 20=300`

`:.D=300/T`

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ T\ \ \rule[-1ex]{0pt}{0pt} & \ \ \ 10\ \ \ & \ \ 15\ \ \ & \ \ \ 20\ \ \ & \ \ \ 25\ \ \ & \ \ \ 30\ \ \ \\
\hline
\rule{0pt}{2.5ex} \ \ D\ \ \rule[-1ex]{0pt}{0pt} & \ \ \ 30\ \ \ & \ \ 20\ \ \ & \ \ \ 15\ \ \ & \ \ \ 12\ \ \ & \ \ \ 10\ \ \ \\
\hline
\end{array}

Show Worked Solution

a. `D \prop 1/T\ \ =>\ \ D=k/T`

	`15`	`=k/20`
	`k`	`=15 xx 20=300`

`:.D=300/T`

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} \ \ D\ \ \rule[-1ex]{0pt}{0pt} & \ \ \ 10\ \ \ & \ \ 15\ \ \ & \ \ \ 20\ \ \ & \ \ \ 25\ \ \ & \ \ \ 30\ \ \ \\
\hline
\rule{0pt}{2.5ex} \ \ T\ \ \rule[-1ex]{0pt}{0pt} & 30 & 20 & 15 & 12 & 10 \\
\hline
\end{array}

Algebra, STD2 A4 2022 HSC 24

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.

Find the equation relating `M` and `T`. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
By first completing this table of values, graph the relationship between temperature and time from `T=5^@C` to `T=30^@ text{C}.` (2 marks)

\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}

--- 0 WORK AREA LINES (style=lined) ---

Show Answers Only

a. `M prop 1/T \ \ =>\ \ M=k/T`

	`12`	`=k/15`
	`k`	`=15 xx 12=180`

`:.M=180/T`

\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}

Show Worked Solution

a.	`M`	`prop 1/T`
	`M`	`=k/T`
	`12`	`=k/15`
	`k`	`=15 xx 12`
		`=180`

`:.M=180/T`

♦♦ Mean mark part (a) 29%.

♦ Mean mark 44%.

Algebra, STD2 A4 2014 HSC 29a

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.

Complete the table below by filling in the THREE missing values. (1 mark)
\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} & & & & 60 & 48\ & 40 \ \\
\hline
\end{array}
Using the values from the table, draw the graph showing the relationship between `n` and `C`. (2 marks)
What equation represents the relationship between `n` and `C`? (1 mark)

--- 1 WORK AREA LINES (style=lined) ---
Give ONE limitation of this equation in relation to this context. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---
Is it possible for the cost per person to be $94? Support your answer with appropriate calculations. (1 mark)

--- 2 WORK AREA LINES (style=lined) ---

Show Answers Only

\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.)`

Show Worked Solution

ii.

♦ Mean mark (iii) 48%

iii. `C = (120\ 000)/n`

♦♦♦ Mean mark (iv) 7%
COMMENT: When asked for limitations of an equation, look carefully at potential restrictions with respect to both the domain and range.

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…`

♦ Mean mark (v) 38%

`:.\ text(C)text(ost cannot be $94 per person,)`

`text(because)\ n\ text(isn’t a whole number.)`