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The graph displays the cost ($c) charged by two companies for the hire of a minibus for x hours.
Both companies charge $360 for the hire of a minibus for 3 hours.
(i) What is the hourly rate charged by Company A? (1 mark)
(ii) Company B charges an initial booking fee of $75.
Write a formula, in the form of c=mx+b, for the cost of hiring a minibus from Company B for x hours. (2 marks)
(iii) A minibus is hired for 5 hours from Company B.
Calculate how much cheaper this is than hiring from Company A. (2 marks)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
i. Hourly rate (A) =360÷3
=$120
ii. m=hourly rate
Find m, given c=360, when x=3 and b=75
360 =m×3+75
3m =285
m =95
∴c=95x+75
iii. Cost (A) =120×5=$600
Cost (B) =95×5+75=$550
∴ Company B′s hiring cost is $50 cheaper.
y = x + 5
y + 2x = 2
Draw these two linear graphs on the number plane below and determine their intersection. (3 marks)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Table of coordinates: y=x+5
Table of coordinates: y+2x=2 ⇒ y=−2x+2
From graph (and table), intersection occurs
at (−1,4).
Fiona and John are planning to hold a fund-raising event for cancer research. They can hire a function room for $650 and a band for $850. Drinks will cost them $25 per person.
(i) Write a formula for the cost ($C) of holding the charity event for x people. (1 mark)
(ii) The graph below shows the planned income and costs if they charge $50 per ticket. Estimate the number of guests they need to break even. (1 mark)
(iii) How much profit will Fiona and John make if 80 people attend their event? (1 mark)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(i) Fixed Costs =650+850
=$1500
Variable Costs=$25x
∴ $C=1500+25x
(ii) From the graph
Costs = Income when x=60
(i.e. where graphs intersect)
(iii) Income when x=80
=80×50
=$4000
$C =1500+25×80
=$3500
∴ Profit =4000−3500
=$500
Draw each graph on the grid below and hence solve the simultaneous equations. (3 marks)
y=2x+1
x−2y−4=0 (3 marks)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Solution is at the intersection: x=−2, y=−3
A student was asked to solve the following simultaneous equations.
Y = 2x − 8
x − 4y + 3 = 0
After graphing the equations, the student found the point of intersection to be (5,2)?
Is the student correct? Support your answer with calculations. (2 marks)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Substitute (5, 2) into y=2x−8
LHS=y=2
RHS=2(5)−8=2
∴ LHS = RHS
Substitute (5, 2) into x−4y+3=0
LHS =5−4(2)+3
=0
= RHS
⇒(5,2) satisfies both equations.
∴ Student is correct.