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A farmer has a garden which is enclosed by 16 metres of fencing.
If the length of the rectangular field is x metres, show that the area of the rectangular garden is given by A = 8x – x^2. (1 mark)
By considering the different possible lengths (x) of rectangular garden, complete the table of values. (1 mark)
Draw a neat graph of the Area (A) of the garden against its length (x). (2 marks)
From the graph, find the dimensions of the garden that will maximise the area. (1 mark)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(i) A=x(8-x)
A=8x-x^2
(ii)
(iii)
(iv) X = 4m x 4m
Solve for x (2 (x – 1) )/3 = (3x + 3)/4. (2 marks)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
8(x-1)=3(3x+3)
8x-8=9x+9
x=-17
When a force is applied to a certain object, its acceleration (a) varies inversely to its mass (m). When the acceleration of an object 12m/s^2, the corresponding mass is 3kg.
(i) Find the constant of variation. (1 mark)
(ii) Find the acceleration of a 1.5 kg object. (1 mark)
(iii) Find the mass of an object when the acceleration is 6 m/s^2. (1 mark)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(i) a=k/M
12=k/3
k=36
(ii) a=36/1.5
a=24
(iii) 6=36/M
M=6kg
Sketch the graph y = 2x + 4 on the number plane below. (1 mark)
Sketch the graph of y = 2 on the number plane above. (1 mark)
Write down the coordinates of the point where y = 2x + 4 and y = 2 intersect. ( 1 mark)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(a), (b)
(c) (-1,2)
The graph shows the average cost of producing a toy car
When the average production cost per doll was $10, determine the number of dolls produced? (1 mark)
If the company produced 22 500 toy dolls, what was average production cost per doll for that year? (1 mark)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(a) 10000 (b) $5.50
Ahmed rents a holiday house for $720 per week. In order to pay the rent he has other people stay with him and divides the expenses evenly. He uses the formula C = 720/n, where C is the cost ($) per person and n is number of people staying per week.
(i) Copy and complete the following table. (2 marks)
(ii) Graph cost (C) versus the number of people (n). Let the vertical axis represent the cost. (2 marks)
(iii) Ahmed and another person stay for a full week. A third person stays for a half a week. Use the graph to estimate the cost per person. (2 marks)
(iv) How many people would need to stay at the holiday house if the cost per person is $90? (2 marks)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(i)
(ii) See below
(iii) $288 (exact) Any answer from $280-$290
(iv) 8
(b)
A year 12 formal reception uses this graph to determine the total cost, C dollars, of catering for N people.
(i) What is the gradient and what does it mean in relation to the question? (2 marks)
(ii) Write a formula relating the total cost, C dollars, they charge for N guests. (1 mark)
(iii) If $14,050 was paid for a Year 12 formal, use a formula to find how many guests were at the formal? (2 marks)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(i) gradient = (11300 – 300)/(200 – 0) = 55, cost per person
(ii) c = 55n + 300
(iii) 14050 = 55n + 300
13750 = 55n
n = 250 therefore 250 quests
250 quests
The cost of manufacturing footballs is given by the formula C = 2n + 10. Where n is the number of footballs made and C is the cost in dollars.
The revenue from selling footballs is given by the function R = 4n. Where n is the number of footballs sold and R is the sales total in dollars.
(i) Graph C = 2n + 10 and R = 4n on the same axes for values of n from 0 to 10, on the graph paper provided. Let the horizontal axis be the number of footballs. (3 marks)
(ii) Write the coordinates of the point of intersection of the two graphs. (1 mark)
(iii) What does this point indicate? (1 mark)
(iv) How many footballs need to be manufactured before a profit can be made? (1 mark)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(i)
(ii) (5,20)
(iii) If 5 footballs are manufactured, the cost of producing 5 footballs ($20) equals the revenue gained by selling 5 footballs (this is called the break even point).
(iv) Six or more.
In class, Tina was asked to set up a table of values and plot the graph of d = -t^2 + 4t + 6. Shauni was asked to do the same thing for the function d = 10 – (t – 2)^2. In each case d represented the distance in metres of a particle from a fixed point at time t seconds. They each came up correctly with the following table of values and graph. (Note that one value in the table has been shaded.)
(i) Find the value that should appear in the shaded cell of the table by finding the value of d when t = 5. (1 mark)
(ii) Explain why Tina and Shauni came up with the same table of values and the same graph. (1 mark)
(iii) What is the maximum distance of the particle from the fixed point? (1 mark)
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(i) 1
(ii) 10 – (t – 2)^2 = 10 – (t^2 – 4t + 4) = 6 – t^2 – 4t + 4
= -t^2 – 4t + 6
i.e Both Tina and Shauni are using the same equation
(iii) 10