1 The rate of flow of water, R mL/min, into a vessel is described by the quartic expression
R= kt3(20− t), 0 ≤ t ≤ 20, where t minutes is the time elapsed from the beginning of the flow. The graph is shown.
0 t
(15, 20)
20 R
a Find the value of k.
b Find the rate of flow when t= 10.
c The flow is adjusted so that a new expression for flow is:
Rnew= 2kt3(20− t), 0 ≤ t ≤ 20
i Sketch the graph of Rnewagainst t for 0≤ t ≤ 20.
ii Find the rate of flow when t= 10.
d Water is allowed to run from the vessel and it is found that the rate of flow from the vessel is given by
Rout = −k(t − 20)3(40− t) for 20 ≤ t ≤ 40 i Sketch the graph of Routagainst t for 20≤ t ≤ 40.
ii Find the rate of flow when t= 30.
Hint: Note that the graph of Rnewagainst t is given by a dilation of factor 2 from the x-axis. The graph of Routagainst t is given by a translation with rule (t, R) → (t + 20, R), followed by a reflection in the t-axis.
2 A large gas container is being deflated. The volume V (in m3) at time t hours is given by:
V = 4(9 − t)3 where 0≤ t ≤ 9 a Find the volume when:
i t= 0 ii t= 9
b Sketch the graph of V against t for 0≤ t ≤ 9.
c At what time is the volume 512 m3?
SAMPLE
Review
3 A reinforced box is made by cutting congruent squares of side length x cm from the four corners of a rectangular piece of cardboard that measures 48 cm by 96 cm. The flaps are folded up.
96 cm
48 cm
a Find an expression for V, the volume of the box formed.
x V (cm3)
20 000 15 000 10 000 5000
0 4 8 12 16 20 24
b A graph of V against x is as shown.
i What is the domain of the function V ?
ii From the graph, find the maximum volume of the box and the value of x for which this occurs
(approximate values are required).
c Find the volume of the box when x= 10.
d It is decided that 0≤ x ≤ 5.
Find the maximum volume possible.
e If 5≤ x ≤ 15, what is the minimum volume of the box?
4 A hemispherical bowl of radius 6 cm contains water. The volume of water in the hemispherical bowl where the depth of the water is x cm and is given by:
V = 1
3x2(18− x) cm3 a Find the volume of water when:
i x= 2 ii x= 3 iii x= 4
b Find the volume when the hemispherical bowl is full.
c Sketch the graph of V against x.
d Find the depth of water when the volume is equal to325 3 cm3. 5 A metal worker is required to cut a circular cylinder from
a solid sphere of radius 5 cm. A cross-section of the sphere and the cylinder is shown in the diagram.
h
r 0
5 a Express r in terms of h, where r cm is the radius of
the cylinder and h cm is the height of the cylinder.
Hence show that the volume, V cm3, of the cylinder is given by V = 1
4h(100 − h2).
SAMPLE
Review
b Sketch the graph of V against h for 0< h < 10.
The coordinates of the maximum point are≈ (5.77, 302.3).
c Find the volume of the cylinder if h= 6.
d Find the height and radius of the cylinder if the volume of the cylinder is 48 cm3. 6 An open tank is to be made from a sheet of
metal 84 cm by 40 cm by cutting congruent squares of side length x cm from each of the corners.
x cm x cm
b State the maximal domain for V when it is considered as a function of x.
c Plot the graph of V against x using a calculator.
d Find the volume of the tank when:
i x= 2 ii x= 6 iii x= 8 iv x= 10
e Find the value(s) of x, correct to two decimal places, for which the capacity of the tank is 10 litres.
f Find, correct to two decimal places, the maximum capacity of the tank in cubic centimetres.
7 The rectangle is defined by vertices B and C on the curve with equation y= 16 − x2 and vertices A and D on the x-axis.
x
a i Find the area, A, of the rectangle in terms of x.
ii State the implied domain for the function defined by the rule given in i.
b Find:
i the value of A when x= 3
ii the value, correct to two decimal places, of x when A= 25 c A cuboid has volume V given by the rule V= xA
i Find V in terms of x.
ii Find the value, correct to two decimal places, of x such that V= 100.
8 The plan of a garden adjoining a wall is shown. The rectangle BCEF is of length y m and width x m. The borders of the two end sections are quarter circles of radius x m and centres at E and F. A fence
is erected along the curves AB and CD and the straight line CB.
A
Review
a Find the area A of the garden in terms of x and y.
b If the length of the fence is 100 m, find:
i y in terms of x ii A in terms of x
iii the maximal domain of the function for which the rule has been obtained inii c Find, correct to two decimal places, the value(s) of x if the area of the garden is to be
1000 m2.
d It is decided to build the garden up to a height of x
50metres. If the length of the fence is 100 m, find correct to two decimal places:
i the volume V m3of soil needed in terms of x
ii the volume V m3of soil needed for a garden of area 1000 m2 iii the value(s) of x for which 500 m3of soil is required
9
x (m) h (m)
A(10, 1)
B(30, 2) C(40, 3)
0 10 20 30 40
A mound of earth is piled up against a wall.
The cross-section is as shown. The
coordinates of several points on the surface are given.
a Find the equation for the cubic function for which the graph passes through the points O, A, B and C.
b For what value of x is the height of the mound 1.5 metres?
c The coefficient of x3for the function is ‘small’. Consider the quadratic formed when the x3term is deleted. Compare the graph of the resulting quadratic function with the graph of the cubic function.
d The mound moves and the curve describing the cross-section now passes through the points O(0, 0), A(10, 0.3), B(30, 2.7) and D(40, 2.8).
Find the equation of the cubic function for which the graph passes through these points.
e Let y= f (x) be the function obtained ina.
i Sketch the graph of the hybrid function:
g(x)=
f (x) 0≤ x ≤ 40 f (80− x) 40 < x ≤ 80 ii Comment on the appearance of the graph of y= g(x)