Maximum and minimum values
d 1 Calculate the time the stone returns to the ground.
d Returns to ground when h = 0 0= 1.5 + 5t − 0.5t2 t2 − 10t − 3 = 0 t2 − 10t = 3 t2 − 10t + 25 = 3 + 25 (t− 5)2 = 28 t= 5 ± 28
t≃ 10.3 (reject negative value) The stone reaches the ground after 10.3 seconds. 2 Sketch the graph, from its initial height to
when the stone hits the ground. label the axes appropriately.
When t= 0,h= 1.5 so stone is thrown from a height of 1.5 metres.
Initial point: (0,1.5)
Maximum turning point: (5,14) Endpoint: (10.3,0) 15 10 (5, 14) (10.3, 0) (0, 1.5) 5 1 2 3 4 5 6 7 8 9 10 11 0 h t h= 1.5 + 5t – 0.5t2
part of the parabola y = 1.2 + 2.2x − 0.2x2 where x (metres) is the horizontal distance travelled by the ball from where it was hit and y (metres) is the vertical height the ball reaches.
a Use completing the square technique to express the equation in the form
y= a(x − b)2 + c.
b How high does the volleyball reach?
c The net is 2.43 metres high and is placed in the centre of the playing court. Show that the ball clears the net and calculate by how much.
5 Georgie has a large rectangular garden area with dimensions
l metres by w metres which she wishes to divide into three sections so she can grow different vegetables. She plans to put a watering system along the perimeter of each section. This will require a total of 120 metres of hosing.
a Show the total area of the three sections, A m2 is given by A= 60w − 2w2 and hence calculate the dimensions when the total area is a maximum.
b Using the maximum total area, Georgie decides she wants the areas of the three sections to be in the ratio 1 : 2 : 3. What is the length of hosing for the watering system that is required for each section?
6 The number of bacteria in a slowly growing culture at time t hours after 8.00 am is given by N = 100 + 46t + 2t2.
a How long does it take for the initial number of bacteria to double?
b How many bacteria are present at 1.00 pm?
c At 1.00 pm a virus is introduced that initially
starts to destroy the bacteria so that t hours after 1.00 pm the number of bacteria is given by N = 380 − 180t + 30t2. What is the minimum number the population of bacteria reaches and at what time does this occur?
7 let z = 5x2 + 4xy + 6y2. Given x+ y = 2, fi nd the minimum value of z and the values of x and y for which z is minimum.
8 A piece of wire of length 20 cm is cut into two sections, and each is used to form a square. The sum of the areas of these two squares is S cm2.
a If one square has a side length of 4 cm, calculate the value of S.
b If one square has a side length of x cm, express S in terms of x and hence determine how the wire should be cut for the sum of the areas to be a minimum.
9 The cost C dollars of manufacturing n dining tables is the sum of three parts. One part represents the fi xed overhead costs c, another represents the cost of raw materials and is directly proportional to n and the third part represents the labour costs which are directly proportional to the square of n.
a If 5 tables cost $195 to manufacture, 8 tables cost $420
to manufacture and 10 tables cost $620 to manufacture, fi nd the relationship w metres
l metres
b If the sum of two non-zero numbers is k:
i express their greatest product in terms of k
ii are there any values of k for which the sum of the squares of the numbers and their product are equal? If so, state the values; if not, explain why.
13 Meteorology records for the heights of tides above mean sea level in Tuvalu predict the tide levels shown in the following table.
a Use CAS technology to fi nd the equation of a quadratic model which fi ts these three data points in the form h = at2 + bt + c where h is the height in metres of the tide t hours after midnight. Express the coeffi cients to 2 decimal places. b Find the greatest height of the tide above sea level and the time of day it is
predicted to occur.
14 A piece of wire of length 20 cm is cut into two sections, one a square and the other a circle. The sum of the areas of the square and the circle is S cm2.
a If the square has a side length of x cm, express S in terms of x.
b Graph the S–x relationship and hence calculate the lengths of the two sections of the wire for S to be a minimum. Give the answer to 1 decimal place.
Master
b Find the maximum number of dining tables that can be manufactured if costs are not to exceed $1000.
10 The arch of a bridge over a small creek is parabolic in shape with its feet evenly spaced from the ends of the bridge. Relative to the coordinate axes, the points A, B and C lie on the parabola.
14 m 2 m 5 m 0 B A C x 5 m 14 m
a If AC = 8 metres, write down the coordinates of the points A, B and C.
b Determine the equation of the parabola containing points A, B and C.
c Following heavy rainfall the creek floods and overflows its bank, causing the water level to reach 1.5 metres above AC. What is the width of the water level to 1 decimal place?
11 The daily cost C dollars of producing x kg of plant fertiliser for use in market gardens is C= 15 + 10x. The manufacturer decides that the fertiliser will be sold for v dollars per kg where v= 50 − x.
Find an expression for the profit in terms of x and hence find the price per kilogram that should be charged for maximum daily profit.
12 a If the sum of two numbers is 16, find the numbers for which:
i their product is greatest ii the sum of their squares is least.
b If the sum of two non-zero numbers is k:
i express their greatest product in terms of k
ii are there any values of k for which the sum of the squares of the numbers and their product are equal? If so, state the values; if not, explain why.
13 Meteorology records for the heights of tides above mean sea level in Tuvalu predict the tide levels shown in the following table.
a Use CAS technology to find the equation of a quadratic model which fits these three data points in the form h= at2 + bt + c where h is the height in metres of the tide t hours after midnight. Express the coefficients to 2 decimal places. b Find the greatest height of the tide above sea level and the time of day it is
predicted to occur.
14 A piece of wire of length 20 cm is cut into two sections, one a square and the other a circle. The sum of the areas of the square and the circle is S cm2.
a If the square has a side length of x cm, express S in terms of x.
b Graph the S–x relationship and hence calculate the lengths of the two sections of the wire for S to be a minimum. Give the answer to 1 decimal place.
c Use the graph to find the value of x for which S is a maximum. Master
Time of day Height of tide (in metres)
10.15 am 1.05
4.21 pm 3.26
10.30 pm 0.94
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