1.
Review of Exercise 6:
Transformations of
Quadratic Functions 3
2.
1. A farmer wishes to build a rectangular
pen along one side of his barn. If he has 80
metres of fencing, find the dimensions that
will yield a maximum area.
w
2w + L = 80
l barn L = 80 - 2w
Area = L * w
w
3.
w
L = 80 - 2w
l barn
Area = L * w
w
A = (80 - 2w) * w
2
A = 80w - 2w
4.
2
A = 80w - 2w
2
A = -2{w - 40w}
2
A = -2{w - 40w + 400 - 400}
2
A = -2{(w-20) - 400}
2
A = -2(w-20) + 800
5.
w
L = 80 - 2w
l barn
Area = L * w
w
2
A = -2(w-20) + 800
Area is a maximum when w = 20.
2
The maximum area is 800 m
800 = L * 20
L = 40 m
6.
2. Find 2 positive numbers whose sum
is 13 if the sum of their squares is a
minimum
2 positive numbers: a, b
a + b = 13
2 2
a +b =y where y is a minimum
7.
2 2
a +b =y where y is a minimum
a + b = 13 a = 13 - b
2 2
a +b =y
2 2
(13 - b) + b = y
8.
2 2
(13 - b) + b = y
2 2
(169 - 26b + b ) + b = y
2
y = 2b - 26b + 169
2
y = 2{b - 13b} + 169
10.
a + b = 13
2 2
a +b =y where y is a minimum
2
y = 2(b - 6.5) + 84.5
y is a minimum when b = 6.5
a + 6.5 = 13
a = 13 - 6.5
a = 6.5
11.
3. A projectile is shot straight up from a
height of 6 m with an initial velocity of80
m/s. Its height in meters above the ground
after t seconds is given by the equation
2
h = 6 + 80t - 5t . After how many seconds
does the projectile reach its max height, and
what is this height?
max height
6m
12.
2
h = 6 + 80t - 5t
2
h = - 5t + 80t + 6
2
h = -5{t - 16t} +6
2
h = -5{t - 16t + 64 - 64} +6
2
h = -5{(t - 8) - 64} + 6
2
h = -5(t - 8) - (-5)(64) +6
13.
2
h = -5(t - 8) - (-5)(64) +6
2
h = -5(t - 8) + 326
The maximum height is reached
after 8 seconds. The maximum
height is 326 metres.
14.
4. A survey found that 400 people will
attend a theatre when the admission price
is 80 cents. The attendance decreases by
40 people for each 10 cents added to the
price. What price admission will yield the
greatest receipt?
Profit = Tickets * Cost
x = number of times the ticket price
is increased
15.
Profit = Tickets * Cost
x = number of times the ticket price
is increased
T = 400 - 40x
C = .8 + .1x
P = (400 - 40x) (.8 + .1x)
16.
P = (400 - 40x) (.8 + .1x)
2
P = 320 + 40x - 32x - 4x
2
P = -4x + 8x + 320
17.
2
P = -4x + 8x + 320
2
P = -4{x - 2x} + 320
P = -4{x 2 - 2x + 1 - 1} + 320
P = -4{(x - 1) 2 - 1} + 320
2 -1(-4) + 320
P = -4(x - 1)
2 + 324
P = -4(x - 1)
18.
P = -4(x - 1) 2 + 324
x = number of times the ticket price
is increased
Profit = Tickets * Cost
Profit will be a maximum when x = 1.
C = .8 + .1x
Cost of each ticket will give the
maximum profit when C = .8 + .1(1)
C = $0.90
19.
5. Find 2 positive numbers whose sum
is 13 and whose product is a maximum.
2 positive numbers: a, b
a + b = 13
a*b=c where c is a maximum
20.
a*b=c where c is a maximum
a + b = 13 a = 13 - b
a*b=c
(13 - b) * b = c
13b - b 2=c
21.
13b - b2=c
c = -b2 + 13b
c = -1{b2 - 13b}
c = -1{b2 - 13b + 6.52 - 6.52}
2 - 6.52}
c = -1{(b - 6.5)
22.
c = -1{(b - 6.5) 2 - 6.52}
c = -1(b - 6.5) 2 - (-1)6.52
c = -1(b - 6.5) 2 + 42.25
23.
c = -1(b - 6.5) 2 + 42.25
c is a maximum when b = 6.5.
a + b = 13
a + 6.5 = 13
a = 6.5
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