Quadratic Word Problems

1
Quadratic Word Problems

• Max/min
• Finding Roots

2
Recap

• You should know the following about quadratic
  functions
• How to graph them
• How to find the vertex
• How to find the x- and y- intercepts
• How to find the equation from the pattern
• How to find the equation from the graph
• How to change from one form to another

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Quadratic Word Problems
There are basically two types of quadratic word
problems
Those that ask you to find the vertex
Those that ask you to find the roots
4
Quadratic Word Problems
There are basically two types of quadratic word
problems
Those that ask you to find the vertex
Those that ask you to find the roots
These are the harder ones!!
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Max/Min Values
We have seen that the vertex of a quadratic in
general form is given by
The y-value of the vertex is either the maximum
value the function can have or its the minimum
value the function can have.
y-values a min when a gt0
y-values a max when alt0
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Max/Min Word Problems
Example 1 A small business profits over the
last year have been related to the price of the
only product. The relationship is R(p) -0.4p2
64p-2400, where R is the revenue measured in
thousands of dollars and p is the price of the
product measured in dollars.
a. What price would maximize the revenue?
The word maximize screams FIND THE VERTEX!!
Thus, a price of 80 would maximize the revenue.
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Max/Min Word Problems
Example 1 A small business profits over the
last year have been related to the price of the
only product. The relationship is R(p) -0.4p2
64p-2400, where R is the revenue measured in
thousands of dollars and p is the price of the
product measured in dollars.
b. What is the maximum revenue possible?
The answer to this question is the y-value of the
vertex.
The maximum revenue is 160 000.
8
Max/Min Word Problems
Example 1 A small business profits over the
last year have been related to the price of the
only product. The relationship is R(p) -0.4p2
64p-2400, where R is the revenue measured in
thousands of dollars and p is the price of the
product measured in dollars.
c. How much money would they lose if they gave
the product away?
This question is talking about a price of 0 or p
0
The business would lose 2 400 000.
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Max/Min Word Problems
Example 2 no equation A farmer needs to fence
off his animals. He bought 800 m of fencing and
would like to maximize the area for his
livestock. According to regulations the pens need
to find these relative dimensions
Kids
Sheep pen
Pig Pen
What are the dimensions of the largest area?
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Max/Min Word Problems
Example 2 no equation A farmer needs to fence
off his animals. He bought 800 m of fencing and
would like to maximize the area for his
livestock. According to regulations the pens need
to find these relative dimensions
Again the word Maximum means VERTEX. BUT
We need the equation!!
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Max/Min Word Problems
Example 2 no equation A farmer needs to fence
off his animals. He bought 800 m of fencing and
would like to maximize the area for his
livestock. According to regulations the pens need
to find these relative dimensions
We know
The total fencing is 800m. This means
3w3w3w8L8Lw2L800
This equation has two unknowns!!
or 10w 18L 800
We need the (other) equation!!
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Max/Min Word Problems
Example 2 no equation A farmer needs to fence
off his animals. He bought 800 m of fencing and
would like to maximize the area for his
livestock. According to regulations the pens need
to find these relative dimensions
The question is asking for the maximum area. So
lets get an area equation
Area of a rectangle A (3w)(8L) A 24wL
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Max/Min Word Problems
The two equations are
A 24wL
10w 18L 800
To solve this system of equations, use
substitution.
Solve the linear equation (the one without the
multiplication of variables) for one of the
unknowns.
10w 18L 800
Substitute this into the area equation.
10w 800 -18L w 80 -1.8L
A 24wL A 24(80-1.8L)L
A 24L(80-1.8L) A 1920L-43.2L2
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Max/Min Word Problems
A 1920L-43.2L2
Remember, were looking for the MAX AREA. Well
find that with the vertex.
The BEST L value to use is 22.22m
This leads to a w value of
w 80 -1.8L w 80-1.8(22.22) w 40.00m
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Max/Min Word Problems
Example 2 no equation A farmer needs to fence
off his animals. He bought 800 m of fencing and
would like to maximize the area for his
livestock. According to regulations the pens need
to find these relative dimensions
The dimensions that would be the best are w
40.00m and L 22.22m
The largest area would be
A 24wL A 24(40.00)(22.22) A 21331.2m2
16
Your Turn
A lifeguard has 75m of rope to section off the
supervised area of the beach. What is the largest
rectangular swimming area possible?
2w L 75
A wL
L 75-2w
Aw(75-2w)
A 75w-2w2
A 75(18.75)-2(18.75)2 A 1406.25-703.125 A
703.125
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Roots of a Quadratic
We know that we can find the roots of a quadratic
function by setting one side equal to zero and
Factoring (sometimes)
Using the quadratic root formula
Completing the square (too long)
This ALWAYS works for a quadratic in general form
and is easy to do.
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Roots Word Problems
We know that we can find the roots of a quadratic
function by setting one side equal to zero and
Factoring (sometimes)
Using the quadratic root formula
Completing the square (too long)
This ALWAYS works for a quadratic in general form
and is easy to do.
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Roots Word Problems
Example 1 A duck dives under water and its path
is described by the quadratic function y 2x2
-4x, where y represents the position of the duck
in metres and x represents the time in seconds.
a. How long was the duck underwater?
The duck is no longer underwater when the depth
is 0. We can plug in y 0 and solve for x.
The duck was underwater for 4 seconds
So x 0 or 4
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Roots Word Problems
Example 1 A duck dives under water and its path
is described by the quadratic function y 2x2
-4x, where y represents the position of the duck
in metres from the water and x represents the
time in seconds.
b. When was the duck at a depth of 5m?
We can plug in y -5 and solve for x.
We cannot solve this because theres a negative
number under the square root.
We conclude that the duck is never 5m below the
water.
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Roots Word Problems
Example 1 A duck dives under water and its path
is described by the quadratic function y 2x2
-4x, where y represents the position of the duck
in metres from the water and x represents the
time in seconds.
We conclude that the duck is never 5m below the
water.
b. When was the duck at a depth of 5m?
We can check this by finding the minimum value of
y.
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Roots Word Problems
Example 1 A duck dives under water and its path
is described by the quadratic function y 2x2
-4x, where y represents the position of the duck
in metres and x represents the time in seconds.
c. How long was the duck at least 0.5m below the
waters surface?
We can plug in y -0.5 and solve for x.
This will give us the times when the duck is at
0.5 m below.
The duck was 0.5m below at t 0.14s and at t
1.87s
Therefore it was below 0.5m for 1.73s
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Roots Word Problems
Example 2 no equation A rectangular lawn
measures 8m by 6m. The homeowner mows a strip of
uniform width around the lawn, as shown. If 40
of the lawn remains unmowed, what is the width of
the strip?
The area of the lawn is 8 x 6 48
-X-
-6-2x-
-X-
40
40 of this is unmowed 48 x 0.40 19.2
-----8-2x-----
The dimensions of this unmowed rectangle are
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Roots Word Problems
Example 2 no equation A rectangular lawn
measures 8m by 6m. The homeowner mows a strip of
uniform width around the lawn, as shown. If 40
of the lawn remains unmowed, what is the width of
the strip?
So 19.2 (8-2x)(6-2x)
-X-
-6-2x-
We need to solve for x
-X-
40
Make one side equal to 0 and use the quadratic
root formula
-----8-2x-----
But first we FOIL it out
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Roots Word Problems
Example 2 no equation A rectangular lawn
measures 8m by 6m. The homeowner mows a strip of
uniform width around the lawn, as shown. If 40
of the lawn remains unmowed, what is the width of
the strip?
So 19.2 (8-2x)(6-2x)
-X-
-6-2x-
-X-
40
-----8-2x-----
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Roots Word Problems
Example 2 no equation A rectangular lawn
measures 8m by 6m. The homeowner mows a strip of
uniform width around the lawn, as shown. If 40
of the lawn remains unmowed, what is the width of
the strip?
0 4x2-28x28.8
-X-
-6-2x-
-X-
40
-----8-2x-----
The mowed strip has a width of 1.25m
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Roots Word Problems
Example 2 no equation A rectangular lawn
measures 8m by 6m. The homeowner mows a strip of
uniform width around the lawn, as shown. If 40
of the lawn remains unmowed, what is the width of
the strip?
Lets check
-X-
-6-2x-
If x 1.25m then the
-X-
40
length is 8 2x 8 -2(1.25) 5.5m
width is 6 2x 6 -2(1.25) 3.5m
-----8-2x-----
A (5.5)(3.5) A 19.2
Which was 40 of the total area!
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Roots Word Problems
Example 3 no equation
Two numbers have a difference of 18. The sum of
their squares is 194. What are the numbers?
Lets define our variables S one of the
numbers G the other number
S G 18
The question indicates two equations relating
these two variables.
S2 G2 194
Again, we have a substitution situation. Solve
the simpler equation for a variable and plug it
in to the other equation.
S18G
(18G)2 G2 194
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Roots Word Problems
Example 3 no equation
Two numbers have a difference of 18. The sum of
their squares is 194. What are the numbers?
(18G)2 G2 194
We need to solve this equation for G. Use the
quadratic root formula
Lets FOIL and make one side equal to 0.
32436G G2 G2 194 2G2 36G324-194 0 2G2
36G1300
S 18 (-5) 13
or S 18(-13)5
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Roots Word Problems
Example 3 no equation
Two numbers have a difference of 18. The sum of
their squares is 194. What are the numbers?
(18G)2 G2 194
The numbers are either -5 and 13 or -13 and 5
31
Roots Word Problems Try one
A rectangle is 8 feet long and 6 feet wide. If
the same number of feet increases each dimension,
the area of the new rectangle formed is 32 square
feet more than the area of the original
rectangle. How many feet increased each dimension?
So 80 (6x)(8x)
The new are is 326x8 3248 80
804814xx2 0x2 14x-32
The dimensions of the new rectangle are
x 2 or x -16
6x and 8x
32
May I have a Word (Problem)
A ball is thrown and follows the path described
by the function h(t) -5t2 20t 1, where h is
the height of the ball and t is the time since
the ball was released.
a) When was the ball at a height of 3.5m?
0 -5t2 20t -2.5
This question is looking for t so it gives a
specific h. In this case h 3.5.
3.5 -5t2 20t 1 We will solve this by
setting one side equal to zero and using the
quadratic root formula
One time is on the way up and the other is on the
way down.
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May I have a Word (Problem)
A ball is thrown and follows the path described
by the function h(t) -5t2 20t 1, where h is
the height of the ball and t is the time since
the ball was released.
a) When was the ball at a height of 3.5m?
0 -5t2 20t -2.5
The ball is at a height of 3.5m at two times at
t 0.129s and at t 3.871s
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May I have a Word (Problem)
A ball is thrown and follows the path described
by the function h(t) -5t2 20t 1, where h is
the height of the ball and t is the time since
the ball was released.
b) How high is the ball after 4.0s?
This question is looking for h given a value of
t. t 4.0
After 4.0 seconds in the air, the ball is 1 m off
the ground.
h -5(4)2 20(4) 1 h -80 80 1 h 1
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May I have a Word (Problem)
A ball is thrown and follows the path described
by the function h(t) -5t2 20t 1, where h is
the height of the ball and t is the time since
the ball was released.
c) What is the balls maximum height?
h -5t2 20t 1
The question is asking for height so I must know
the time. Do I?
The max height
The word MAXIMUM screams VERTEX!! I do know the
time value Its
The ball reaches its maximum height 2.0 seconds
after being thrown
The maximum height is 21 m
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May I have a Word (Problem)
A ball is thrown and follows the path described
by the function h(t) -5t2 20t 1, where h is
the height of the ball and t is the time since
the ball was released.
d) When does the ball hit the ground?
0 -5t2 20t 1
This question is asking for the time so I must
know the height. The height is 0 hitting the
ground!
Time cant be negative so this cannot be an
answer.
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May I have a Word (Problem)
A ball is thrown and follows the path described
by the function h(t) -5t2 20t 1, where h is
the height of the ball and t is the time since
the ball was released.
e) From what height was the ball thrown?
This question is asking for the height so I must
know the time. The time is 0 just before it is
thrown!
h -5(0)2 20(0)1 h 1