2. Maxima & Minima
Problems
When solving word problems we need to;
3. Maxima & Minima
Problems
When solving word problems we need to;
(1) Reduce the problem to a set of equations
(there will usually be two)
4. Maxima & Minima
Problems
When solving word problems we need to;
(1) Reduce the problem to a set of equations
(there will usually be two)
a) one will be what you are maximising/minimising
5. Maxima & Minima
Problems
When solving word problems we need to;
(1) Reduce the problem to a set of equations
(there will usually be two)
a) one will be what you are maximising/minimising
b) one will be some information given in the problem
6. Maxima & Minima
Problems
When solving word problems we need to;
(1) Reduce the problem to a set of equations
(there will usually be two)
a) one will be what you are maximising/minimising
b) one will be some information given in the problem
(2) Rewrite the equation we are trying to minimise/maximise with
only one variable
7. Maxima & Minima
Problems
When solving word problems we need to;
(1) Reduce the problem to a set of equations
(there will usually be two)
a) one will be what you are maximising/minimising
b) one will be some information given in the problem
(2) Rewrite the equation we are trying to minimise/maximise with
only one variable
(3) Use calculus to solve the problem
8. e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
9. e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
10. e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
s
s
11. e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
s r
s s
12. e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
s r
s s
A s 2 rs1
13. e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
s r
s s
A s 2 rs1 6s 2r 362
14. e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
s r
s s
A s 2 rs1 6s 2r 362
make r the subject in 2
15. e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
s r
s s
A s 2 rs1 6s 2r 362
make r the subject in 2
2r 36 6s
r 18 3s
16. e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
s r
s s
A s 2 rs1 6s 2r 362
make r the subject in 2
2r 36 6s
r 18 3s
substitute into 1
17. e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
s r
s s
A s 2 rs1 6s 2r 362
make r the subject in 2
2r 36 6s
r 18 3s
substitute into 1
A s 2 18 3s s
18. e.g. A rope 36 metres in length is cut into two pieces. The first is bent
to form a square, the other forms a rectangle with one side the
same length as the square.
Find the length of the square if the sum of the areas is to be a
maximum.
s r
s s
A s 2 rs1 6s 2r 362
make r the subject in 2
2r 36 6s
r 18 3s
substitute into 1
A s 2 18 3s s
A s 2 18s 3s 2
A 18s 2s 2
22. A 18s 2s 2
dA
18 4s
ds
d2A
2
4
ds
dA
Stationary points occur when 0
ds
23. A 18s 2s 2
dA
18 4s
ds
d2A
2
4
ds
dA
Stationary points occur when 0
ds
i.e. 18 4s 0
9
s
2
24. A 18s 2s 2
dA
18 4s
ds
d2A
2
4
ds
dA
Stationary points occur when 0
ds
i.e. 18 4s 0
9
s
2
9 d2A
when s , 2 4 0
2 ds
25. A 18s 2s 2
dA
18 4s
ds
d2A
2
4
ds
dA
Stationary points occur when 0
ds
i.e. 18 4s 0
9
s
2
9 d2A
when s , 2 4 0
2 ds
1
when the side length of the square is 4 metres, the area is a maximum
2
26. (ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is 12 units 2, find the height,
radius and volume.
27. (ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is 12 units 2, find the height,
radius and volume.
r
h
28. (ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is 12 units 2, find the height,
radius and volume.
r V r 2 h1
h
29. (ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is 12 units 2, find the height,
radius and volume.
r V r 2 h1
12 r 2 2rh2
h
30. (ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is 12 units 2, find the height,
radius and volume.
r V r 2 h1
12 r 2 2rh2
make h the subject in 2
h
31. (ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is 12 units 2, find the height,
radius and volume.
r V r 2 h1
12 r 2 2rh2
make h the subject in 2
h
2rh 12 r 2
12 r 2
h
2r
32. (ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is 12 units 2, find the height,
radius and volume.
r V r 2 h1
12 r 2 2rh2
make h the subject in 2
h
2rh 12 r 2
12 r 2
h
2r
substitute into 1
33. (ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is 12 units 2, find the height,
radius and volume.
r V r 2 h1
12 r 2 2rh2
make h the subject in 2
h
2rh 12 r 2
12 r 2
h
2r
substitute into 1
12 r 2
V r 2
2r
34. (ii) An open cylindrical bucket is to made so as to hold the largest
possible volume of liquid.
If the total surface area of the bucket is 12 units 2, find the height,
radius and volume.
r V r 2 h1
12 r 2 2rh2
make h the subject in 2
h
2rh 12 r 2
12 r 2
h
2r
substitute into 1
12 r 2
V r 2
2r
1
V 6r r 3
2
37. 1
V 6r r 3
2
dV 3
6 r 2
dr 2
d 2V
2
3r
dr
38. 1
V 6r r 3 Stationary points occur when
dV
0
2 dr
dV 3
6 r 2
dr 2
d 2V
2
3r
dr
39. 1
V 6r r 3 Stationary points occur when
dV
0
2 dr
dV 3 3
6 r 2 i.e. 6 r 2 0
dr 2 2
d 2V 3 2
2
3r r 6
dr 2
r2 4
r 2
40. 1
V 6r r 3 Stationary points occur when
dV
0
2 dr
dV 3 3
6 r 2 i.e. 6 r 2 0
dr 2 2
d 2V 3 2
2
3r r 6
dr 2
r2 4
d 2V r 2
when r 2, 2 6 0
dr
41. 1
V 6r r 3 Stationary points occur when
dV
0
2 dr
dV 3 3
6 r 2 i.e. 6 r 2 0
dr 2 2
d 2V 3 2
2
3r r 6
dr 2
r2 4
d 2V r 2
when r 2, 2 6 0
dr
when r 2, V is a maximum
42. 1
V 6r r 3 Stationary points occur when
dV
0
2 dr
dV 3 3
6 r 2 i.e. 6 r 2 0
dr 2 2
d 2V 3 2
2
3r r 6
dr 2
r2 4
d 2V r 2
when r 2, 2 6 0
dr
when r 2, V is a maximum
12 2 2
when r 2, h
22
2
43. 1
V 6r r 3 Stationary points occur when
dV
0
2 dr
dV 3 3
6 r 2 i.e. 6 r 2 0
dr 2 2
d 2V 3 2
2
3r r 6
dr 2
r2 4
d 2V r 2
when r 2, 2 6 0
dr
when r 2, V is a maximum
12 2 2
when r 2, h V 2 2
2
22
8
2
44. 1
V 6r r 3 Stationary points occur when
dV
0
2 dr
dV 3 3
6 r 2 i.e. 6 r 2 0
dr 2 2
d 2V 3 2
2
3r r 6
dr 2
r2 4
d 2V r 2
when r 2, 2 6 0
dr
when r 2, V is a maximum
12 2 2
when r 2, h V 2 2
2
22
8
2
maximum volume is 8 units 3 when the radius and height are both 2 units