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Bài tập Toán DIFFERENTIATION OPTIMIZATION 03
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Created by T. Madas
Question 6 (***)
A
8x
6x
E
B
y
D
10 x
C
The figure above shows a pentagon ABCDE whose measurements, in cm , are given in
terms of x and y .
a) If the perimeter of the pentagon is 120 cm , show clearly that its area, A cm 2 , is
given by
A = 600 x − 96 x 2 .
b) Use a method based on differentiation to calculate the maximum value for A ,
fully justifying the fact that it is indeed the maximum value.
Amax = 937.5
Created by T. Madas
Created by T. Madas
Question 7 (***)
x
A
y B
G
F
1c
E
x
y C
D
The figure above shows a clothes design consisting of two identical rectangles attached
to each of the straight sides of a circular sector of radius x cm .
The rectangles measure x cm by y cm and the circular sector subtends an angle of one
radian at the centre.
The perimeter of the design is 40 cm .
a) Show that the area of the design, A cm 2 , is given by
A = 20 x − x 2 .
b) Determine by differentiation the value of x for which A is stationary.
c) Show that the value of x found in part (b) gives the maximum value for A .
d) Find the maximum area of the design.
x = 10 , Amax = 100
Created by T. Madas
Created by T. Madas
Question 8 (***+)
r
h
The figure above shows a closed cylindrical can of radius r cm and height h cm .
a) Given that the surface area of the can is 192π cm 2 , show that the volume of the
can, V cm3 , is given by
V = 96π r − π r 3 .
b) Find the value of r for which V is stationary.
c) Justify that the value of r found in part (b) gives the maximum value for V .
d) Calculate the maximum value of V .
r = 4 2 ≈ 5.66 , Vmax = 256π 2 ≈ 1137
Created by T. Madas
