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Bài tập Toán DIFFERENTIATION OPTIMIZATION 10
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Created by T. Madas
Question 26
(****)
2c
h
r
The figure above shows solid right prism of height h cm .
The cross section of the prism is a circular sector of radius r cm , subtending an angle of
2 radians at the centre.
a) Given that the volume of the prism is 1000 cm3 , show clearly that
S = 2r 2 +
4000
,
r
where S cm 2 is the total surface area of the prism.
b) Hence determine the value of r and the value of h which make S least, fully
justifying your answer.
r = 10 , h = 10
Created by T. Madas
Created by T. Madas
Question 27
(****)
A tank is in the shape of a closed right circular cylinder of radius r m and height h m .
The tank has a volume of 16π m3 and is made of thin sheet metal.
Given the surface area of the tank is a minimum, determine the value of r and the value
of h .
r=2 , h=4
Created by T. Madas
Created by T. Madas
Question 28 (****+)
12 x
H
G
D
C
y
E
F
x
A
B
10 x
The figure above shows the design of a baking tray with a horizontal rectangular base
ABCD , measuring 10 x cm by y cm .
The faces ABFE and DCGH are isosceles trapeziums, parallel to each other.
The lengths of the edges EF and HG are 12 x cm .
The faces ADHE and BCGF are identical rectangles.
The height of the tray is x cm .
The capacity of the tray is 1980 cm3 .
a) Show that the surface area, A cm 2 , of the tray is given by
A = 22 x 2 +
360
5+ 2 .
x
(
)
b) Determine the value of x for which A is stationary, showing that this value of
x minimizes the value for A .
c) Calculate the minimum surface area of the tray.
x ≈ 3.744 , Amin ≈ 925
[solution overleaf]
Created by T. Madas
