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Bài tập Toán DIFFERENTIATION OPTIMIZATION 06
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Created by T. Madas
Question 15 (***+)
The figure below shows a large tank in the shape of a cuboid with a rectangular base
and no top.
Two of the vertical opposite faces of the cuboid are square and the height of the cuboid
is x metres.
x
a) Given that the surface area of the tank is 54 m 2 , show that the capacity of the
tank, V m3 , is given by
V = 18 x − 2 x3 .
3
b) Find the maximum value for V , fully justifying the fact that it is indeed the
maximum value.
Vmax = 36
Created by T. Madas
Created by T. Madas
Question 16 (***+)
x
x
x
θ
x
x
A wire of total length 60 cm is to be cut into two pieces. The first piece is bent to form
an equilateral triangle of side length x cm and the second piece is bent to form a
circular sector of radius x cm . The circular sector subtends an angle of θ radians at the
centre.
a) Show that
xθ = 60 − 5 x .
The total area of the two shapes is A cm 2 .
b) Show clearly that
A=
1
4
(
)
3 − 10 x 2 + 30 x .
c) Use differentiation to find the value of x for which A is stationary.
d) Find, correct to three significant figures, the maximum value of A , justifying the
fact that it is indeed the maximum value of A .
x ≈ 7.26 , Amax ≈ 109
Created by T. Madas
Created by T. Madas
Question 17 (****)
2x
y
The figure above shows the design of a theatre stage which is the shape of a semicircle
attached to rectangle. The diameter of the semicircle is 2 x m and is attached to one side
of the rectangle also measuring 2 x m . The other side of the rectangle is y m .
The perimeter of the stage is 60 m .
a) Show that the total area of the stage, A m 2 , is given by
A = 60 x − 2 x 2 − 1 π x 2 .
2
b) Show further, by using a differentiation method, that the maximum area of the
stage is
1800
m2 .
π +4
proof
Created by T. Madas
