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Bài tập CALCULUS 70
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Created by T. Madas
Question 240
(****)
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 .
C2Z , r = 2 , h = 4
Question 241
(****+)
The curve C has equation
y = x3 + ax 2 + bx − 10 ,
where a and b are constants.
The curve has two stationary points P and Q .
Given the coordinates of P are ( −1, −5 ) , find the coordinates of Q and use
d2y
dx 2
to
determine its nature.
C2I , Q ( 3, −37 ) , min
Created by T. Madas
Created by T. Madas
Question 242
(****+)
It is given that
2
kx 2 + a dx = 11
and
k
1
1
6
x2
dx = a ,
where a and k are constants.
Determine the possible values of k .
MP1-U , k = 3 ∪ k = − 6
7
Created by T. Madas
Created by T. Madas
Question 243
(****+)
The profit of a small business, £ P is modelled by the equation
P=
( 54 x + 6 y − xy − 324 )2 ,
3x
where x and y are positive variables associated with the running of the company.
It is further known that x and y constrained by the relation
3x + y = 54 .
a) Show clearly that
P = 108 x − 36 x 2 + 3 x3 .
b) Hence show that the stationary value of P produces a maximum value of £96 .
The owner is very concerned about the very small profit and shows the calculations to
a mathematician. The mathematician agrees that the calculations are correct but he
asserts that the profit is substantially higher.
c) Explain, by calculations, the mathematician’s reasoning.
C2X , proof
Created by T. Madas
