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Bài tập Toán DIFFERENTIATION OPTIMIZATION 16
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Created by T. Madas
Question 41 (*****)
A right circular cone of radius r and height h is to be cut out of a sphere of radius R .
It is a requirement that the circumference of the base of the cone and its vertex lie on the
surface of the sphere.
Determine, in exact form in terms of R , and with full justification, the maximum
volume of the cone that can be cut out of this sphere.
Vmax =
Created by T. Madas
32π R 3
81
Created by T. Madas
Question 42 (*****)
A mobile phone wholesaler buys a certain brand of phone for £35 a unit and sells it to
shops for £100 a unit.
In a typical week the wholesaler expects to sell 500 of these phones.
Research however showed that on a typical week for every £1 reduced of the selling
price of this phone, an extra 20 sales can be achieved.
Determine the selling price for this phone if the weekly profit is to be maximized, and
find this maximum weekly profit.
£80, maximum profit £40500
Created by T. Madas
Created by T. Madas
Question 43 (*****)
R
H
R
The figure above shows a hollow container consisting of a right circular cylinder of
radius R and of height H joined to a hemisphere of radius R .
The cylinder is open on one of the circular ends and the hemisphere is also open on its
circular base. The cylinder is joined to the hemisphere at their open ends so that the
resulting object is completely sealed.
Given that volume of the container is V , show the surface area of the container is
minimised when R = H , and hence show further that this minimum surface area is
3
45π V 2 .
proof
Created by T. Madas
