Bài tập Toán DIFFERENTIATION OPTIMIZATION 06

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 hei

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

a) Given that the surface area of the tank is 54 2

m , show that the capacity of the

tank, V m3, is given by

3 2 18 3

V = x− x

b) Find the maximum value for V , fully justifying the fact that it is indeed the

maximum value

max 36

x

Question 16 (***+)

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 xcm The circular sector subtends an angle of θ radians at the centre

a) Show that

60 5

xθ = − x

The total area of the two shapes is Acm 2

b) Show clearly that

1

4

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

7.26

x ≈ , Amax ≈109

x

x x

x

x

θ

Question 17 (****)

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 2x m and is attached to one side

of the rectangle also measuring 2x 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 2

m , is given by

2 1 2

2

A= x− x − πx

b) Show further, by using a differentiation method, that the maximum area of the

stage is

1800 4

π +

2

m

proof

2x

y