[PDF] [PDF] DIFFERENTIATION OPTIMIZATION PROBLEMS - MadAsMaths

[PDF] Find the volume of each prism 1 SOLUTION: The volume V of a

The volume V of a prism is V = Bh, where B is the 3 the oblique rectangular prism shown The planter is to be filled inches below the top, so Find each measure to the nearest tenth the base area and height of each triangular prism

[PDF] Find the volume of each prism 1 SOLUTION: The volume V of a

SOLUTION: The volume V of a prism is V = Bh, where B is the 3 the oblique rectangular prism shown SOLUTION: inches below the top? Find each measure to the nearest tenth 30 the base area and height of each triangular prism

[PDF] Quiz-12-Review-GH_Solutionspdf

2 Find the volume of the composite space figure to the nearest whole number The right prism below has bases which are equilateral triangles of side length 4 to the nearest tenth, if necessary Rect Prism + Triangular Prism V=41004)

[PDF] Surface area and volume

Questions 1 to 3 refer to the prism below 4 Calculate the surface area of each of the following triangular prisms 2 Find the surface areas of these closed cylinders to the nearest whole number The volume of a right prism (or cylinder) is given by: V = A × h where A is the area of the base (or cross-sectional area) and h 

[PDF] Chapter 10

The volume of a three-dimensional figure is the amount of space it round to the nearest whole number V = 56 ft3 12 7 yd w 9 yd V = 189 yd3 13 Find the volume of a rectangular prism with length 9 meters, width To find a missing dimension in a triangular prism, you will solve one or two equations shown below

[PDF] Surface Area and Volume

or your volume is increasing more?” Find the surface area of the solid shown by the net Finding the Surface Area of a Triangular Prism 2 nearest tenth Words The volume V of a prism is the product of the area of the base and the nearest whole number A rectangular prism and its dimensions are shown below

[PDF] Algebra_with_Finance_BTC_Activity_4pdf

Volume of the cylinder: V = Bh Sometimes the height of a triangular base in a triangular prism is not given Find the volume of each triangular prism to the nearest tenth 10 Find the volume of each composite figure to the nearest whole number From the figures shown below, choose the pyramid with volume closest

[PDF] Geometry test review answer key - Commack Schools

Radius = + (diameter) Volume of prisms and cylinder: V = Bh Pyramid: SA = (# of triangular or lateral faces) ( bh )+ area of the base a 1) Find the area of the following figure 5m A=bh Round to the nearest tenth d=13 card 6) A) Find the volume of the pyramid below B) Find the surface area of the following V=Bh H

[PDF] Volumes of Prisms and Cylinders 115 - Big Ideas Math

V = Bh Finding Volume Work with a partner Consider a stack of square How can you find the volume of a prism or cylinder that is not a right prism volume The prisms below have equal heights h and equal cross-sectional Density is the amount of matter that an object has in a given unit of volume nearest gram

[PDF] DIFFERENTIATION OPTIMIZATION PROBLEMS - MadAsMaths

a) Show that the volume of the box, V 3 cm , is given by 3 2 4 176 1536 V d) Find the maximum value for V , fully justifying the fact that it is the maximum 12 The figure above shows a solid triangular prism with a total surface area of 3600 2 this maximum value of V , correct to the nearest 3 where k is an integer

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Created by T. Madas

Created by T. Madas

DIFFERENTIATION

OPTIMIZATION

PROBLEMS

Created by T. Madas

Created by T. Madas

Question 1 (***)

An open box is to be made out of a rectangular piece of card measuring

64cm by

24cm. Figure 1 shows how a square of side length xcm is to be cut out of each corner

so that the box can be made by folding, as shown in figure 2. a) Show that the volume of the box, V3cm, is given by

3 24 176 1536V x x x= - +.

b) Show further that the stationary points of Voccur when

23 88 384 0x x- + =.

c) Find the value of x for which V is stationary. (You may find the fact

24 16 384× = useful.)

d) Find, to the nearest 3cm, the maximum value for V, justifying that it is indeed the maximum value.

163x=, max3793V≈

xx 24cm
x64cm figure 1figure 2

Created by T. Madas

Created by T. Madas

Question 2 (***)

The figure above shows the design of a fruit juice carton with capacity of

10003cm.

The design of the carton is that of a closed cuboid whose base measures xcm by

2xcm, and its height is h cm.

a) Show that the surface area of the carton, A2cm, is given by

230004A xx= +.

b) Find the value of x for which A is stationary. c) Calculate the minimum value forA, justifying fully the fact that it is indeed the minimum value of A.

3375 7.21x= ≈, min624A≈

h 2xx

Created by T. Madas

Created by T. Madas

Question 3 (***)

The figure above shows a

solid brick, in the shape of a cuboid, measuring 5xcm by xcm by h cm. The total surface area of the brick is 7202cm. a) Show that the volume of the brick, V3cm, is given by

3253006V x x= -.

b) Find the value of x for which V is stationary. c) Calculate the maximum value for V, fully justifying the fact that it is indeed the maximum value.

2 6 4.90x= ≈, max400 6 980V= ≈

h x 5x

Created by T. Madas

Created by T. Madas

Question 4 (***)

The figure above shows a box in the shape of a cuboid with a rectangular base xcm by

4xcm and no top. The height of the box is h cm.

It is given that the surface area of the box is

21728 cm.

a) Show clearly that

2864 2

5 xhx b) Use part (a) to show that the volume of the box , V3cm, is given by ()384325V x x= -. c) Find the value of x for which V is stationary. d) Find the maximum value for V, fully justifying the fact that it is the maximum.

12x=, max5529.6V=

4x x h

Created by T. Madas

Created by T. Madas

Question 5 (***)

The figure above shows the design of a large water tank in the shape of a cuboid with a square base and no top.

The square base is of length

x metres and its height is h metres.

It is given that the volume of the tank is

5003m.

a) Show that the surface area of the tank, A2m, is given by

22000A xx= +.

b) Find the value of x for which A is stationary. c) Find the minimum value forA, fully justifying the fact that it is the minimum.

10x=, min300A=

x h x

Created by T. Madas

Created by T. Madas

Question 6 (***)

The figure above shows a pentagon

ABCDE whose measurements, in cm, are given in

terms of x and y. a) If the perimeter of the pentagon is 120cm, show clearly that its area, A2cm, is given by

2600 96A x x= -.

b) Use a method based on differentiation to calculate the maximum value forA, fully justifying the fact that it is indeed the maximum value. max937.5A= 10x 8x6x y A B CD E

Created by T. Madas

Created by T. Madas

Question 7 (***)

The figure above shows a clothes design consisting of two identical rectangles attached to each of the straight sides of a circular sector of radius xcm.

The rectangles measure

xcm by ycm and the circular sector subtends an angle of one radian at the centre.

The perimeter of the design is

40cm.
a) Show that the area of the design, A2cm, is given by

220A x x= -.

b) Determine by differentiation the value of x for which A is stationary. c) Show that the value of x found in part (b) gives the maximum value for A. d) Find the maximum area of the design.

10x=, max100A=

xC x D E A B F G y y c1

Created by T. Madas

Created by T. Madas

Question 8 (***+)

The figure above shows a

closed cylindrical can of radius rcm and height hcm. a) Given that the surface area of the can is 192π2cm, show that the volume of the can,

V3cm, is given by

396V r rπ π= -.

b) Find the value of r for which V is stationary. c) Justify that the value of r found in part (b) gives the maximum value for V. d) Calculate the maximum value of V.

4 2 5.66r= ≈, max256 2 1137Vπ= ≈

h r

Created by T. Madas

Created by T. Madas

Question 9 (***+)

A pencil holder is in the shape of a right circular cylinder, which is open at one of its circular ends.

The cylinder has radius

r cm and height h cm and the total surface area of the cylinder, including its base, is

3602cm.

a) Show that the volume, V3cm, of the cylinder is given by

311802V r rπ= -.

b) Determine by differentiation the value of r for which V has a stationary value. c) Show that the value of r found in part (b) gives the maximum value for V. d) Calculate, to the nearest 3cm, the maximum volume of the pencil holder.

1206.18rπ= ≈, max742V≈

r h

Created by T. Madas

Created by T. Madas

Question 10 (***+)

The figure above shows a solid triangular prism with a total surface area of 36002cm. The triangular faces of the prism are right angled with a base of

20xcm and a height of

15xcm. The length of the prism is ycm.

a) Show that the volume of the prism, V3cm, is given by

39000 750V x x= -.

b) Find the value of x for which V is stationary. c) Show that the value of x found in part (b) gives the maximum value for V. d) Determine the value of y when V becomes maximum.

2x=, 20y=

25x
y15x 20x

Created by T. Madas

Created by T. Madas

Question 11 (***+)

The figure above shows a

closed cylindrical can, of radius rcm and height hcm. a) If the volume of the can is 3303cm, show that surface area of the can, A2cm, is given by

26602A rrπ= +.

b) Find the value of r for which A is stationary. c) Justify that the value of r found in part (b) gives the minimum value for A. d) Hence calculate the minimum value of A.

3.745r≈, min264A≈

h r

Created by T. Madas

Created by T. Madas

Question 12 (***+)

The figure above shows

12 rigid rods, joined together to form the framework of a

storage container, which in the shape of a cuboid.

Each of the four upright rods has height

hm. Each of the longer horizontal rods has length lm and each of the shorter horizontal rods have length ()2l-m. a) Given that the total length of the 12 rods is 36m show that the volume, V3m, of the container satisfies

3 22 15 22V l l l= - + -.

b) Find, correct to 3 decimal places, the value of l which make Vstationary. c) Justify that the value of l found in part (b) maximizes the value of V, and find this maximum value of

V, correct to the nearest 3m.

d) State the three measurements of the container when its volume is maximum.quotesdbs_dbs21.pdfusesText_27