Download problem-solving guidelines

Problem-solving
guidelines:
 Understanding the
problem
 Developing a plan and
strategy
 Carrying out the plan
 Checking the answer
Problem-Solving
Strategies:
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Guessing and checking
Drawing a table/diagram
Writing an equation
Simplifying the problem
Looking for patterns
Using logical reasoning
EXAMPLE 1
Drawing a Diagram
Silage, fodder preserved and stored by farmers to
provide nutritious feedstuff for livestock in the
winter. The large, round, brick or metal storage
container commonly found on farms in the U.S. is
known as a silo. Originally, corn, sorghum, and
sunflower were staple silage crops in the U.S., but
these have been replaced by the more protein-rich
alfalfa. A silo shown below has been built from
metal. The top part of the silo is a cylinder with a
diameter of 4 m and the height of 8 m. The bottom
part of the silo is a cone of slant height of 3 m. The
silo has a circular opening of a radius of 30 cm on
the top. (Look at the diagram below)
a. What area of metal (to the nearest m2) which is needed to build the silo?
b. If it costs Rp 25,000.00 per m2 to cover the surface with an anti-rust material, how
much will it cost to cover the silo completely?
Mathematics for Junior High School Grade 9 / 67
Solution:
1. Understanding the problem
a. What is the unknown?
The area of the metal and its cost to cover the surface
b. What are the data?
A cylinder with a diameter of 4 m and height of 8 m.
The bottom part of the silo is a cone with a slant height of 3 m.
The silo has a circular opening with a radius of 30 cm on the top.
It costs Rp 25,000.00 per m2 to cover the surface with an anti-rust material
2. Developing a plan and strategy
a. Find the surface area of the top face.
b. Find the area of the curved surface of the cylinder.
c. Find the area of the curved surface of a cone.
d. Find the area of the silo by adding the area of the top face, the curved
surface of the cylinder and the curved surface of a cone.
e. The total cost can be obtained by multiplying total area by the cost per m2.
3. Carrying out the plan
a. The area of the top face = the area of a large circle – the area of a small
circle (the opening).
= 3.14(22) - 3.14(0.32)
= 3.14(4 - 0.09)
= 12.28 m2
b. The area of the curved surface of the cylinder = 2(3.14)(2)(8) = 100.48 m2
c. The area of the curved surface of cone = 3.14rs = 3.14 (2)(3) = 18.84 m2
d. The area of the silo = 12.28 m2 + 100.48 m2 + 18.84 m2 = 131.6 m2
e. Total cost = 132 x (Rp 25,000.00) = Rp 3,300,000.00
68 / / Student’s Book – Problem Solving
An object is made of a cuboid at the bottom and a hemisphere on the top (see picture
below). Find the volume of the object. Give your answer to the nearest tenth.
EXAMPLE 2
Solution:
1. Understanding the problem
a. What is the unknown?
The volume of the cuboid and hemisphere
b. What are the data?
A cuboid having the edge of 4 cm and the height of 2 cm.
The radius of hemisphere is 2 cm.
2. Developing a plan and strategy
The object is made up of a cuboid and a hemisphere.
Find the volume of the cuboid first, then the volume of the hemisphere. The
volume of the object is the total volume of both the cuboid and the
hemisphere.
3. Carrying out the plan
a. The volume of the cuboid is 4(4)(2) = 32 cm3
b. The volume of the hemisphere is 0.5(4/3)(3.14)(23) = 16.77 cm3
c. The total volume is 32 cm3 + 16.77 cm3 = 48.77 cm3
4. Checking the answer
The answer refers to the original question. Recheck the volume of the cuboid
and the hemisphere, and the total volume found.
Mathematics for Junior High School Grade 9 / 69
Use the strategy drawing diagram and/or writing equation to solve each
problem.
1.
A right triangle, whose right sides are 15 cm and 20 cm, is revolved with
respect to its hypotenuse. Find the volume and the surface area of the double
cone object which is formed.
2.
A warehouse building is in the form as shown in the figure below.
7m
3m
10 m
The building consists of two parts: a cuboid shape and a half cylinder shape. If
the radius of the circular cylinder base is 3.5 m and the dimension of the cuboid
is 10 m x 7 m x 3 m, find the volume of the building and the inner surface area of
the building excluding the floor.
3.
A cylinder whose radius of its base is 7 cm and whose height is 10 cm. If
another cylinder is made with the radius of its base is doubled, what is the
volume of the second cylinder compared to the first one ? (take  = 22/7)
4.
A sphere is inside a cylinder as given on the figure below. The sphere touches
the lateral face, the top, and the base of the cylinder. Show that the area of the
lateral face of the cylinder is the same as the surface area of the sphere.
70 / / Student’s Book – Problem Solving
5.
A cylinder has a base whose radius is r. It is reduced in such a way that its
radius is ½ r. If the initial volume of the cylinder is 480 cm3, find its volume
after the reduction.
6.
A metal sphere is put into a full-of-water cylinder. The radius of the cylinder is
the same as that of the sphere, namely 10 cm, and  = 3.14. Find how much
water is left in the cylinder after the sphere is put inside.
7.
A toy is in the form of a cone mounted on a hemisphere with the same radius.
The diameter of the base of the conical part is 6 cm and its height is 4 cm.
Determine the surface area of the toy (using  = 3.14).
8.
A solid is composed from a cylinder and two hemispheres at the ends of the
cylinder. If the total length of the solid is 104 cm and the radius of each of the
hemisphere is 7 cm, find the cost of polishing the surface of the solid at the rate
of Rp 100 per cm2.
9.
Find the mass of a 3.5 m long pipe, if the external diameter of the pipe is 2.4 cm,
the thickness of the pipe is 2 mm and the mass of 1 cm3 of pipe is 11.4 kg.
10. A solid of toy is in the form of a hemisphere surmounted by a right circular
cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. If a
right circular cylinder circumscribes the solid, find how much more space it
will cover.
Mathematics for Junior High School Grade 9 / 71