using the identity (a-b)cube =a cube-3a square b+3ab+b cube. 1, (5-x)cube. 2,(2x-4y) cube. 3,(47)cube. 4,(48) cube.please give the correct answers.
Answers
Answer:
First we will make a cube of dimensions (a+b) i.e., (3+1 = 4 units). For this, we will use 64 unit cubes and arrange them as shown in fig.(i).
cbse-class-9-maths-lab-manual-algebraic-identity-a-b3-a3-b3-3a2b-3ab2-1
From other set of 64 cubes we will make arrangements as shown in figures.
Arrange 27 cubes such that a x a x a cube is formed i.e., 3 x 3 x 3 set is formed in fig.(ii).
cbse-class-9-maths-lab-manual-algebraic-identity-a-b3-a3-b3-3a2b-3ab2-2
Arrange 9 cubes such that 3 columns of 3 cubes is formed as shown in fig. (iii). Make 3 sets of such arrangement.
cbse-class-9-maths-lab-manual-algebraic-identity-a-b3-a3-b3-3a2b-3ab2-3
Arrange 3 cubes such that 1 column of 3 cubes is formed as shown in fig. (iv). Make 3 sets of such arrangement.
cbse-class-9-maths-lab-manual-algebraic-identity-a-b3-a3-b3-3a2b-3ab2-4
Last arrangement consists of only 1 cube of volume b3 [fig (v)].
cbse-class-9-maths-lab-manual-algebraic-identity-a-b3-a3-b3-3a2b-3ab2-5
Observation and Calculation
In fig. (i) we have used 64 cubes i.e., (a+b)3
Other set of 64 cubes are arranged in following manner:
Volume of cube in fig. (ii) = a3
Volume of cuboids in fig.(iii) = 3(a)(a)(b) = 3ba2
Volume of cuboids in fig. (iv) = 3(a)(b)(b) = 3ab2
Volume of cube in fig. (v) = b3
Total volume of cube in fig. (i) = Total volume of cubes and cuboids in fig. (ii), (iii), (iv) and (v)
(a+b)3 = a3 + b3 + 3a2b + 3ab2
Hence, this identity is verified geometrically.
Result
The identity (a+b)3 = a3 + b3 + 3a2b + 3ab2 is verified geometrically by using cubes and cuboids.
Learning Outcome
In this way, students can learn the concept of verifying the identity geometrically by adding volume of cubes and cuboids.
Activity Time
Students must perform this activity for any other values of a and b, e.g., a=4, b=1 and find volumes of different cubes and cuboids used in this activity.
Viva Voce
Question 1.
Find 113 by using the formula (a+b)3 = a3 + b3 + 3a2b + 3ab2.
Answer:
(10 + 1)3 = (10)3 + 3 x (10)2 x 1 + 3 x 10 x 1 + (1)3 = 1331.
Question 2.
Find 233 by using the formula (a+b)3 = a3 + b3 + 3a2b + 3ab2
Answer:
(20 + 3)3 = (20)3 + (3)3 + 3.(20)2.(3) + 3.(20)(3)2
= 8000 + 27 + 9(400) + 60(9) = 8000 + 27 + 3600 + 540 = 12167.
Question 3.
Find (5x + 5y)3
Answer:
27x3 + 125y3 + 135x2y + 225xy2.
Question 4.
Find (10 + 2)3.
Answer:
103 + 23 +600+ 120 = 1728.
Question 5.
In the activity of (a + b)3, what do you mean by 3a2b, 3ab2?
Answer:
3 set of cuboids with volumes a x a x b and 3 sets of cuboids with volumes a x b x b.
Question 6.
If one side of a cube is (a – b), then what is the volume of the cube ?
Answer:
Volume of cube = (a – b)3.