Download ae-8-Cubes-and-Cube-Root

ID : ae-8-Cubes-and-Cube-Root [1]
Grade 8
Cubes and Cube Root
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Answer t he quest ions
(1)
If the surf ace areas of two cubes are in the ratio 9:25, and the volume of the f irst cube is 216
m3, then what is the volume of the second cube?
(2)
If one f ace of a cube has an area of 25 m2, then what is the volume of the cube?
(3)
If you add a number x with another number that is 13 times the value of x, and then take the
cube of the sum, what will be the result?
(4) Calculate the cube root of 64
(5)
T he number of people living in a small town is f ound to be a perf ect cube. We know that the
number of men in the town is 65231, and the number of women in the town is 93944.
By an odd coincidence, we count and f ind that the number of children in the town is also a
perf ect cube. If we give you a hint that this number is more than 35930, then what is the smallest
possible number of children in the town?
(6) T here are two numbers such that sum of the numbers is 26 and their dif f erence is 4. Find the
sum of their cubes .
(7) What is the value of
(8)
What is the value of (
11
)3
15
Choose correct answer(s) f rom given choice
(9) T ake a number x, and multiply it with 3. T ake the cube of the resulting number. What is the ratio
of this number to the cube of the original number?
a. 27:3
b. 9:1
c. 27:1
d. 3:1
(10) What is the value of 14 3
a. 2744
b. 2730
c. 2940
d. 3150
(11) Which of the f ollowing numbers is a perf ect cube?
a. 998
b. 8006
c. 4099
d. 4913
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ID : ae-8-Cubes-and-Cube-Root [2]
(12) A number n is called a perf ect cube if there exists:
a. a natural number m such that m = n + n + n
b. a natural number m such that n = m x m
c. a natural number m such that m = n x n x n
d. a natural number m such that n = m x m x m
(13) Find the value of A if
a. 2
b. 5
c. 8
d. 7
(14) If you subtract a number x f rom 40 times that number, and then take the cube of the dif f erence,
what will be the result?
a. 59395x3
b. 1611x2
c. 37x3
d. 59319x3
Fill in t he blanks
(15)
T he value of
is
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ID : ae-8-Cubes-and-Cube-Root [3]
Answers
(1)
1000 m3
Step 1
Let us assume the side of the f irst cube is a .
Let us assume the side of the second cube is b.
Surf ace area of f irst cube = 6a2
Surf ace area of second cube = 6b2
Ratio of surf ace area of two cubes = 9:25
Step 2
Volume of f irst cube, a3 = 216 m3
⇒ a3 = 63
⇒a=6
Step 3
Now the ratio of surf ace area of cubes,
6a2
=
25
6b2
⇒
9
62
9
=
25
b2
⇒ 36 × 25 = 9 × b2
⇒
900
= b2
9
⇒ 100 = b2
⇒ 102 = b2
⇒ b = 10
Step 4
Volume of second cube, b3 = (10)3
= 10 × 10 × 10
= 1000
Step 5
T heref ore, the volume of the second cube is 1000 m3.
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ID : ae-8-Cubes-and-Cube-Root [4]
(2)
125 m3
Step 1
Let us assume the length of a side of the cube is a.
Step 2
T he area of one f ace of cube is 25 m2.
Area of cube of one f ace = a2
⇒ 25 = a2
⇒ 52 = a2
⇒a=5m
Step 3
Volume of the cube,
V = a3
= (5)3
=5×5×5
= 125 m3
Step 4
T heref ore, the volume of the cube is 125 m3.
(3)
2744x3
Step 1
According to the question,
the f irst number is x and the second number is 13 x.
Step 2
Sum of the two numbers = (x + 13 x)
Step 3
Cube of the sum = (x + 13 x)3
= (14 x)3
= 2744 x3
Step 4
T heref ore, the result will be 2744x3.
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ID : ae-8-Cubes-and-Cube-Root [5]
(4) 4
Step 1
We have been asked to f ind the cube root of 64.
Step 2
Prime f actors of 64 are,
2 × 2 × 2 × 2 × 2 × 2.
Step 3
Now, make group of three prime f actors.
(2 × 2 × 2) × (2 × 2 × 2)
Step 4
T here is no prime f actor lef t which is not part of any group. Cube root is given by the
multiplication of a element f rom each group of prime f actors. i.e.
2×2
=4
Step 5
T heref ore, the cube root of 64 is 4.
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ID : ae-8-Cubes-and-Cube-Root [6]
(5)
35937
Step 1
T otal population of the town is a perf ect cube.
Number of men in the town = 65231
Number of women in the town = 93944
Step 2
It is given that the number of children in the village is also the perf ect cube, and is more
than 35930. Let us f irst f ind the values of perf ect cubes more than 35930.
Step 3
Cube root of 35930 = 32.99785722236 and the perf ect cubes more than 35930 will be the
cubes of all integers more than 32.99785722236, i.e. 333, 34 3, 353, etc.
Step 4
T he population of children will be the smallest value out of 333, 34 3, 353, etc, f or which the
total population of the village is a perf ect cube.
Step 5
Let us see if the population of the children can be 333 (= 35937). In this case, the total
population of the village will be = 65231 + 93944 + 333
= 65231 + 93944 + 35937
= 195112
Since the cube root of 195112 is 58, which is an integer, the total population of the village
in this case is indeed a perf ect cube.
Step 6
T heref ore, the smallest possible number of children in the village is 333 = 35937.
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ID : ae-8-Cubes-and-Cube-Root [7]
(6) 4706
Step 1
Let us assume that x and y are the two numbers.
Step 2
According to the question, the sum of two numbers is 26, that is,
x + y = 26 ........... (1)
Step 3
And the dif f erence of two numbers is 4, that is,
x - y = 4 ........... (2)
Step 4
On adding equations (1) and (2),
x + y + x - y = 26 + 4
⇒ 2x = 30
⇒ x = 15
Step 5
Now using equation (1),
x + y = 26
⇒ 15 + y = 26
⇒ y = 26 - 15 = 11
Step 6
Now, sum of their cubes = x3 + y3
= (15)3 + (11)3
= 4706
Step 7
T heref ore, the value of the sum of their cubes is 4706.
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ID : ae-8-Cubes-and-Cube-Root [8]
(7)
6
17
Step 1
We have been asked to f ind the value of
.
Step 2
Now,
(6 × 6 × 6)
=(
)
(17 × 17 × 17)
(6)3
=(
)
(17)3
6
= ((
)3)
17
1
3
1
3
1
3
6
=
17
Step 3
T heref ore, the value of
is
6
.
17
(8)
1331
3375
Step 1
We have been asked to f ind the value of (
11
)3.
15
Step 2
Now,
(
11
)3=
15
=
113
153
=
11 × 11 × 11
15 × 15 × 15
1331
3375
Step 3
T heref ore, the value of (
11
15
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)3 is
1331
.
3375
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ID : ae-8-Cubes-and-Cube-Root [9]
(9) c. 27:1
Step 1
If we multiply the number x with 3, the resulting number is: 3x.
Step 2
T he cube of the resulting number 3x = (3x)3
= 27x3
Step 3
T he original number is x.
Cube of the original number = x3.
Step 4
T he ratio of the new number to the original number =
New number
Original number
=
27x3
x3
=
27
1
Step 5
T heref ore, the ratio of the new number to the cube of the original number is 27:1.
(10) a. 2744
Step 1
According to the question, we have to f ind the value of 14 3.
Step 2
Now,
14 3 = 14 × 14 × 14
= 196 × 14
= 2744
Step 3
T heref ore, the value of 14 3 is 2744.
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ID : ae-8-Cubes-and-Cube-Root [10]
(11) d. 4913
Step 1
T he def inition of a perf ect cube is a number that is the result of multiplying an integer by
itself three times.Steps to f ind the perf ect cube are as f ollows.
i. Resolve the given number into prime f actors.
ii. Group the f actors in 3 in such a way that each number of the group is same.
iii. If there is nothing lef t with the group of prime numbers so the number is perf ect cube
otherwise not.
Step 2
Prime f actors of 4913 = 17 x 17 x 17
T here is no prime f actor lef t with the group of prime f actors. So, 4913 is a perf ect cube.
Step 3
Prime f actor of 4099 = 4099
T here is a prime f actor lef t. So, 4099 is not a perf ect cube.
Step 4
Prime f actors of 998 = 2 x 499
T here is a prime f actor lef t. So, 998 is not a perf ect cube.
Step 5
Prime f actors of 8006 = 2 x 4003
T here is a prime f actor lef t. So, 8006 is not a perf ect cube.
Step 6
T heref ore, 4913 is the perf ect cube.
(12) d. a natural number m such that n = m x m x m
Step 1
A perf ect cube is a natural number that has a natural number as its cube root.
In other words, a perf ect cube is the result of multiplying a natural number three times by
itself .
For example: 5 × 5 × 5 = 53 = 125, which is a perf ect cube.
Step 2
For any natural number m, n = m × m × m is a perf ect cube.
Step 3
T heref ore, the number n is called a perf ect cube if there exists a natural number m such
that n = m × m × m.
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ID : ae-8-Cubes-and-Cube-Root [11]
(13) b. 5
Step 1
We have been asked to f ind the value of A f rom the f ollowing equation,
³√(200A) = 10.
Step 2
³√(200A) = 10
T aking cube both side,
⇒ (³√(200A))3 = (10)3
⇒ 200A = 10 × 10 × 10
⇒A=
2×2×2×5×5×5
200
⇒A=
2×2×2×5×5×5
2×2×2×5×5
⇒A=
2×2×2×5×5×5
2×2×2×5×5
⇒A= 5
Step 3
T heref ore, the value of A is 5.
(14) d. 59319x3
Step 1
According to the question,
the f irst number is x and
the second number is 40 x.
Step 2
Dif f erence of the numbers = (40 x - x)
Step 3
Cube of the dif f erence = (40 x - x)3
= (39 x)3
= 59319 x3
Step 4
T heref ore, the result will be 59319x3.
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ID : ae-8-Cubes-and-Cube-Root [12]
(15)
1.2
Step 1
Let x = ³√(1.728)
Step 2
Multiply both lef t and right hand side by 10,
10x = 10 × ³√(1.728)
⇒ 10x = ³√(1.728 × 1000)
⇒ 10x = ³√(1728)
Step 3
Now f ind the cube root of 1728
⇒ 10x = ³√(1728) = 12
⇒ x = 12/10
⇒ x = 1.2
Step 4
T heref ore, the value of ³√(1.728) is 1.2.
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