\(V_1, V_2, V_3\) and \(V_4\) are the volumes of four cubes of side lengths x cm, 2x cm, 3x cm and 4x cm respectively. The following statements regarding these volumes are given below.
(A) \(V_1+V_2+2V_3\lt V_4\)
(B) \(V_1+3V_2\gt V_3+V_4\)
(C) \(2(V_1+V_3)+V_2=V_4\)
(D) \(V_1+4V_2+V_3\lt V_4\)
Which of these statements is/are correct?
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (C), (D), and (A) only
The volume of a cube of side \(a\) is \(a^3\), so each volume is a multiple of \(x^3\). Every statement can then be checked with these multiples alone.
\(V_1=x^3\), \(V_2=(2x)^3=8x^3\), \(V_3=(3x)^3=27x^3\), \(V_4=(4x)^3=64x^3\)
In units of \(x^3\):
(A) \(V_1+V_2+2V_3\lt V_4\) Correct.
\(1+8+2\times27=63\), which is less than 64
(B) \(V_1+3V_2\gt V_3+V_4\) Incorrect.
Left side: \(1+3\times8=25\)
Right side: \(27+64=91\), so the left side is smaller
(C) \(2(V_1+V_3)+V_2=V_4\) Correct.
\(2(1+27)+8=56+8=64\), which equals \(V_4\)
(D) \(V_1+4V_2+V_3\lt V_4\) Correct.
\(1+4\times8+27=60\), which is less than 64