Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
MHT CET202613 April 2026Morning ShiftMathematicsApplication of DerivativesActual

If a spherical balloon has a variable diameter 3x + 9 2 units, then the rate of change of its volume with respect to x is

Options

  1. A27 4 (2x + 3)^2
  2. B2 3 (2x + 3)^2
  3. C27 8 (2x + 3)^2
  4. D27 2 (2x + 3)^2

Correct answer

C. 27 8 (2x + 3)^2

Step-by-step solution

Given the diameter of the spherical balloon is D = 3x + 9 2 = 3(2x + 3) 2 The radius of the balloon is r = D 2 = 3(2x + 3) 4 The volume of the spherical balloon is V = 4 3 r^3 Substituting the value of r : V = 4 3 ( 3(2x + 3) 4 )^3 V = 4 3 ( 27(2x + 3)^3 64 ) V = 9 16 (2x + 3)^3 Differentiating the volume with respect to x : dV dx = 9 16 3(2x + 3)^2 d dx (2x + 3) dV dx = 27 16 (2x + 3)^2 2 dV dx = 27 8 (2x + 3)^2

Practice Application of Derivatives on Quantrex Academy →

More from Application of Derivatives

Consider the quadratic equation a x^2+b x+c=0 , where 2 a+3 b+6 c=0 and let g(x)= a x^3 3 + b x^2 2 +c x . Statement-I : The given quadratic equation ax ^2+ bx + c =0 has at least 2025The difference between the absolute maximum and absolute minimum values of the function f(x)=2 x^3-15 x^2+36 x-30 on [-1,4] is 2025If f(x)=x e^ x(1-x) , x R , then f(x) is 2025The angle between the curves y ^2= x and x ^2= y at the point (1,1) is 2025If the tangent of the curve 4 y^3=3 a x^2+x^3 drawn at the point (a, a) forms a triangle of area 25 24 sq.units with the coordinate axes then a = 2025If the function f(x)= x- ^2 x is defined on the interval [- , ] , then f is strictly increasing in the interval 2025If the Lagrange's mean value theorem is applied to the function f(x)=e^x defined on the interval [1,2] and the value of c (1,2) is k , then e^ k-1 = 2025If the tangent to the curve x y+a x+b y=0 at (1,1) makes an angle Tan ⁻¹ 2 with X -axis, then ab a + b = 2025 Full Application of Derivatives list All MHT CET PYQs