Theory question 110
DifferentiationA solid rectangular block has a base which measures 3x cm by 2x cm. The height of the block is y cm and its volume is 72 cm3. (a) Express y in terms of x. (b) Find: (i) an expression for the total surface area of the block in terms of x only; (ii) the value of x for which the total surface area has a stationary value.
Show the model answer
Model answer
(a) 6x²y = 72, so y = 12/x².
(b)(i) A = 2(6x²) + 2(3xy) + 2(2xy) = 12x² + 10xy = 12x² + 120/x.
(ii) dA/dx = 24x − 120/x² = 0 gives x³ = 5, x = ∛5 ≈ 1.71 cm.
Explanation
Eliminate y using the volume before differentiating.