A solid rectangular block has a base that measures 3x cm by 2x cm. The height of the block is ycm and its volume is 72cm\(^3\).
i. Express y in terms of x.
ii. An expression for the total surface area of the block in terms of x only;
iii. the value of x for which the total surface area has a stationary value.
Explanation
volume of rectangular block(cuboid) = l × b × h
V = 3x × 2x × y
72 = 6x\(^2\)y
y = \(\frac{72}{6x^2}\)
y = \(\frac{12}{x^2}\)
ii.
Area of cuboid = 2(lb + lh + bh)
l = 3x, b = 2x, h = y
A = 2(3x(2x) + 3x(y) + 2x(y))
A = 2(6x\(^2\) + 3xy + 2xy)
A = 12x\(^2\) + 6xy + 4xy
A = 12x\(^2\) + 10xy
where y = \(\frac{12}{x^2}\)
A = 12x\(^2\) + \(\frac{120}{x}\)
iii.
At stationary point \(\frac{dy}{dx}\) = 0
\(\frac{dA}{dx} = 2(12x^2-1) + -120x^{-1-1}\)
24x - 120x\(^{-2}\)
\(\frac{dA}{dx}\) = 0
24x- \(\frac{120}{x^2}\) = 0
multiply through by x\(^2\)
24x\(^3\) - 120 = 0
24x\(^3\) = 120
x\(^3\) = 5
x = ∛5
x = 1.71