In one line
Write the quantity to optimise as a function, use the given condition to reduce it to two variables, then find and classify the stationary point.
Explained simply
Engineers constantly ask: what shape uses the least material, or gives the most volume, or costs the least? A tank, a box, a pipe, a beam: each is an optimisation problem. Calculus answers them by turning "least" or "most" into "where the slopes are zero".
Technically, if you must optimise \(F(x,y,z)\) and a condition \(g(x,y,z)=c\) links the variables, solve the condition for one variable, substitute to get \(f(x,y)\), and apply the two-variable test. (Lagrange's multiplier method is an alternative that avoids substitution.)
How it works
- Name the variables (dimensions) and write the quantity to optimise.
- Write the constraint and use it to eliminate one variable.
- Find \(f_x=0\), \(f_y=0\) and solve.
- Check \(rt-s^2>0\) and the sign of \(r\) to confirm max or min.
- Find the eliminated variable and the optimum value, with units.
Key formula
Open box with base \(x\times y\) and height \(z\): volume \(V=xyz\), surface area (no lid) \(S=xy+2xz+2yz\). Lengths in m, area in m², volume in m³.
Quick example
An open box must hold 32 m³. Find dimensions for the least surface area.
\(z=\dfrac{32}{xy}\), so \(S=xy+\dfrac{64}{y}+\dfrac{64}{x}\).
\(S_x=y-\dfrac{64}{x^2}=0\), \(S_y=x-\dfrac{64}{y^2}=0\). So \(x^2y=64\) and \(xy^2=64\); dividing gives \(x=y\), so \(x^3=64\), \(x=y=4\) m and \(z=\dfrac{32}{16}=2\) m.
\(r=\dfrac{128}{x^3}=2\), \(s=1\), \(t=\dfrac{128}{y^3}=2\): \(rt-s^2=3>0\), \(r>0\), so minimum.
Least area \(=16+2(8)+2(8)=48\) m². The base is square and the height is half the side.
Common mistakes
- Forgetting that an open box has no top, so the area has \(xy\) only once.
- Not substituting the constraint and trying to differentiate with three variables.
- Not confirming it is a minimum, or leaving out units.
How MU asks it
Applied 6-mark questions such as "Find the dimensions of a rectangular box, open at the top, of maximum capacity whose surface area is 108 sq. cm" (answer \(6\times6\times3\) cm) or "Divide 24 into three parts such that the continued product of the first, square of the second and cube of the third is maximum" (answer 4, 8, 12) are standard. Show the function, the substitution, the stationary point, the test, and the final answer with units.