Applications of Derivatives
Applied Maxima and Minima — Volume/Surface Area
nta_pyq_2023_apr
Grade 12
Question:
A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm$^2$) is equal to
Step-by-Step Solution
Key Concept: Let $x$ be the side of corner squares cut. Volume $V(x)=(30-2x)^2 x$. Differentiate and set $V'(x)=0$.
$V(x)=(30-2x)^2x$, $V'(x)=(30-2x)(30-6x)=0\Rightarrow x=5$ cm. Surface area $=(30-10)^2+4(20)(5)=400+400=800$ cm$^2$.
Correct Answer: 1