Question Type 3: Working with shapes to find the largest (or smallest) area (or volume) of specific 2D (or 3D) shapes Exercises

A right circular cylinder must have volume $1000 \text{ cm}^3$. Find the radius $r$ and height $h$ that minimize its total surface area.

An open-top box with square base has volume $32 \text{ m}^3$. Find the base side $x$ and height $h$ minimizing the surface area.

A fence encloses a shape formed by a rectangle topped with a semicircle on one of its longer sides. If the total fencing length is $100$ m, find the rectangle’s dimensions (width $w$ and height $h$) that maximize the enclosed area.

A farmer has $100 \text{ m}$ of fencing and wants to build a rectangular pen against a straight river (no fence along the river). Find the dimensions that maximize the area.

An isosceles triangle has perimeter $30$ cm. Find its side lengths that maximize the area.

Find the dimensions of the rectangle with perimeter $20$ that give the maximum area.

Given a fixed area of $100$ m$^2$, find the rectangle dimensions that minimize the perimeter.

A rectangle is inscribed in a circle of radius $5$. Determine the rectangle’s dimensions that maximize its area.

A right circular cylinder is inscribed in a sphere of radius $6$ cm. Find the cylinder dimensions that maximize its volume.

A right circular cone has fixed slant height $l=10\,\text{cm}$. Determine the radius $r$ and height $h$ that maximize its volume.

A right circular cylinder has total surface area $600 \, \text{cm}^2$. Determine its radius and height that maximize the volume.