Question Type 1: Finding the optimal area or volume of specific 2D or 3D shapes Exercises

A rectangle has a perimeter of 60 cm. Find its dimensions that maximize the area and determine this maximum area.

The profit from selling $x$ items is given by $P(x)=50x-0.5x^2$. Find $x$ that maximizes profit and the maximum profit.

A company earns $18 per product sold and has a cost function $C(q)=5q^2-22q$. Find the production quantity $q$ that maximizes profit.

A rectangular garden has its width equal to three times its length. If the perimeter is 80 m, find the garden's dimensions that maximize its area.

A farmer wants to fence a rectangular field along a river. No fence is needed along the river, and he has 200 m of fencing for the other three sides. Find the dimensions that maximize the area enclosed.

A firm has cost function $C(q)=0.5q^2+10q+200$ and revenue $R(q)=25q$. Find the quantity $q$ that maximizes profit.

A closed cuboid has width $w=3l$ and the sum of all its edges is 120 cm. Find $l$, $w$, and $h$ that maximize its volume.

A closed cuboid has width $w=3l$ and total surface area 600 cm$^2$. Find $l$, $w$, and height $h$ that maximize its volume.

For a cylindrical can with top and bottom, the total surface area is $50\pi\,$cm$^2$. Determine the radius $r$ and height $h$ that maximize its volume.