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

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 closed cuboid has width $w = 3l$ and total surface area $600\text{ cm}^2$. Find the values of $l$, $w$, and height $h$ that maximize its volume.

A rectangular garden is designed such that its width is three times its length. Given that the perimeter of the garden is $80\text{ m}$, find the dimensions of the garden and calculate its area.

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

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

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

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

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

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.