Question Type 4: Working with more complex, 3D shapes that are harder to optimize Bootcamps

An open-top box has width $w$, length $l$, height $h$ with $w:h=1:2$ and surface area (base + sides) $100\text{ m}^2$. Find $l,w,h$ that maximize the volume and give that volume.

A closed rectangular box has width $w=2l$ and total surface area $150\text{ cm}^2$. Find the dimensions that maximize its volume and give the maximum volume.

A closed rectangular box has width $w=3l$ and total surface area $120\text{ m}^2$. Find the dimensions $l,w,h$ that maximize its volume and determine the maximum volume.

An open-top box has width $w=3l$, length $l$ and height $h$, and its total surface area (including base but excluding top) is $80\text{ cm}^2$. Find the dimensions that maximize its volume and compute the maximum volume.

A closed rectangular box has width $w$, length $l$ and height $h$ with ratio $w:h=2:3$ and total surface area $180\text{ m}^2$. Find $l,w,h$ that maximize volume and the maximum volume.

A closed rectangular box has width $w=2l$ and height $h=l$. Its space diagonal is $30\text{ cm}$. Find the value of $l$ that maximizes the volume, and determine the maximum volume.

A closed right circular cylinder has height $h=2r$, where $r$ is its radius. If its total surface area is $150\pi\text{ cm}^2$, determine $r$ and $h$ that maximize the volume and compute that volume.

A closed right circular cylinder has height $h$ and radius $r$ satisfying $h + 2r = 20\text{ cm}$. Find $r$ and $h$ that maximize the volume, and state the maximum volume.