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Trigonometry and Calculus: Volume

The concept of volume is important at all levels of mathematics. Some of the volume problems below, such as The Super-Ultra-Mega-Big Cup, are more easily solved by using calculus. Others, like That's A Lot of Rock, can be solved by using area and volume formulas.

For background information elsewhere on our site, explore the High School Trigonometry and Calculus areas of the Ask Dr. Math archives. To find relevant sites on the Web, browse and search Trigonometry and Calculus (Single Variable) in our Internet Mathematics Library.


Access to these problems requires a Membership.

Building Boxes - Leigh Nataro
Trig/Calc, difficulty level 2. Find the maximum volume of a rectangular prism constructed from an 8.5-inch by 11-inch piece of paper. ... more>>

Cup Capacity - Leigh Nataro
Trig/Calc, difficulty level 3. Find the volume of a cup based on its dimensions. ... more>>

Doughnut Dilemma - Leigh Nataro
Trig/Calc, difficulty level 2. Which doughnut shop makes a doughnut with a greater volume? ... more>>

Five Golden Rings - Leigh Nataro
Trig/Calc, difficulty level 5. Based on the given information, find the volume of five golden rings. ... more>>

The General Box Company - Leigh Nataro
Trig/Calc, difficulty level 3. A rectangular piece of cardboard is folded up to make a box. Find a general formula for the maximum volume of the box. ... more>>

Googol the Giant's Gargantuan Goblet - Leigh Nataro
Trig/Calc, difficulty level 2. Find the volume of Googol the Giant's elliptical paraboloid goblet. ... more>>

Job Interview at the Can-Can Company - Leigh Nataro
Trig/Calc, difficulty level 3. Find the smallest amount of tin needed to make a can that can hold 12 cubic inches. ... more>>

Shipping Packages - Leigh Nataro
Trig/Calc, difficulty level 2. Find the maximum volume of a box based on the given information. ... more>>

The Super-Ultra-Mega-Big Cup - Alvirne High School Calculus Class
Trig/Calc, difficulty level 3. What is the minimum cost to make a cup based on given information? ... more>>

That's A Lot of Rock! - Leigh Nataro
Trig/Calc, difficulty level 3. Find the volume of a tunnel through a mountain. ... more>>

Time to Make the Doughnuts - David Turner
Trig/Calc, difficulty level 4. Find the radius of a doughnut with center at (0, 10) and a volume of 3158 cubic centimeters. ... more>>

Your Interview Question Is... - Leigh Nataro
Trig/Calc, difficulty level 3. If you want to minimize the surface area of a can with a fixed volume, what should be the ratio of the height to the radius? ... more>>

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