Use the method of Lagrange Multipliers to solve it.

A rectangular tank with a bottom and sides but no top is to have volume 500 cubic feet. Determine the dimensions (length, width, height) with the smallest possible surface area.

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Nelly StrackeLv2

28 Oct 2019

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Your firm is designing and manufactoring a large tank for water storage. The proposed tank is to be a right circular cylinder with a hemisphere as its top. The total volume of the tank with both parts included is to be 10,000 cubic meters. Your firm also wants to use the smallest amount of material possible in building the tank, which means it must minimize the total surface area of the tank, including the surface at the bottom of the tank. Use LaGrange Multipliers as part of your solution.

a) Find the dimensions of the tank which has the smallest possible surface area. Show work

b) Also find the surface area of the resulting tank. Show work.

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A cargo container (in the shape of a rectangular solid) must have a volume of 480 cubic feet. The bottom will cost $5 per square foot to construct, and the sides and the top will cost $3 per square foot to construct. Use Lagrange multipliers to find the dimensions of the container of this size that has minimum cost.

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Use the method of Lagrange Multipliers to solve it.

A rectangular tank with a bottom and sides but no top is to have volume 500 cubic feet. Determine the dimensions (length, width, height) with the smallest possible surface area.

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Calculus

Single Variable Calculus: Early Transcendentals

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Use the method of Lagrange Multipliers to solve it. A rectangular tank with a bottom and sides but no top is to have volume 500 cubic feet. Determine the dimensions (length, width, height) with the smallest possible surface area.

Unlock all answers

Related textbook solutions

Calculus

Single Variable Calculus: Early Transcendentals

CALCULUS:EARLY TRANSCENDENTALS

Related questions

Weekly leaderboard

Use the method of Lagrange Multipliers to solve it.

A rectangular tank with a bottom and sides but no top is to have volume 500 cubic feet. Determine the dimensions (length, width, height) with the smallest possible surface area.