An aquarium is being built in the shape of a rectangular box with a square bottom and no top.

Let represent the length of one side of the bottom of the aquarium and let represent the height of the aquarium.

Suppose we wish to minimize the amount of material needed to create the aquarium. If the finished aquarium must hold 32 ft of water, what is the equation we should optimize?

Question

An aquarium is being built in the shape of a rectangular box with a square bottom and no top.

Let  represent the length of one side of the bottom of the aquarium and let   represent the height of the aquarium.

Suppose we wish to minimize the amount of material needed to create the aquarium. If the finished aquarium must hold 32 ft  of water, what is the equation we should optimize?

BlackTom AI · Accepted Answer

First, introduce variables consistent with the problem: let the square base have side length x feet and the height be y feet.
Next, write the volume constraint for the aquarium. Since the base is square, the volume V is V = base area × height = x^2 × y, and we’re given V = 32 ft^3. This yields a relation between x and y: y = 32 / x^2.
Now express the surface area to be minimized. The aquarium has a square bottom and four side faces (no top). So the surface area S consists of the bottom area plus the lateral area: S = (base area) + (lateral area) = x^2 + (4 × x × y).
Substitute the height expression from the volume constraint into the surface area: S(x) = x^2 + 4x(32 / x^2) = x^2 + 128/x.
To find the minimization, analyze S as a function of x > 0. Compute the derivative: S'(x) = 2x − 128/x^2. Set the derivative equal to zero to locate critical points: 2x − 128/x^2 = 0.
Multiply both sides by x^2 to clear the denominator: 2x^3 − 128 = 0, which simplifies to x^3 = 64. From here you would determine x by taking the cube root (ensuring x > 0).
Once x is found, use the volume constraint to determine y: y = 32 / x^2. These steps yield the dimensions that minimize the material used, and the corresponding minimal surface area can then be computed by substituting x and y back into S = x^2 + 4xy.
In summary, the equation to optimize is S(x) = x^2 + 128/x with x > 0, derived by substituting y = 32/x^2 into the surface-area expression S = x^2 + 4xy, and the minimization proceeds by solving S'(x) = 0.

MATH_1225_17255_202501 4.7 Optimization

查看解析

登录即可查看完整答案

类似问题

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

A product has a selling price of $20, a contribution margin ratio of 40% and fixed cost of $120,000. To make a profit of $30,000. The number of units that must be sold is: Type the number without $ and a comma. Eg: 20000

Which of the following statement regarding cost is correct:

Under the assumptions used in cost-volume-profit analysis, as the activity increases:

The CVP and break-even analysis are supported by a range of important assumptions. Which of the following is not an assumption of the CVP:

The break-even point is that at which:

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单