a rectangular storage container without a lid is to have a volume of 10 m3. the length of its base is twice the width. material

Question

1 year ago

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a rectangular storage container without a lid is to have a volume of 10 m3. the length of its base is twice the width. material

for the base costs $20 per square meter. material for the sides costs $12 per square meter. find the cost (in dollars) of materials for the least expensive such container. (round your answer to the nearest cent.)

Mathematics

1 answer:

victus00 [196] · Answer 1 · 2024-06-26T19:56:09+03:00

245.31 (dollars) is the cost (in dollars) of materials for the least expensive such container.

<h3>What is maxima and minima ? </h3>

Calculus maxima and minima are found using the concept of derivatives. Knowing that the derivative concept gives information about the slope/slope of a function, we find the point where the slope is zero. These points are called inflection points/stationary points. These are the points associated with the maximum or minimum (local) values of the function.

Knowledge of maxima and minima is essential for our everyday problems. In addition, this article also explains how to find the absolute maximum and minimum values.

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<h3>Calculation</h3>

Suppose the width is x (m), length of the base is 2x (m), the base area is 2x^2 (m^2).

Since the volume is 10 (m^3), the height has to be 10/2x^2 (m) = 5/x^2.

The cost of making such container is

cost of base: 2x^2*15 = 30x^2

cost of sides: (2*2x*5/x^2 + 2*x*5/x^2)*9 = 270/x

The overall cost is hence the sum of the base and the sides: f(x) = 30x^2 + 270/x

The get the minimum,

df(x)/dx = 30*(2x - 9/x^2) = 0

so x = (9/2)^(1/3) = 1.651 (m)

f(x) = 245.31 (dollars)