The length of one side, s, of a shipping box is s(x) = ^3√2x, where x is the volume of the box in cubic inches. A manufacturer needs the volume of the box to be between 108 in.3 and 256 in.3. What are the minimum and maximum possible lengths of s?

The required minimum value of the length is 6 inches and the maximum value is 8 inches.

Given that,

The length of one side, s, of a shipping box is ,

Where x is the volume of the box in cubic inches.

A manufacturer needs the volume of the box to be between 108 inches and 256 inches

We have to determine,

The minimum and maximum possible lengths of s.

According to the question,

To obtain the minimum length of s, using the minimum volume of the shipping box in the equation.

Where, x = 108

Substitute the value of x in the given equation,

Then,

And, Where x = 256

Substitute x = 256 in the given equation,

Hence, The required minimum value of the length is 6 inches and the maximum value is 8 inches.

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Minimum value: 6 inches,

Maximum value: 8 inches.

Step-by-step explanation:

To find the minimum length of s, we need to use the minimum volume of the shipping box in the equation, so:

s_minimum = ^3√(2*108) = ^3√216 = 6 inches

The maximum value of the volume will give us the maximum value of the length:

s_maximum = ^3√(2*256) = ^3√512 = 8 inches

So the minimum value of the length is 6 inches and the maximum value is 8 inches.

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