I need this question solved with derivatives please: A rectangular piece of cardboard, 100 cm by 40 cm, is going to be used to
make a rectangular box with an open top by cutting congruent squares from
the corners. Calculate the dimensions (to one decimal place) for a box with
the largest volume.

16,242. 7 cm^3.

Step-by-step explanation:

We need to cut off a square piece at the 4 corners of the cardboard.

Let the length of their edges be x cm.

The volume of the box will be:

V = height * width * length

V = x(100-2x)(40-2x)

V = x(4000 - 200x - 80x + 4x^2)

V = x(4x^2 - 280x + 4000)

V =  4x^3 + - 280x^2 + 4000x

Finding the derivative:

dV / dx =  12x^2 - 560x + 4000  = 0  ( when  V is a maxm or minm.)

4(3x^2 - 140x + 1000) = 0

x =  37.86, 8.80.

Looks like x = 8.80 is the right value but we can check this out be looking at the sign of the second derivative:

V" =  24x - 560, when x = 8.8   V" is negative so this is a Maximum for V.

So the maximum volume of the box is when x = 8.8 so we have

V =  8.8(100-2(8.8)(40 - 2(8.8)

=  16,242. 7 cm^3.

introduction, methodology, results, discussion

step-by-step explanation:

an observational study is one which consists of actual observation to record data and then analyse the data.

there are a number of factors that should be analyzed before starting an observational study.

the main four factors are:

i) introduction:   this should be analysed to check whether the purpose of the study is correctly established and the data recorded are appropriate for the study

methodology:   the method of collecting the data, such as whether randomly selected, and whether sample size matches with the population to be studied, whether questions asked are easy to answer by common men, etc.

results: then results such as summary of data with comments should be studied thoroughly

discussion:   the last is the discussion, this can be either by management people or persons who are all interested , etc.

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step-by-step explanation: