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3 years ago

I need this question solved with derivatives please:

Mathematics

1 answer:

Shkiper50 [21]3 years ago

5 0

Answer:

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.

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2 years ago

In a survey, the planning value for the population proportion is p* = 0.35. How large a sample should be taken to provide a 95%

lions [1.4K]

Answer:

$n=\frac{0.35(1-0.35)}{(\frac{0.05}{1.96})^2}=349.59$

And rounded up we have that n=350

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$p$ represent the real population proportion of interest

$\aht p$ represent the estimated proportion for the sample

n is the sample size required (variable of interest)

$z$ represent the critical value for the margin of error

Solution to the problem

The population proportion have the following distribution

$p \sim N(p,\sqrt{\frac{\hat p(1-\hat p)}{n}})$

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by $\alpha=1-0.95=0.05$ and $\alpha/2 =0.025$ . And the critical value would be given by: