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Murrr4er [49]
3 years ago
15

A rectangular bird sanctuary is being created with one side along a straight riverbank. The remaining three sides are to be encl

osed with a protective fence. If there are 12 km of fence available, find the dimension of the rectangle to maximize the area of the sanctuary.
Mathematics
1 answer:
shusha [124]3 years ago
6 0

Answer:

For a rectangle of length L and width W, the perimeter is:

P = 2*L + 2*W

And the area is:

A = L*W.

Now, we know that the sanctuary has one side along a straight riverbank, then we will not fence that side. if L > W, then makes sense to put one of the "length" sides in the riverbank, in that way, we are saving more fence for the other 3 sides.

Then we will have that, the total length of fence used is:

12km = 2*W + L (two times the width and only one time the length are fenced).

From this, we can isolate one of the variables:

L = 12km - 2*W

And replace that in the area equation:

A(W) = (12km - 2*W)*W = 12km*W - 2*W^2.

To maximize this function, we must see when the first derivate is equal to zero:

A'(W) = 12km - 4*W = 0

            12km = 4*W

             12km/4 = 3km = W

Then the width that maximizes the area is W = 3km.

And the maximum area will be:

A'(3km) = 12km*3km - 2*(3km)^2 = 18km^2

And the length can be computed with the equation:

L = 12km - 2*W = 12km - 2*3km = 6km

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Answer:

We conclude that this is an unusually high number of faulty modems.

Step-by-step explanation:

We are given that while conducting a test of modems being manufactured, it is found that 10 modems were faulty out of a random sample of 367 modems.

The probability of obtaining this many bad modems (or more), under the assumptions of typical manufacturing flaws would be 0.013.

Let p = <em><u>population proportion</u></em>.

So, Null Hypothesis, H_0 : p = 0.013      {means that this is an unusually 0.013 proportion of faulty modems}

Alternate Hypothesis, H_A : p > 0.013      {means that this is an unusually high number of faulty modems}

The test statistics that would be used here <u>One-sample z-test</u> for proportions;

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where, \hat p = sample proportion faulty modems= \frac{10}{367} = 0.027

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So, <u><em>the test statistics</em></u>  =  \frac{0.027-0.013}{\sqrt{\frac{0.013(1-0.013)}{367} } }

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The value of z-test statistics is 2.367.

Since, we are not given with the level of significance so we assume it to be 5%. <u>Now at 5% level of significance, the z table gives a critical value of 1.645 for the right-tailed test.</u>

Since our test statistics is more than the critical value of z as 2.367 > 1.645, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which <u><em>we reject our null hypothesis</em></u>.

Therefore, we conclude that this is an unusually high number of faulty modems.

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ok done. Thank to me :>

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