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miv72 [106K]
3 years ago
6

A fireplace arch is to be constructed in the form of a semiellipse. The opening is to have a height of 3 feet at the center and

a width of 8 feet along the base. The contractor cuts a string of a certain length and nails each end of the string along the base in order to sketch the outline of the semiellipse.a.) What is the total length of the string?b.) How far from the center should the string be nailed into the base?
Mathematics
1 answer:
USPshnik [31]3 years ago
8 0

Answer:

Step-by-step explanation:

Given

Fireplace is in the form of semi-ellipse  with height of 3 ft from center

Length of base=8 ft

if the arch is in the form of ellipse then it can be considered as horizontal ellipse with 2 a and 2 b be length of major and minor axis

2 a =8

a=4 ft

If contractor cover the whole base

then total length of string=2a =8 ft

(b)String should be nailed at focus

focus of ellipse is a e

where e=eccentricity

e=\sqrt{1-(\frac{b}{a})^2}

e=\sqrt{1-(\frac{3}{4})^2}=\frac{\sqrt{7}}{4}

Focus =ae =\sqrt{7}

Thus string should be nailed at 2.64 m from center

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Answer:

n=(\frac{1.64(4)}{0.25})^2 =688.537 \approx 689

So the answer for this case would be n=689 rounded up to the nearest integer

Step-by-step explanation:

Assuming this complete question: "Suppose that the minimum and maximum ages for typical textbooks currently used in college courses are 0 and 8 years. Use the range rule of thumb to estimate the standard deviation.

Estimate the minimum and maximum ages for typical textbooks currently used in college courses, then use the range rule of thumb to estimate the standard deviation. Next, find the size of the sample required to estimate the mean age (in years) of textbooks currently used in college courses. Use a 90% confidence level and assume that the sample mean will be in error by no more than 0.25 year."

Solution for the problem

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Important 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 (ME) 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".

The margin of error is given by this formula:

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And on this case we have that ME =\pm 0.25 and we are interested in order to find the value of n, if we solve n from equation (1) we got:

n=(\frac{z_{\alpha/2} \sigma}{ME})^2   (2)

We can assume that the estimator for the population deviation from the rule of thumb is \hat \sigma = s= 4

The critical value for 90% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.05;0;1)", and we got z_{\alpha/2}=1.64, replacing into formula (2) we got:

n=(\frac{1.64(4)}{0.25})^2 =688.537 \approx 689

So the answer for this case would be n=689 rounded up to the nearest integer

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