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victus00 [196]
3 years ago
11

A certain right triangle has area 30 in. squared . one leg of the triangle measures 1 in. less than the hypotenuse. let x repres

ent the length of the hypotenuse. ​a) express the length of the leg mentioned above in terms of x. give the domain of x. ​b) express the length of the other leg in terms of x. ​c) write an equation based on the information determined thus far. square both sides and then write the equation with one side as a polynomial with integer​ coefficients, in descending​ powers, and the other side equal to 0. divide out any common factors to find the most simplified form. ​d) solve the equation in part​ (c) graphically. find the lengths of the three sides of the triangle.

Mathematics
1 answer:
Brut [27]3 years ago
6 0
A) The length of the longer leg is x-1

b) Based on the area, the other leg is 2*30/(x -1). Based on the Pythagorean theorem, the other leg is √(x^2 -(x -1)^2).

c) Equating the two expressions for the shorter leg, we have
.. 60/(x -1) = √(2x -1)
.. 3600/(x -1)^2 = (2x -1)
.. (2x -1)(x^2 -2x +1) = 3600
.. 2x^3 -5x^2 +4x -3601 = 0

d) There is one positive real root, at x=13. A graphical solution works well.

The three sides of the triangle are 5 in, 12 in, 13 in.


_____
5-12-13 is a well-known Pythagorean triple. It is the next smallest one after 3-4-5.

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Suppose we are interested in analyzing the weights of NFL players. We know that on average, NFL players weigh 247 pounds with a
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Answer:

SE= \frac{\sigma}{\sqrt{n}} = \frac{47}{\sqrt{30}}= 8.58

ME= 1.64 *\frac{\sigma}{\sqrt{n}} = 1.64*\frac{47}{\sqrt{30}}= 14.073

\bar X - ME = 237- 14.073 = 222.927

\bar X + ME = 237+ 14.073 = 251.073

n=(\frac{1.640(47)}{5})^2 =237.65 \approx 238

So the answer for this case would be n=238 rounded up to the nearest integer

Null hypothesis:\mu \geq 247  

Alternative hypothesis:\mu  

z=\frac{237-247}{\frac{47}{\sqrt{30}}}=-1.165  

p_v =P(z  

If we compare the p value and the significance level given \alpha=0.1 we see that p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can't conclude that the true mean is lower than 247 at 10% of significance.

Step-by-step explanation:

For this case we have the following data given:

\bar X =237 represent the sample mean

\sigma = 47 represent the population deviation

n =30 represent the sample size selected

\mu_0 = 247 represent the value that we want to test.

The standard error for this case is given by:

SE= \frac{\sigma}{\sqrt{n}} = \frac{47}{\sqrt{30}}= 8.58

For the 90% confidence the value of the significance is given by \alpha=1-0.9 = 0.1 and \alpha/2 = 0.05 so we can find in the normal standard distribution a quantile that accumulates 0.05 of the area on each tail and we got:

z_{\alpha/2}= 1.64

And the margin of error would be:

ME= 1.64 *\frac{\sigma}{\sqrt{n}} = 1.64*\frac{47}{\sqrt{30}}= 14.073

The confidence interval for this case would be given by:

\bar X - ME = 237- 14.073 = 222.927

\bar X + ME = 237+ 14.073 = 251.073

The margin of error is given by this formula:

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

And on this case we have that ME =5 and we are interested in order to find the value of n, if we solve n from equation (a) we got:

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

Replacing into formula (b) we got:

n=(\frac{1.640(47)}{5})^2 =237.65 \approx 238

So the answer for this case would be n=238 rounded up to the nearest integer

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the true mean is lower than 247 pounds, the system of hypothesis would be:  

Null hypothesis:\mu \geq 247  

Alternative hypothesis:\mu  

Since we know the population deviation, is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:  

z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}} (1)  

z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic  

We can replace in formula (1) the info given like this:  

z=\frac{237-247}{\frac{47}{\sqrt{30}}}=-1.165  

P-value  

Since is a left tailed test the p value would be:  

p_v =P(z  

Conclusion  

If we compare the p value and the significance level given \alpha=0.1 we see that p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can't conclude that the true mean is lower than 247 at 10% of significance.

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