MATHEMATICS COLLEGE

The integral \pi\int\limits^2_1 {(1-(lnx)^(2)) } \, dx represents the volume of a solid obtained by rotating a region around y=-1. Evaluate.

Answers

Answer 1
Answer:

Answer:

V = π (-2 (ln 2)² + 4 ln 2 − 1)

V ≈ 2.55

Step-by-step explanation:

V = π ∫₁² (1 − (ln x)²) dx

V/π = ∫₁² (1 − (ln x)²) dx

V/π = ∫₁² dx − ∫₁² (ln x)² dx

V/π = x |₁² − ∫₁² (ln x)² dx

V/π = 1 − ∫₁² (ln x)² dx

To evaluate the second integral, integrate by parts.

If u = (ln x)², then du = 2 (ln x) / x dx.

If dv = dx, then v = x.

∫ u dv = uv − ∫ v du

= (ln x)² x − ∫ x (2 (ln x) / x) dx

= x (ln x)² − 2 ∫ ln x dx

Integrate by parts again.

If u = ln x, then du = 1/x dx.

If dv = dx, then v = x.

∫ u dv = uv − ∫ v du

= x ln x − ∫ x (1/x dx)

= x ln x − ∫ dx

= x ln x − x

Substitute:

∫ (ln x)² dx = x (ln x)² − 2 ∫ ln x dx

∫ (ln x)² dx = x (ln x)² − 2 (x ln x − x)

∫ (ln x)² dx = x (ln x)² − 2x ln x + 2x

Substitute again:

V/π = 1 − ∫₁² (ln x)² dx

V/π = 1 − (x (ln x)² − 2x ln x + 2x) |₁²

V/π = 1 + (-x (ln x)² + 2x ln x − 2x) |₁²

V/π = 1 + (-2 (ln 2)² + 4 ln 2 − 4) − (-1 (ln 1)² + 2 ln 1 − 2)

V/π = 1 − 2 (ln 2)² + 4 ln 2 − 4 + 2

V/π = -2 (ln 2)² + 4 ln 2 − 1

V = π (-2 (ln 2)² + 4 ln 2 − 1)

V ≈ 2.55

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The average speed of greyhound dogs is about 18.4 meters per second. A particular greyhound breeder claims that her dogs are faster than the average greyhound. In a sample of 35 of her dogs, they ran, on the average, 18.7 m/s. Assuming a population standard deviation of 1.5 m/s and a .05 significance level. Required:
Does this sample data provide evidence that the breeder's claim is correct?

Answers

Answer:

The calculated value    Z = 1.183 < 1.96 at 0.05 level of significance

Null hypothesis is accepted

A particular greyhound breeder claims that her dogs are faster than the average greyhound

Step-by-step explanation:

Step(i):-

Given the average speed of greyhound dogs is about 18.4 meters per second.

Size of the sample 'n' = 35

mean of the sample x⁻ = 18.7

Population standard deviation = 1.5m/s

level of significance (∝)  = 0.05

Step(ii):-

Null hypothesis : H₀ : μ = 18.4

Alternative  hypothesis H₁ :  μ ≠ 18.4

Test statistic

              Z = (x^(-) - mean)/((S.D)/(√(n) ) )

             Z = (18.7-18.4)/((1.5)/(√(35) ) )

            Z = 1.183

Conclusion:-

The calculated value    Z = 1.183 < 1.96 at 0.05 level of significance

Null hypothesis is accepted

A particular greyhound breeder claims that her dogs are faster than the average greyhound

Final answer:

Evaluate the breeder's claim by using a one-sample z-test. The null hypothesis states that the breeder's dogs aren't faster than average, while the alternative hypothesis states that they're faster. If the calculated Z is greater than Zcritical, it supports the breeder's claim.

Explanation:

In this scenario, you would carry out a one-sample z-test to evaluate the breeder's claim. Given an average speed of greyhounds as 18.4 m/s, the breeder's dogs with an average of 18.7 m/s could be faster or this could just be due to statistical fluctuation. Hence, we need to statistically test it to see if this evidence is strong enough (with a significance level of .05) to support the breeder's claim.

Our null hypothesis (H0) is that the breeder's dogs are not faster than the average greyhound (µ = 18.4 m/s), and our alternative hypothesis (Ha) is that they are faster (µ > 18.4 m/s). Using the Z-test formula:

Z = (Xbar - µ) / (σ/√n)

 

where Xbar is sample mean, µ is population mean, σ is standard deviation, and 'n' is sample size. Applying the provided figures, we get:

Z= (18.7-18.4)/(1.5/√35)

Give the calculated Z value and compare it to the Z critical value for 0.05 significance level (1.645 for one-tail). If the calculated Z is greater than Zcritical, we reject the null hypothesis, providing evidence that the population mean is greater than 18.4 m/s, supporting the breeder's claim.

Learn more about One-sample z-test here:

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Find all missing angles in the diagrams below, please help​

Answers

Answer:

x= 80°, y= 100°

Step-by-step explanation:

Please see the attached picture for the full solution.

Answer:

x = 80,

y = 80

Step-by-step explanation:

This is difficult to explain, so here are the theorems used

two lines parallel, with a transversal; alternate interior angles,

verticle angles theorem,

parts - whole postulate,

verticle angles theorem,

parts- whole postulate,

sum angles in a triangle,

supplementary angle, definition straight angle,

parts- whole postulate,

there are many other ways to reach the answer, I just whent this way.

Cost of a dozen pens is ₹ 180 and cost of 8 ball pens is ₹ 56. Find the ratio of the cost of a pen to the cost of a ball pen. [1 ½ m]

​

Answers

Answer:

Ratio of cost of a pen to cost of a ball pen is 15:7

Step-by-step explanation:

Given that:

Cost of dozen pens = ₹180

Unit rate = (180)/(12)

Unit rate = ₹ 15 per pen

Cost of 8 ball pens = ₹ 56

Unit rate = (56)/(8)

Unit rate = ₹7 per ball pen

Ratio of cost of a pen to cost of a ball pen,

15 : 7

Hence,

Ratio of cost of a pen to cost of a ball pen is 15:7

What is the total interest paid on an $185,000 loan at 6.25% APR over 15 years (monthly payments)? ​

Answers

Answer:

$100,522

Step-by-step explanation:

Answer:

Total Interest $100,522

Step-by-step explanation:

A local doctor’s office logged the number of patients seen in one day by the doctor for ten days. Find the means, median, range, and midrange of the patients seem in 10 days. 27 31 27 35 35 25 28 35 33 24

Answers

Answer:

Mean = 30, Median = 29.5, Range = 9 and Mid-range = 29.5.

Step-by-step explanation:

We are given that a local doctor’s office logged the number of patients seen in one day by the doctor for ten days.

Arranging the given data in ascending order we get;

24, 25, 27, 27, 28, 31, 33, 35, 35, 35.

(a) Mean is calculated by using the following formula;

         Mean, \bar X  =  \frac{\text{Sum of all values}}{\text{Total number of observations}}

                          =  (27+ 31+ 27+ 35+ 35+ 25+ 28+ 35+ 33+ 24)/(10)

                          =  (300)/(10)  = 30

So, the mean of the given data is 30.

(b) For calculating the median, we have to first have to observe that the number of observations (n) in the data is even or odd.

  • If n is odd, then the formula for calculating median is given by;

                     Median  =  ((n+1)/(2))^(th) \text{ obs.}

  • If n is even, then the formula for calculating median is given by;

                     Median  =  \frac{((n)/(2))^(th) \text{ obs.}+ ((n)/(2)+1)^(th) \text{ obs.} }{2}

Here, the number of observations is even, i.e. n = 10.

So,  Median  =  \frac{((n)/(2))^(th) \text{ obs.}+ ((n)/(2)+1)^(th) \text{ obs.} }{2}

                     =  \frac{((10)/(2))^(th) \text{ obs.}+ ((10)/(2)+1)^(th) \text{ obs.} }{2}

                     =  \frac{(5)^(th) \text{ obs.}+ (6)^(th) \text{ obs.} }{2}

                     =  (28+31)/(2)

                     =  (59)/(2)  =  29.5

So, the median of the data is 29.5.

(c) The range of the data is given by = Highest value - Lowest value

                                                        = 35 - 24 = 9

So, the range of the data is 9.

(d) Mid-range of the data is given by the following formula;

                   Mid-range  =  \frac{\text{Highest value}+\text{Lowest value}}{2}

                                      =  (35+24)/(2) = 29.5

Final answer:

The mean of the patients seen in 10 days is 30, the median is 29.5, the range is 11, and the midrange is 29.5.

Explanation:

To find the mean, median, range, and midrange of the numbers, we first need to understand what these terms mean. The mean is the average, the median is the middle number in a sorted set, the range is the difference between the highest and lowest values, and the midrange is the average of the highest and lowest values.

First, let's sort the numbers: 24 25 27 27 28 31 33 35 35 35.

To calculate the mean, add all the numbers and divide by the count (10): (24+25+27+27+28+31+33+35+35+35)/10 = 30.

The median is the average of the two middle numbers (which are 28 and 31 in this case): (28 + 31) / 2 = 29.5.

The range is the highest number minus the lowest number: 35 - 24 = 11.

The midrange is the average of the highest and lowest values: (35 + 24) / 2 = 29.5.

Learn more about Mean, Median, Range

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The slope of a line is -, and the y-intercept is 10/3. What is the equation of the line written in general form?

Answers

I think it should be y=-x+10/3
As slope is y=mx+b so substitute your slope for the coefficient of x and then add your y intercept to the end.
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