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The reduction form is [tex]I_4 i= cot^3 x =ln |sin x| - 3 cot x + 3x + C[/tex].

To find the reduction formula for ∫cot^n x dx, we can use integration by parts. Let u = cot^(n-1) x and dv = cot x dx, then[tex]du = (n-1)cot^(n-2) x csc^2[/tex]x dx and v = ln |sin x|. By the formula for integration by parts, we have:

∫cot^n x dx = ∫u dv = uv - ∫v du

= [tex]cot^(n-1) x ln |sin x| - (n-1) ∫cot^(n-2) x csc^2 x ln |sin x| dx.[/tex]

This gives us the reduction formula:

[tex]I_n = ∫cot^n x dx = cot^(n-1) x ln |sin x| - (n-1) I_(n-2).[/tex]

Using this formula, we can find I_4 as follows:

[tex]I_4 = ∫cot^4 x dx = cot^3 x ln |sin x| - 3 I_2\\= cot^3 x ln |sin x| - 3 ∫cot^2 x dx\\= cot^3 x ln |sin x| - 3 (cot x - x) + C,\\[/tex]

where C is the constant of integration. Therefore, the solution for I_4 is [tex]cot^3 x ln |sin x| - 3 cot x + 3x + C.[/tex]

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Answer:

[tex]\overline{\sf QR}=28\sqrt{2}[/tex]

Step-by-step explanation:

If ΔPQR is a right triangle, where angles P and R both measure 45°, then the triangle is a special 45-45-90 triangle.

The measure of the sides of a 45-45-90 triangle are in the ratio 1 : 1 : √2.

This means that the length of each leg is equal, and the length of the hypotenuse is equal to the length of a leg multiplied by √2.

The legs of ΔPQR are segments PQ and QR.

The hypotenuse of ΔPQR is segment PR.

Therefore, to find the length of the leg QR, divide the length of the hypotenuse PR by √2.

[tex]\begin{aligned}\implies \overline{\sf QR}&=\dfrac{\overline{\sf PR}}{\sqrt{2}}\\\\&= \dfrac{56}{\sqrt{2}} \\\\&=\dfrac{56}{\sqrt{2}}\cdot \dfrac{\sqrt{2}}{\sqrt{2}}\\\\&=\dfrac{56\sqrt{2}}{2}\\\\&=28\sqrt{2}\end{aligned}[/tex]

Therefore, the length of segment QR is 28√2.

The function f(x) = 8x + 3x^(-1) has a local maximum at x = 2 with a value of f(2) = 19, and a local minimum at x = -2 with a value of f(-2) = -19.Explanation:

To find the local extrema of a function, we need to find the critical points of the function, which are the points where the derivative is either zero or undefined. In this case, the derivative of f(x) is f'(x) = 8 - 3x^(-2), which is undefined at x = 0.Setting the derivative equal to zero, we get:8 - 3x^(-2) = 0Solving for x, we get:x = ±2

These are the critical points of the function. To determine whether each critical point is a local maximum or a local minimum, we need to examine the second derivative of the function.

The second derivative of f(x) is f''(x) = 6x^(-3), which is negative for x > 0 and positive for x < 0.Therefore, x = 2 is a local maximum of the function with a value of f(2) = 19, and x = -2 is a local minimum of the function with a value of f(-2) = -19. These are the only local extrema of the function, since the function is increasing for x < -2 and decreasing for -2 < x < 0, and then increasing again for x > 0.

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She will need 4 quarts of water.

Given Hannah decided to make finger gelatin for a huge children’s party. She had to make 8 packs of gelatin. Each pack needed 2 cups of water.

Since each pack of gelatin requires 2 cups of water, Hannah will need a total of:

8 packs x 2 cups/pack = 16 cups of water

To convert cups to quarts, we need to divide the number of cups by 4 (since there are 4 cups in a quart):

16 cups ÷ 4 cups/quart = 4 quarts

Therefore, Hannah will need 4 quarts of water to make 8 packs of finger gelatin for the children’s party.

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The two-way table is of the class survey is:
                         Vanilla Ice Cream  |  Strawberry Ice Cream  |  Total
Chocolate Cake      |          12                   |             3                        |     15
Yellow Cake           |          11                   |             4                        |     15
Total                     |          23                   |             7                        |     30

A two-way table summarizing the data from Mrs. Robinson's class survey on cake and ice cream preferences can be constructed as follows.

1: Create a table with rows for Chocolate Cake and Yellow Cake, and columns for Vanilla Ice Cream, Strawberry Ice Cream, and Total.

2: Fill in the given information:

3: Complete the table using the given information:

So, the completed two-way table is:
                         Vanilla Ice Cream  |  Strawberry Ice Cream  |  Total
Chocolate Cake      |          12                   |             3                        |     15
Yellow Cake           |          11                   |             4                        |     15
Total                     |          23                   |             7                        |     30

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The exponential decay function to model this situation where t represents the number of years since 2007 and y is the amount of people is y = 19265 * (1 - 0.015)^t. The population for 2031 will be approximately 14,814 people.

To write an exponential decay function for this situation, you can use the formula:

y = P * (1 - r)^t

where y is the population at time t, P is the initial population, r is the annual decrease rate, and t represents the number of years since 2007.

In this case, P = 19265, r = 0.015 (1.5% expressed as a decimal), and t represents the number of years since 2007.

So, the exponential decay function is:

y = 19265 * (1 - 0.015)^t

To estimate the population for 2031, find the difference in years between 2031 and 2007 (2031 - 2007 = 24 years), and plug it into the formula as t:

y = 19265 * (1 - 0.015)^24

y ≈ 14814

So, the estimated population in 2031 will be approximately 14,814 people, rounded to the nearest person.

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Answer: 210 yards of fencing will be needed

Step-by-step explanation: well the perimeter of this rectangular field is 21 + 63 + 21 + 63 yards or 2(21) + 2(63) yards which equals 168 yards.
To divide the field into 3 equal parts, u need to divide the length (63 yards) into 3 parts which also requires two more lines of fencing.
63/3=21 which means u get squares perfect squares when u divide. now that means that's an additional 21*2 yards of fencing since you need two more rows of fencing in the middle of the field to divide the length into three equal parts. 21*2 = 42 so thats an  additional 42 yards. The total amount of fencing is 168 + 42 = 210 yards.

Answer:

option c

Step-by-step explanation:

n²-1/2

pls mrk me brainliest (⁠≧⁠(⁠ｴ⁠)⁠≦⁠ ⁠)

The shape of the distribution in the histogram can tell us about the distribution of weights of fish caught by the fisherman.

Looking at the given data set, we can see that the weights of the fish caught vary from as low as 2 pounds to as high as 15 pounds. The histogram of this data set can help us to  fantasize the distribution of these weights.   Grounded on the shape of the histogram, we can see that the distribution is  kindly slanted to the right, with a long tail extending towards the advanced end of the weights.

This suggests that there were  further fish caught that counted  lower than the mean weight of the catch, with smaller fish caught that counted  further than the mean weight. also, the presence of a many outliers(  similar as the fish that counted 15 pounds) suggests that there may have been some larger or unusual fish caught on the trip.

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The percentage of the data values that are greater than 16, as shown in the box plot is: 75%.

A box plot shows how the data points of a data set are distributed, in such a way that, 25% of the data points lie below the lower quartile, % lie below the median, and 75% lie below the upper quartile.

In the box plot given, the values that are greater than 16 lie above the upper quartile, which equals about 75% of the data values.

Therefore, the percentage of the data values that are greater than 65, as shown in the box plot is: 75%.

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The true statements about the rectangular prism are Prism B has a greater volume than Prism C, Prism B has the greatest volume, and Prism A has the least volume.

To understand, the differences between the prisms and therefore to verify the statements about them, let's calculate the volume of each by using the formula length x width x height.

Prism A: 2 x 2 x  3= 12 cubic units

Prism B: 2 x 3 x 4 = 24 cubic units

Prism C: 3 x 2 x 3 = 18cubic units

Based on this, the statements A, D, and E are correct.

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The critical value for this hypothesis test is 1.676.

To test the claim that manufacturing workers in Racine and Kenosha had the same salary as workers across the state, we can conduct a hypothesis test with the following null and alternative hypotheses:

We can use a t-test for the sample mean to test this hypothesis. At a 90% confidence level, we have a significance level of alpha = 0.10. Since this is a two-tailed test (we are testing for a difference in either direction), we will split the significance level evenly between the two tails, so alpha/2 = 0.05.

We need to calculate the critical value of the t-distribution with n-1 degrees of freedom, where n is the sample size. In this case, n = 54, so the degrees of freedom is 53. We can use a t-distribution table or a calculator to find the critical value. For a two-tailed test with alpha/2 = 0.05 and 53 degrees of freedom, the critical value is approximately 1.676.

Therefore, the critical value for this hypothesis test is 1.676.

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The amount which is greater than the given amount four hundred forty-five and fifty-seven hundredths is given by option b. 445.7.

Amount representing the  number is 445.57.

Amount greater than this number,

Compare the decimal parts of the numbers given in the options.

445.5 has a decimal part of 0.5, which is not greater than 0.57.

Option a is not greater than 445.57.

445.7 has a decimal part of 0.7, which is greater than 0.57.

Option b is greater than 445.57.

445.005 has a decimal part of 0.005, which is less than 0.57.

Option c is not greater than 445.57.

445.057 has a decimal part of 0.057, which is not greater than 0.57.

Option d is not greater than 445.57.

Therefore, the only option that is greater than 445.57 is option b. 445.7.

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Answer:

The price of the meal after the discount was (A)$48.11 and the final price of the meal was (B)$57.73.

Step-by-step explanation:

Let us consider the amount of money $56.60 as the cost of the meal.

So the amount of money is 56 dollars and 60 cents.

A: The family has a 15% off coupon.

Discount = 15% of 56.60=0.15x56.60=8.49

Now we subtract the discount amount from the original amount.

Cost of the meal = 56.60-8.49=48.11

So the final price of the meal for the family is $48.11 or 48 dollars and 11 cents.

B: The family gave a tip of 20%.

Amount of money tipped= 20% of 48.11=0.20 x 48.11=9.622

To get the final amount we add the tip to the discounted price.

The final cost of the meal= 48.11+9.622=57.732=57.73

Therefore the family paid a total of 57 dollars and 73 cents for the final cost of the meal after the discount and tip.

Hope this helps :)

The calculated value of the circumference of the circle is 31.4 units

From the question, we have the following parameters that can be used in our computation:

Radius, r = 5

Using the above as a guide, we have the following:

Circumference = 2 * π * r

Substitute the known values in the above equation, so, we have the following representation

Circumference = 2 * 5 * 3.14

Evaluate the products

Circumference = 31.4

HEnce, the value of the circumference is 31.4 units

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The company’s earnings per share reported in its financial statements for the year ended December 31, 2020 is $1.45. The correct option is c.

To calculate earnings per share, we need to take the company's net income and divide it by the weighted average number of shares outstanding during the year.

First, let's adjust for the stock dividend on July 31. Since the dividend was 100%, we can double the number of shares outstanding to 416,000:

Jan. 1: 200,000 shares
Feb. 1: 8,000 new shares
July 31: 200,000 shares doubled to 400,000 shares
Dec. 31: 416,000 shares

Next, we need to calculate the weighted average number of shares outstanding during the year. We can do this by taking the number of shares outstanding for each period and multiplying it by the number of months those shares were outstanding:

Jan. 1 to Jan. 31: 200,000 shares x 1 month = 200,000
Feb. 1 to July 31: (200,000 + 8,000) shares x 6 months = 1,248,000
Aug. 1 to Dec. 31: 416,000 shares x 5 months = 2,080,000

Total weighted average shares outstanding: 3,528,000

Finally, we can divide the net income of $560,000 by the weighted average shares outstanding of 3,528,000 to get earnings per share of $0.1585. Multiplying this by 9 (since there are 9 months of the year remaining after February 1) gives us earnings per share of $1.4265. Rounded to the nearest penny, the answer is $1.45.

Thus, The correct option is c.

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The length of the string needed is 65 meters

An equation is an expression that shows how numbers and variables using mathematical operators.

Pythagoras theorem shows the relationship between the sides of a right angle triangle.

To find the length of string, Mia needs. A triangle is formed with hypotenuse (l) represent the length of string. The height of the kite (h) = 33 m which is the triangle height; while the 56 m is the base of the triangle (b). Hence:

l² = b² + h²

l² = 33² + 56²

l = 65 meters

The length of the string needed is 65 meters

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The value of median m that makes Q₃ equal to 128 is approximately 18.75.

The median is the value that divides the higher half of a population, a probability distribution, or a sample of data from the lower half. It can be conceptualised as a data set's "middle" value to put it simply.

To find the value of m, we need to first calculate the median and third quartile of the data.

To calculate the median, we need to find the value that splits the data into two halves. Since the data is already sorted into intervals, we can find the cumulative frequency for each interval and use it to determine the median interval. The median interval is the interval that contains the median. We can then use the formula for the median of grouped data to calculate the median value.

Cumulative frequency for each interval:

- Interval 0-30: 2

- Interval 30-60: 2+8=10

- Interval 60-90: 10+22=32

- Interval 90-120: 32+24=56

- Interval 120-150: 56+m

- Interval 150-180: 56+m+9=65+m

Since there are 6 intervals, the median interval is the 3rd interval, which is 60-90. The lower limit of this interval is 60, and the cumulative frequency up to this interval is 32. The frequency of this interval is 22. Using the formula for the median of grouped data:

Median = L + ((n/2 - CF) / f) * w

where L is the lower limit of the median interval, CF is the cumulative frequency up to the median interval, n is the total sample size, f is the frequency of the median interval, and w is the width of the interval.

Plugging in the values, we get:

Median = 60 + ((50 - 32) / 22) * 30

Median = 60 + (18 / 22) * 30

Median = 60 + 15.45

Median ≈ 75.45

Now, to find the third quartile (Q₃), we need to find the value that splits the upper 50% of the data. Since Q₃ is the 75th percentile, the cumulative frequency up to Q₃ is 0.75 times the total sample size:

Q₃ = L + ((0.75 * n - CF) / f) * w

We know that Q₃ is 128, and we can plug in the values for L, n, CF, f, and w that correspond to the interval that contains Q₃:

128 = 120 + ((0.75 * 85 - 56 - m) / (24)) * 30

Simplifying and solving for m, we get:

m = 120 + ((0.75 * 85 - 56) / (24)) * 30 - 128

m ≈ 18.75

Therefore, the value of m that makes Q₃ equal to 128 is approximately 18.75.

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The limit of the sequence a_n is 0. The sequence a_n = (cos n)/[tex]7^n[/tex] oscillates between -1/[tex]7^n[/tex] and 1/[tex]7^n[/tex] since the cosine function is bounded between -1 and 1. Therefore, by the squeeze theorem, the limit of the sequence is 0 as n approaches infinity.

The cosine function oscillates between -1 and 1, so we have:

-1/[tex]7^n[/tex] ≤ cos(n)/7^n ≤ 1/[tex]7^n[/tex]

Dividing each term by [tex]7^n[/tex], we obtain:

-1/[tex]7^n[/tex] ≤ a_n ≤ 1/[tex]7^n[/tex]

By the squeeze theorem, since -1/[tex]7^n[/tex] and 1/[tex]7^n[/tex] both approach zero as n approaches infinity, we have:

lim a_n = 0

Therefore, the limit of the sequence a_n is 0.

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To find the velocity and acceleration 8 seconds after the rocket is launched, we need to find the first and second derivatives of the height function with respect to time.

The first derivative of h(t) gives the velocity function v(t):
v(t) = h'(t) = 3t^2 + 1.2t

Substituting t = 8 into this equation gives us the velocity of the rocket at 8 seconds after launch:
v(8) = 3(8)^2 + 1.2(8) = 204.8 feet per second

So the velocity of the rocket 8 seconds after launch is 204.8 feet per second.

The second derivative of h(t) gives the acceleration function a(t):
a(t) = h''(t) = 6t + 1.2

Substituting t = 8 into this equation gives us the acceleration of the rocket at 8 seconds after launch:
a(8) = 6(8) + 1.2 = 49.2 feet per second squared

So the acceleration of the rocket 8 seconds after launch is 49.2 feet per second squared.
To find the velocity and acceleration of the rocket 8 seconds after it is launched, we need to determine the first and second derivatives of the height function h(t) with respect to time t.

Given h(t) = t^3 + 0.6t^2, let's find its first and second derivatives:

1. Velocity (first derivative of h(t)):
v(t) = dh/dt = 3t^2 + 1.2t

2. Acceleration (second derivative of h(t)):
a(t) = d^2h/dt^2 = d(v(t))/dt = 6t + 1.2

Now, let's evaluate the velocity and acceleration at t = 8 seconds:

Velocity at t=8:
v(8) = 3(8^2) + 1.2(8) = 192 + 9.6 = 201.6 ft/s

Acceleration at t=8:
a(8) = 6(8) + 1.2 = 48 + 1.2 = 49.2 ft/s^2

So, 8 seconds after the rocket is launched, its velocity is 201.6 ft/s and its acceleration is 49.2 ft/s^2.

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Answer:

y=1.2x-6.4

Step-by-step explanation:

get slope

y2-y1/x2-x1

6.4 - 4.6/0-(-1.5)

1.8/1.5=1.2

get formula

y-y1=m(x-x1)

y-4.6=1.2(x-1.5)

y-4.6=1.2x-1.8

y-4.6=1.2x-1.8

y=1.2x-6.4

We know that the P(Marching Band and Team Sport) ≠ P(Marching Band) * P(Team Sport), the two events are dependent and associated.

A) The conditional probability of being on the marching band given that the student plays a team sport can be calculated using the formula:

P(Marching Band | Team Sport) = P(Marching Band and Team Sport) / P(Team Sport)

where P(Marching Band and Team Sport) is the probability of being on the marching band and playing a team sport, and P(Team Sport) is the probability of playing a team sport.

Let's say that out of a total of 500 students, 100 students play a team sport and 50 of them are also on the marching band. Then,

P(Marching Band and Team Sport) = 50/500 = 0.1
P(Team Sport) = 100/500 = 0.2

Plugging these values into the formula, we get:

P(Marching Band | Team Sport) = 0.1 / 0.2 = 0.5

Therefore, the conditional probability of being on the marching band given that the student plays a team sport is 0.5 or 50%.

b. The probability of being on the marching band can be calculated as:

P(Marching Band) = (Number of students on the marching band) / (Total number of students)

Let's say that out of the same 500 students, 75 students are on the marching band. Then,

P(Marching Band) = 75/500 = 0.15 or 15%

The difference between part (a) and part (b) is that in part (a), we are given additional information (the student plays a team sport) and we want to find the probability of being on the marching band. In part (b), we are simply asked for the probability of being on the marching band without any other information.

c. The two events, {on the marching band} and {on a team sport}, may or may not be associated. We can use probabilities to determine whether they are associated or not.

If the probability of being on the marching band and playing a team sport is different from the product of the probabilities of being on the marching band and playing a team sport separately, then the events are dependent and associated. If they are the same, then the events are independent and not associated.

Let's calculate the probabilities:

P(Marching Band and Team Sport) = 50/500 = 0.1
P(Marching Band) = 75/500 = 0.15
P(Team Sport) = 100/500 = 0.2

Product of the probabilities:

P(Marching Band) * P(Team Sport) = 0.15 * 0.2 = 0.03

Since P(Marching Band and Team Sport) ≠ P(Marching Band) * P(Team Sport), the two events are dependent and associated. This means that knowing whether a student is on the marching band affects the probability of them playing a team sport, and vice versa.

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the correct ratio to represent the relationship between Hallie's donation of jeans and Mattie's donation of jeans is 2:3. Thus, option B is correct.

To write a ratio to represent the relationship between Hallie's donation of jeans and Mattie's donation of jeans, we need to find the common factor between the number of pairs of blue jeans each person donated.

Hallie donated 4 pairs of blue jeans, and Mattie donated 6 pairs of blue jeans.

The common factor between 4 and 6 is 2. We can divide both 4 and 6 by 2 to get:

Hallie donated 2 pairs of blue jeans.

Mattie donated 3 pairs of blue jeans.

Now we can write the ratio of Hallie's donation to Mattie's donation as:

2:3

Therefore, the correct ratio to represent the relationship between Hallie's donation of jeans and Mattie's donation of jeans is 2:3.

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The correct option is option 3.

Based on the given integral, region D is in the first quadrant, and its boundaries are not explicitly given. However, we can deduce the boundaries of D by looking at the integrand. Since the integrand is f(x,y), we can see that we are integrating over the entire region D, which means that D must be the rectangle that contains all the other regions mentioned in the options.

Therefore, option 1 is not correct, as D is not bounded from the right by x=2, but rather extends indefinitely to the right. Option 2 is also not correct, as D extends beyond the line y=x. Option 4 is not correct either, as D is not bounded above by y=x^2, but rather extends beyond it. Options 5 and 6 are also not correct, as D extends beyond the lines y=0 and y=2.

Therefore, the correct option is option 3, which states that D is the region in the first quadrant that is bounded from the left by the line x=0. This is correct, as D extends indefinitely to the right, and is bounded from the left by x=0.

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The number of coffee drinks sold on a day that 32 ice cream treats were sold using the equation of line of best fit is y = 32m + b, where y represents the number of coffee drinks sold and ice cream treats that  were sold = 32.

To estimate the number of coffee drinks sold on a day that 32 ice cream treats were sold, we would need to use the equation of the line of best fit. This equation represents the trend of the data collected and can be used to make predictions based on that trend.

Assuming that the data collected shows a positive correlation between the number of ice cream treats sold and the number of coffee drinks sold, we can use the equation of the line of best fit to estimate the number of coffee drinks sold on a day that 32 ice cream treats were sold.

Let's say that the equation of the line of best fit is y = mx + b, where y represents the number of coffee drinks sold and x represents the number of ice cream treats sold. Using the data collected, we can find the values of m and b that best fit the trend.

Once we have the equation, we can substitute x = 32 into the equation and solve for y. This will give us an estimate of the number of coffee drinks sold on a day that 32 ice cream treats were sold.

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The basketball coach can order approximately 34 shooting shirts for $300.

To determine the number of shooting shirts the basketball coach can order for $300, we need to solve the equation C = 0.2x^2 + 1.6x + 15, where C represents the cost and x represents the number of shooting shirts.

The equation is given as C = 0.2x^2 + 1.6x + 15.

To find the number of shooting shirts for $300, we set the cost C equal to 300 and solve for x:

0.2x^2 + 1.6x + 15 = 300

0.2x^2 + 1.6x + 15 - 300 = 0

0.2x^2 + 1.6x - 285 = 0

Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

For this equation, a = 0.2, b = 1.6, and c = -285. Plugging in these values into the quadratic formula:

x = (-1.6 ± sqrt(1.6^2 - 4 * 0.2 * -285)) / (2 * 0.2)

Simplifying the equation further:

x = (-1.6 ± sqrt(2.56 + 228)) / 0.4

x = (-1.6 ± sqrt(230.56)) / 0.4

x = (-1.6 ± 15.18) / 0.4

Now we have two solutions:

x1 = (-1.6 + 15.18) / 0.4 = 33.95

x2 = (-1.6 - 15.18) / 0.4 = -44.95

Since the number of shooting shirts cannot be negative, we discard the negative solution.

Therefore, the basketball coach can order approximately 34 shooting shirts for $300.

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The equation of demand of candle sticks can be modeled by

y = 14 - 0.02x while the revenue function will be xy = 14x - 0.02x².

Here we are given the information that

200 candles are sold for $10 and,

250 candles are sold for $9

Let the Price be y while the Quantity sold be x

Hence, by one unit decrease in price P, the quantity sold is increased by 50 units.

Here the slope of the function will be

(10 - 9)/(200 - 250)

= - 1/50

= - 0,02

Now we will use the formula of the equation of a straight line

(y - y₁) = m(x - x₁)

where, m is the slope and x₁ , y₁ are some point on line

Hence we get

(y - 10) = -0.02(x - 200)

or, y - 10 = -0.02x + 4

or, y = 14 - 0.02x

The revenue function will be xy = 14x - 0.02x²

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Correct Question

A candlemaker prices one set of scented candles at $10 and sells an average of 200 sets each week. he finds that when he reduces the price by $1, he then sells 50 more candle sets each week. a function can be used to model the relationship between the candlemaker's weekly revenue, r(x), after one-dollar decrease in price. this situation can be modeled by the equation y =

With regard to the similar triangles,

In ΔACD

BE ∥ CD

In ΔACD and ΔABE

BE ∥ CD

∠ACD =∠ABE           (corresponding angles)

∠ADC = ∠AEB          (corresponding angles)

∠A = ∠A                    (common angle)

ACD ∼ ΔABE

So,  The corresponding sides are in proportion.

Now, find ED

AB/BC = AE/ED
ED = AE (BC/AB)
ED = 26(5/20)
ED = 6.5cm


For BE

AB/AC = BE/CD
BE = CD (AB/AC)
BE = 18 (20/25)
BE  = 14.4cm

Now, find BE

Therefore, the length of ED is 6.5cm and the length of BE is 14.4 cm.

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To find the estimated probability that it will take at least five patients to find one patient on whom the medicine would not be effective, we need to find the probability of getting NNNNN as the first five patients. Since the success rate is 0.3 and the failure rate is 0.7, the probability of getting NNNNN is:

0.7 x 0.7 x 0.7 x 0.7 x 0.7 = 0.16807

Therefore, the estimated probability that it will take at least five patients to find one patient on whom the medicine would not be effective is 0.16807.

To find the estimated probability that the medicine will be effective on exactly three out of four randomly selected patients, we need to count the number of ways we can select three patients out of four and multiply it by the probability of getting EEE and NEEE for the selected patients and non-selected patients, respectively. The number of ways to select three patients out of four is:

4C3 = 4

The probability of getting EEE and NEEE for the selected patients and non-selected patients, respectively, is:

(0.7)^3 x (0.3) x (0.7) = 0.1029

Therefore, the estimated probability that the medicine will be effective on exactly three out of four randomly selected patients is 4 x 0.1029 = 0.4116 (rounded to 4 decimal places).

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The polar coordinates (r, θ) for point B with rectangular coordinates (-5, 12) are (13, 112.62°).

The polar coordinates (r, θ) for point B with rectangular coordinates (-5, 12) can be determined as follows.

1. Calculate the radius r:

r = √(x² + y²) = √((-5)² + 12²) = √(25 + 144) = √169 = 13.

2. Calculate the angle θ in radians:

θ = arctan(y/x) = arctan(12/-5) ≈ -1.176 radians.

3. Convert θ from radians to degrees:

θ = (-1.176 * 180) / π ≈ -67.38 degrees.

4. Adjust the angle to the proper quadrant (since point B is in the second quadrant):

θ = 180 - 67.38 = 112.62 degrees.

So, the polar coordinates (r, θ) for point B are (13, 112.62°).

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