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1. Identify each premise and the conclusion in the following argument. Determine whether the argument is an example of inductive or deductive reasoning.

Our house is made of brick. Both of my next-door neighbors have brick houses. Therefore, all houses in our neighborhood are made of brick.

Premises:

Conclusion:

Inductive or deductive?


2. Use the list of equations and inductive reasoning to predict the next multiplication fact in the list:

37 × 3 = 111, 37 × 6 = 222 , 37 × 9 = 333, 37 × 12 = 444 , ....

Use inductive reasoning to determine the probable next number in the list below. An arithmetic sequence adds or subtracts a fixed amount (the common difference) to get the next term in the sequence.

5, 9, 13, 17, 21, 25, 29, .....


3. Successive Differences

Use the method of successive differences to find the next number in the sequence.

(a) 20, 31, 45, 62, 82, ... (Hint: Start with 31 - 20 = 11, use the subsequent term subtract the previous term)

(b) 8, 6, 3, 1, 2, 8, 21, ... (Hint: start with 6 - 8= -2, use the subsequent term subtract the previous term)

4. Find the sum using the special sum formula.

1 + 2 + 3 + ... + 48


5. Use the formula given on the first section to find

a) the sixth pentagonal number.

b) the eighth triangular number.

6. A vending machine accepts nickels, dimes, and quarters. Exact change is needed to make a purchase. How many ways can a person with four nickels, three dimes, and two quarters make a 30-cent purchase from the machine?


7. Ronnie goes to the racetrack with his buddies on a weekly basis. One week he tripled his money, but then lost $12.

He took his money back the next week, doubled it, but then lost $40. The following week he tried again, taking his money back with him. He quadrupled it, and then played well enough to take that much home, a total of $224. How much did he start with the first week?


8. Use inductive reasoning to determine the units digit (or ones digit):

Example: Use inductive reasoning to determine the units digit of the number

3^55 .


The question is not about giving the exact value, but rather than applying the inductive reasoning to figure out the units digit.


The units digit in base 3 are 3, 9, 7, 1, 3, 9, 7, 1, ... repeating meaning that every power that is divisible by 4 will end in 1. 56 is divisible by 4. Using this inductive reasoning 3 to the 56th power would end in a 1, then moving backward to the 55th power would give you a units digit of 7. This is inductive reasoning because we are using the smaller sample size of 4 repeating unit digits to make broad theories about all powers of 3.

Another similar approach: The units digit in base 3 are 3, 9, 7, 1, 3, 9, 7, 1, ... repeating. If the power is a multiple of four, the units digit is 1. Power 52 is a multiple of 4 and it's close to power 55.

Power 52 ends with units digit of 1, next 53 ends with 3, 54 ends with 9, and 55 ends with 7

(a) The ones digit (also called the units digit) in 2^4000 . The question is not about giving the value to the expression,

but rather than applying the inductive reasoning to figure out the units digit. The answer is just a single digit.

(b) The units digit of the number 3^41

9. Kathy stood on the middle rung of a ladder. She climbed up 2 rungs, moved down 5, and then climbed up 6. Then she climbed up the remaining 4 rungs to the top of the ladder. How many rungs are there in the whole ladder?

10. If you ask Batman’s nemesis, Catwoman, how many cats she has, she answers with a riddle: “Five-sixth of my cats plus seven.” How many cats does Catwoman have?

11. A birdhouse for swallows can accommodate up to 8 nests. How many birdhouses would be necessary to accommodate 58 nests?

12. Use the circle graph below to determine how many of the 140 students made an A or a B.

(see picture of circle graph attached below)

13. The bar graph shows the number of cups of coffee, in hundreds of cups, that a professor had in a given year.

a) Estimate the number of cups in 2004.

b) What year shows the greatest decrease in cups?

(see picture of bar graph below)

14. The line graph shows the average class size of a first grade class at a grade school for years 2001 through 2005. (see picture line graph below)

In which years did the average class size increase from the previous year?

How much did the average size increase from 2001 to 2003?

Answer: 16.5 gallons of water.

Step-by-step explanation:

If it was me. I would be setting up as a table to keep my work organized.

So first we find how much 1 minute is.

15g : 10m

15/10 : 10m/10

1.5g : 1m

Then I multiply how many minutes there are.

1.5g x 11 : 1m x 1

16.5g : 11m

And there we find the answer of 16.5 gallons.

Happy Solving

Answer:16.5

Step-by-step explanation:

Answer:

[tex]3x-3y=31[/tex]

Step-by-step explanation:

Adding the equations yields [tex]3x-3y=20+11=31[/tex].

The cumulative probabilities for the given probability distribution were calculated, and the median of the discrete random variable was found to be 20.

To find the median, we need to find the smallest value of the random variable for which the cumulative probability equals or exceeds 0.5.

The cumulative probabilities are:

(≤0) = 0.02

(≤5) = 0.07

(≤10) = 0.15

(≤15) = 0.31

(≤20) = 0.58

(≤25) = 1

The cumulative probability is the sum of the probabilities of all events that have an outcome less than or equal to a given value. For example, the cumulative probability for the event of collecting 5 dollars or less is the sum of the probabilities for collecting 0 dollars and 5 dollars, which is 0.02 + 0.05 = 0.07. Similarly, the cumulative probability for the event of collecting 10 dollars or less is the sum of the probabilities for collecting 0 dollars, 5 dollars, and 10 dollars, which is 0.02 + 0.05 + 0.08 = 0.15. The same process is used to calculate the cumulative probabilities for all other values. The median is the smallest value of the random variable for which the cumulative probability is greater than or equal to 0.5.

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option C is correct. The range of penguin heights is greater at Cityview Zoo than at Countryside Zoo.

How to calculate data?

Based on the given data, we can see that the penguin heights at Countryside Zoo are distributed more evenly across the range of heights, with heights ranging from 35cm to 45cm, and a fairly consistent distribution across this range. In contrast, the penguin heights at Cityview Zoo are more concentrated around the height of 38cm, with fewer penguins at the extremes of the range.

Therefore, option C is correct. The range of penguin heights is greater at Cityview Zoo than at Countryside Zoo.

Option A is incorrect because the range of penguin heights is greater at Countryside Zoo than at Cityview Zoo.

Option B is incorrect because the ranges of penguin heights are not the same.

Option D is also incorrect because there is no mode of penguin heights at either zoo. A mode is defined as the value that appears most frequently in a distribution. In this case, no height appears more frequently than any other.

In conclusion, we can say that the penguin heights are distributed differently at the two zoos, with Countryside Zoo having a more evenly distributed range of heights, while Cityview Zoo has a more concentrated distribution around a particular height.

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First, 6.7 % can be written in decimal form as 0.067  (6.7 / 100 = 0.067).

Let's use the variable x to represent the number of copies sold to date.

Then we can write and solve the following equation to represent 6.7% of the total sold to date:

0.067 • x = 33600

You can solve this equation by dividing both sides of the equation by 0.067:

0.067 • x  =  33600

0.067             0.067

x = 501493

To date, 500000 copies would have been sold rounded to the nearest whole.

Answer:

you just add a 0 at the end of the answer of what 32 divided by 4 is, so in this case 320 divided by 4 is 80

Step-by-step explanation:

32 divided by 4 is 8.

320 divided by 4 is 80.

To get from 32 to 320 all you need is a 0 at the end, so you can just add the 0 the end of the answer. This means you're going from an 8, to an 80.

OR

Another way you can look at it is 32 multiplied by 10 to get 320. So you need to mutiple your answer by 10 to get the right answer.

32*10=320

8*10=80

Hope this helps!

The f(x) of the linear function is:

f(x) = -4x - 7

Since f is a linear function. The general form of a linear function is:

y = mx + b

where y = f(x), m is the slope and b is the y-intercept

Since f (−3) = 5, we have:

x = -3 and y = 5

Substitute into y = mx + b:

y = mx + b

5 = -3m + b ---- (1)

Also, f(5) = −27, we have:

x = 5 and y = -27

Substitute into y = mx + b:

y = mx + b

-27 = 5m + b ---- (2)

Solving (1) and (2) simultaneously by elimination method:

-3m + b = 5

5m + b = -27

-8m = 32

m = 32/(-8)

m = -4

Put m = -4 in (1) and solve for b:

-3x + b = 5

-3(-4) + b = 5

12 + b = 5

b = 5 - 12

b = -7

Put m and b into f(x) = mx + b to get f(x). That is:

f(x) = mx + b

f(x) = -4x - 7

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Answer:

The given line has a slope of -3/2, since it is in the form y = mx + b, where m is the slope. Any line that is parallel to this line will also have a slope of -3/2.

To find the equation of the line that passes through the point (-2,3) and has a slope of -3/2, we can use the point-slope form of the equation of a line:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope. Substituting in the values we have:

y - 3 = (-3/2)(x - (-2))

y - 3 = (-3/2)x - 3

y = (-3/2)x - 3 + 3

y = (-3/2)x

Therefore, the equation of the line that is parallel to y = -3/2x - 8 and passes through the point (-2,3) is y = (-3/2)x.

Note that this line does not have a y-intercept, since it passes through the point (-2,3) and has a slope of -3/2.

The values of function f(x) positive, and negatives are (-infinity, infinity )

A function is a procedure or link that connects every element of one non-empty set A to at least one element of another non-empty set B. The phrases "domain" and "co-domain" are used in mathematics to define a function f between two sets, A and B. The constraint F = (a,b)| is satisfied by all values of a and b.

In the case of the question,

f (x) = x²

f (x) will be positive for all x values. As a result of the function:

x² = x × x

That is, when any number or integer is multiplied by itself, the result is positive. (For example, - - = + and + + = +)

As a result, f (x) = x2 will be positive for (,).

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As a result, there is a 26% chance that two red marbles will be chosen at random, or around 0.26.

The area of mathematics known as probability is concerned with analysing the results of random events. It represents a probability or likelihood that a specific occurrence will occur. A number in 0 and 1 is used to represent probability, with 0 denoting an event's impossibility and 1 denoting its certainty. In order to produce predictions and guide decision-making, probability is employed in a variety of disciplines, such science, finance, economics, architecture, and statistics.

given

Given that there are 12 red marbles and a total of 24 marbles in the jar, the likelihood of choosing the first red marble is 12/24.

There are 11 red marbles and a total of 23 marbles in the jar after choosing the first red marble.

As a result, the likelihood of choosing a second red marble is 11/23.

We compound the probabilities to determine the likelihood of both outcomes occurring simultaneously (i.e., choosing two red marbles):

P(choosing 2 red marbles) = (12/24) x (11/23) = 0.2609, which is roughly 0.26.

As a result, there is a 26% chance that two red marbles will be chosen at random, or around 0.26.

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Answer:

O = { -4,-3,-2,-1,0,-1 }

The perimeter is 27 feet and the area is 35 square feet

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

The figure

The perimeter is the sum of tthe side lengths

So, we have

Perimeter = 7.5 + 6 + (6 - 2.5) + 4 + 2.5 + 3.5

Evaluate

Perimeter = 27

The area is calculated as

Area = 6 * 3.5 + 4 * (6 - 2.5)

Evaluate

Area = 35

Hence, teh area is 35 square feet

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Check the picture below.

[tex]\cfrac{3^2}{4^2}=\cfrac{27}{A}\implies \cfrac{9}{16}=\cfrac{27}{A}\implies 9A=432\implies A=\cfrac{432}{9}\implies A=48[/tex]

The correct statement that compares the distributions is:

D On average, nonsmokers weighed 13 lbs less than smokers.

What is the variability?

Variability refers to the amount of spread or dispersion in a set of data. It is a measure of how much the data values in a sample or population differ from each other.

One commonly used measure of variability is the standard deviation, which is the square root of the variance. The variance is the average of the squared differences from the mean.

Looking at the graph, we can see that the center of the distribution of smokers is around 178 lbs, while the center of the distribution of nonsmokers is around 165 lbs. This means that, on average, nonsmokers weigh less than smokers.

Option A is incorrect because the range is not a good measure of variability, and it does not necessarily mean that there is more variability in the weights of nonsmokers.

Option B is incorrect because the graph clearly shows that nonsmokers weigh less on average than smokers.

Option C is incorrect because we cannot make any conclusion about half of the smokers weighing more than all of the nonsmokers from the graph.

Option E is incorrect because the graph shows that the variability in the weights of smokers is greater than that of nonsmokers.

Hence, The correct statement that compares the distributions is:

D On average, nonsmokers weighed 13 lbs less than smokers.

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Answer:

yes.

Step-by-step explanation:

cause yes.

The probability that a randomly selected light bulb will last between 750 and 900 hours is given as follows:

P = 47.5%.

The Empirical Rule states that, for a normally distributed random variable, the symmetric distribution of scores is presented as follows:

In the context of this problem, we have that:

The normal distribution is symmetric, hence the probability of an observation between the mean and two standard deviations above the mean is given as follows:

0.5 x 95 = 0.475 = 47.5%.

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Answer:

[tex]tan(x)=\frac{4}{3}[/tex]

Step-by-step explanation:

In the unit circle,

- [tex]cos(a)=\frac{x}{r}[/tex] where [tex]a[/tex] is the degree measure, [tex]x[/tex] is the x-coordinate of the triangle, and [tex]r[/tex] is the radius of the circle

- [tex]sin(a)=\frac{y}{r}[/tex] where [tex]a[/tex] is the degree measure, [tex]y[/tex] is the y-coordinate of the triangle, and [tex]r[/tex] is the radius of the circle

Thus, since tangent is equal to sine over cosine, we can simplify our knowledge to:  [tex]tan(a)=\frac{sin(a)}{cos(a)}=\frac{y}{x}[/tex]

In this problem, [tex]sin(x)=\frac{4}{5}[/tex]. We can conclude from our previous knowledge that [tex]y=4[/tex] and the radius is 5.

Similarly, [tex]cos(x)=\frac{3}{5}[/tex], which means [tex]x=3[/tex] and the radius is the same, at 5.

Since we know that [tex]x=3[/tex] and [tex]y=4[/tex], we can find the value of  [tex]tan(x)[/tex] by using the formula [tex]tan(x)=\frac{y}{x}[/tex] and plug in the numbers.

Therefore, [tex]tan(x)=\frac{4}{3}[/tex].

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Answer:

a) To find the probability that X1 is less than or equal to 0.5 and X2 is greater than or equal to 0.25, we need to integrate the given density function over the region where X1 ≤ 0.5 and X2 ≥ 0.25.

P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫∫(x1,x2) f(x1,x2) dxdy

where the limits of integration are:

0.25 ≤ x2 ≤ 1

0 ≤ x1 ≤ 0.5

Substituting the given density function:

P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫0.25^1 ∫0^0.5 (x1 + x2) dx1 dx2

Evaluating the inner integral:

P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫0.25^1 [(x1^2/2) + x1x2] |0 to 0.5 dx2

Simplifying the expression:

P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫0.25^1 [(0.125 + 0.25x2)] dx2

Evaluating the upper and lower limits:

P(X1 ≤ 0.5, X2 ≥ 0.25) = [0.125x2 + 0.125x2^2] |0.25 to 1

Substituting the limits:

P(X1 ≤ 0.5, X2 ≥ 0.25) = [(0.125 + 0.125) - (0.03125 + 0.015625)]

Solving for the final answer:

P(X1 ≤ 0.5, X2 ≥ 0.25) = 21/64

Therefore, the probability that X1 is less than or equal to 0.5 and X2 is greater than or equal to 0.25 is 21/64.

b) To find the probability that X1 + X2 is less than or equal to 1, we need to integrate the given density function over the region where X1 + X2 ≤ 1.

P(X1 + X2 ≤ 1) = ∫∫(x1,x2) f(x1,x2) dxdy

where the limits of integration are:

0 ≤ x1 ≤ 1

0 ≤ x2 ≤ 1-x1

Substituting the given density function:

P(X1 + X2 ≤ 1) = ∫0^1 ∫0^(1-x1) (x1 + x2) dx2 dx1

Evaluating the inner integral:

P(X1 + X2 ≤ 1) = ∫0^1 [(x1x2 + 0.5x2^2)] |0 to (1-x1) dx1

Simplifying the expression:

P(X1 + X2 ≤ 1) = ∫0^1 [(x1 - x1^2)/2 + (1-x1)^3/6] dx1

Evaluating the integral:

P(X1 + X2 ≤ 1) = [x1^2/4 - x1^3/6 - (1-x1)^4/24] |0 to 1

Substituting the limits:

P(X1 + X2 ≤ 1) = (1/4 - 1/6 - 1/24) - (0/4 - 0/6 - 1/24)

Solving for the final answer:

P(X1 + X2 ≤ 1) = 1/8

Therefore, the probability that X1 + X2 is less than or equal to 1 is 1/8.

Answer:

P=2(l+w)=2·(9+10)=38ft

The smallest that will qualify the window opening per the code is 0.71

What is rectangular?

A quadrilateral with four right angles is a rectangle. It can alternatively be described as a parallelogram with a right angle or an equiangular quadrilateral, where equiangular denotes that all of its angles are equal. A square is a rectangle with four equally long sides.

Here, we have

Given: a basement bedroom must have a window with an opening area of at least 5.7 square feet per the international residential code. A rectangular basement window opening 0.75 meters wide.

First, we convert square feet into square meters.

5.7 square feet = 0.53 square meters

Now,

0.53 / 0.75 = 0.71

Hence, the smallest that will qualify the window opening per the code is 0.71

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The probability that the mean of a sample of 90 computers would be greater than 117.13 months, if the quality control manager is correct, is approximately 0.9955 or 99.55%.

The sampling distribution of the sample mean follows a normal distribution with a mean of 120 and a standard deviation of 10/sqrt(90) = 1.0541 months (using the formula for the standard deviation of the sample mean).

To find the probability that the mean of a sample of 90 computers would be greater than 117.13 months, we can standardize the sample mean using the formula:

z = (sample mean - population mean) / (standard deviation of sample mean) = (117.13 - 120) / 1.0541 = -2.6089

Using a standard normal distribution table or calculator, we can find that the probability of obtaining a z-score greater than -2.6089 is approximately 0.9955.

Therefore, the probability that the mean of a sample of 90 computers would be greater than 117.13 months, if the quality control manager is correct, is approximately 0.9955 or 99.55%.

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Answer:

The travel bag that can hold exactly 3,150 cubic inches is the bag that is 21 inches long.

Step-by-step explanation:

The travel bags can be modelled as rectangular prisms.

The formula for the volume of a rectangular prism is:

[tex]\boxed{\textsf{Volume} = l \times w\times d}[/tex]

where:

Given that all three bags have a depth of 10 inches and a width of 15 inches, we can create an equation for the volume of any of the bags by substituting d = 10 and w = 15 into the formula:

[tex]\begin{aligned}\sf Volume &=l \times w \times d\\&= l \times 15\times10\\&=l \times 150\\&=150\;l\end{aligned}[/tex]

To determine which travel bag can hold exactly 3,150 cubic inches, substitute volume = 3150 into the formula and solve for length, l:

[tex]\begin{aligned}\sf Volume &=150\;l\\\\ \implies 3150&=150\;l\\\\\dfrac{3150}{150}&=\dfrac{150\;l}{150}\\\\21&=l\\\\l&=21\;\sf in\end{aligned}[/tex]

Therefore, the travel bag that can hold exactly 3,150 cubic inches is the bag that is 21 inches long.

Answer:

The average temperature on the Equator is 137°F warmer than the average temperature at the South Pole.

76°c

Step-by-step explanation:

[tex] \: [/tex]

((x/y)^(n+1)) > ((x/y)^n) for all n ≥ 1 and x > y > 0.

To prove this statement by induction, we will use the principle of mathematical induction.

Base case: When n = 1, we have:

((x/y)^(1+1)) = ((x/y)^2) = (x^2)/(y^2)

((x/y)^1) = (x/y)

Since x > y > 0, we have x/y > 1. Therefore, (x^2)/(y^2) > (x/y), which means that the base case is true.

Inductive step: Assume that ((x/y)^(k+1)) > ((x/y)^k) for some arbitrary positive integer k. We want to prove that this implies that ((x/y)^(k+2)) > ((x/y)^(k+1)).

Starting with ((x/y)^(k+2)), we can write:

((x/y)^(k+2)) = ((x/y)^(k+1)) * ((x/y)^1)

Using the induction hypothesis, we know that ((x/y)^(k+1)) > ((x/y)^k), and we also know that x/y > 1. Therefore, we have:

((x/y)^(k+2)) > ((x/y)^k) * (x/y)

Simplifying this expression, we get:

((x/y)^(k+2)) > ((x/y)^k) * (x/y)

((x/y)^(k+2)) > ((x^k)/(y^k)) * (x/y)

((x/y)^(k+2)) > ((x^(k+1))/(y^(k+1)))

Therefore, we have shown that ((x/y)^(k+2)) > ((x/y)^(k+1)) for all positive integers k, which completes the inductive step.

By the principle of mathematical induction, we have proven that ((x/y)^(n+1)) > ((x/y)^n) for all n ≥ 1 and x > y > 0.

The polynomial expression (−4b² + 8b) + (−4b³ + 5b² – 8b) when evaluated is −4b³ + b²

We can start by combining like terms.

The first set of parentheses has two terms: -4b² and 8b. The second set of parentheses also has three terms: -4b³, 5b², and -8b.

So we can first combine the like terms in the set of parentheses:

(−4b² + 8b) + (−4b³ + 5b² – 8b) = −4b³ + b²

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Answer:

5/12

Step-by-step explanation:

We are only looking at people who have a dog, which is 12

Now we need to determine if they have a cat

P( cat given that they have a dog)

            = number of people with a cat/ they have a dog

           = 5/ (5+7)

          = 5/12

After factoring the given expression -2xy³ - 8xy² + xy, the resultant answer is -2xy²(xy-4).

The concept of algebraic expressions is the use of letters or alphabets to represent numbers without providing their precise values.

A group of terms coupled with the operations +, -, x, or form an expression, such as 4 x 3 or 5 x 2 3 x y + 17.

A statement with an equal sign, such as 4 b 2 = 6, asserts that two expressions are equal in value and is known as an equation.

So, we have the expression:

-2xy³ - 8xy² + xy

Now, factor out as follows:

=  -2xy³ - 8xy² + xy

= xy(-2xy² - 8xy)

= -2xy²(xy-4)

Therefore, after factoring the given expression -2xy³ - 8xy² + xy, the resultant answer is -2xy²(xy-4).

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1. The equation for the line, which is tangent to the lemniscate at the point P is y = -√3x + (5/4 + √3/8). The equation for the line which is tangent to the lemniscate at Q is y = (-5/3)x + 11/3.

The pace at which a function is changing at a specific point is known as its derivative. It shows the angle at which the tangent line to the curve at that location slopes. A key idea in calculus, the derivative can be utilised to tackle a range of issues, such as curve analysis, rates of change, and optimisation.

The tangent line to the lemniscate at point P, is determined using the derivative of the function.

(x² + y²)² = 4(x² - y²)

Taking the derivative on both sides we have:

2(x² + y²)(2x + 2y(dy/dx)) = 8x - 8y(dy/dx)

dy/dx = (x² + y²)/(y - x)

Substituting  P= (√5/8, √3/8) for the x and y we have:

dy/dx = (√5/8)² + (√3/8)²) / (√3/8 - √5/8) = -√3

Thus, the slope of the tangent line at point P is -√3.

Using the point slope form:

y - y1 = m (x - x1)

Substituting the values we have:

y - (√3/8) = -√3(x - √5/8)

y = -√3x + (5/4 + √3/8)

Hence, equation for the line, which is tangent to the lemniscate at the point P is y = -√3x + (5/4 + √3/8).

Bonus question:

The equation of tangent for the lemniscate at point Q = (2,1) is:

dy/dx = (2² + 1²)/(1 - 2) = -5/3

Using the point slope form:

y - 1 = (-5/3)(x - 2)

y = (-5/3)x + 11/3

Hence, equation for the line which is tangent to the lemniscate at Q is y = (-5/3)x + 11/3.

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