i need help with this ive been stuck on it

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.

Learn more about derivative here:

https://brainly.com/question/25752367

#SPJ1

Answer:

P=2(l+w)=2·(9+10)=38ft

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

Read more about polynomial at

https://brainly.com/question/7693326

#SPJ1

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:

[tex]\boxed{8-6i}[/tex]

Step-by-step explanation:

First, we developed the square binomial [tex](-3+\mathrm{i})^2[/tex].

[tex]\implies (-3+\mathrm{i})(-3+\mathrm{i})\\9-3\mathrm{i}-3\mathrm{i}+i^2\\9-6\mathrm{i}+\mathrm{i}^2[/tex]

Remember the next product:

[tex]i^2= \mathrm{i} \times \mathrm{i} = -1[/tex]

then:

[tex]9-6\mathrm{i}+ (-1)\\8-6i[/tex]

Hope it helps

[tex]\text{-B$\mathfrak{randon}$VN}[/tex]

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]

Answer:

1025

Step-by-step explanation

The formula to find the sum of the first n terms of an arithmetic sequence is

Sn = n/2 * [2a1 + (n-1)d]

Where

a1 = the first term of the sequence

d = the common difference between consecutive terms

n = the number of terms we want to sum

Substituting the given values,  we get

a1 = 5

d = 3

n = 25

S25 = 25/2 * [2(5) + (25-1)3]

= 25/2 * [10 + 72]

= 25/2 * 82

= 25 * 41

= 1025

Answer:

{0, 1, 2}

Step-by-step explanation:

4x<8x+2

-4x<2

x<-1/2

Only {0, 1, 2} meets the critera.

Answer:

We can start by simplifying the expression inside the parentheses using the identity:

cos 2x = 2 cos² x - 1

Substituting this in, we get:

1 – 3 cos 2x + 3 cos² 2x − cos³ 2x

= 1 – 3(2 cos² x - 1) + 3(2 cos² x - 1)² − (2 cos² x - 1)³

= 1 – 6 cos² x + 9 cos⁴ x - 4 cos⁶ x

Therefore, we can rewrite f(x) as:

f(x) = 4 cos x (1 – 6 cos² x + 9 cos⁴ x - 4 cos⁶ x)

Next, we can use the trigonometric identity:

sin 2x = 2 cos x sin x

to express cos x in terms of sin x:

cos x = √(1 - sin² x)

Substituting this in, we get:

f(x) = 4 sin x cos³ x (1 – 6 cos² x + 9 cos⁴ x - 4 cos⁶ x)

= 4 sin x (√(1 - sin² x))³ (1 – 6 (2 sin² x - 1) + 9 (2 sin² x - 1)² - 4 (2 sin² x - 1)³)

= 4 sin x (1 - sin² x)^(3/2) (16 sin⁶ x - 48 sin⁴ x + 36 sin² x - 8)

Next, we can use the substitution u = 1 - sin² x, du = -2 sin x cos x dx, to obtain:

f(x) dx = -2 du (u^(3/2)) (16 - 48u + 36u² - 8u³)

Integrating, we get:

f(x) dx = 2/3 (1 - sin² x)^(5/2) (8 - 36(1 - sin² x) + 36(1 - sin² x)² - 8(1 - sin² x)³) + C

Now, we can use the trigonometric identity:

sin² x = (1 - cos 2x)/2

to simplify the expression inside the parentheses. After some algebra, we obtain:

f(x) dx = 3/2 sin 7x + C

where C is the constant of integration. Since m is a positive real constant, we can set:

7x = m

and solve for x:

x = m/7

Substituting this in, we get:

f(x) dx = 3/2 sin(7m/7) = 3/2 sin m

Therefore, we have shown that:

f(x) dx = 3/2 sin m, where m is a positive real constant.

Our assumption that I U{—A} is satisfiable must be false. Hence, I U{—A} is not satisfiable if I = A.

A set of propositional formulae, sometimes referred to as a propositional theory, can be satisfiable in terms of propositional logic by having the quality of being true or untrue according to a certain interpretation or model. If there is at least one interpretation that makes all of a set of formulae true, the set is said to be satisfiable.

Using the proof by contradiction we have:

Assume that I U{—A} is satisfiable.

Then, by definition of satisfiability, every formula in the set I U{—A} is true in M.

Since I = A, every formula in I is also in A. Therefore, every formula in I is true in M, since A is true in M.

Consider the formula —A, which is in {—A}. Since M satisfies {—A}, —A is true in M.

But this contradicts the fact that A is true in M, since —A is the negation of A.

Therefore, our assumption that I U{—A} is satisfiable must be false. Hence, I U{—A} is not satisfiable if I = A.

Learn more about proof by contradiction here:

https://brainly.com/question/8062770

#SPJ1

The amount Angela earned this month is $2,618.

Percentage can be described as a fraction of an amount expressed as a number out of hundred.

Angela's earnings = percentage commission x worth of goods sold

[tex]11\% \times 23,800[/tex]

[tex]0.11 \times 23,800 = \bold{\$2618}[/tex]

To learn more about percentages, please check: brainly.com/question/25764815

Answer:

3r^2+10r+3

Step-by-step explanation:

Answer:

Let Haley be represented as x

Now Mateo has 8 more coins than haley

Mateo = 8 + x

total number of coins is Mateo coins and Haley coins.

x + 8 + x

2x + 8

Answer:

136 units

Step-by-step explanation:

All sides are equal in a rectangle:

Value of b : 16-8 = 8 units

h = 13-7 = 6 units.

So Area of triangle= bh/2 = 8*6/2 = 24 units

Area of rectangle = lb = 16*7 = 112 units

So Area of figure= 112+24 units = 136 units

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:

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

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.

To know more about probability visit:

https://brainly.com/question/11234923

#SPJ1

Answer:

x > 2

Hope this helps!

Step-by-step explanation:

3x - 4 > 2

3x - 4 ( + 4 ) > 2 ( + 4 )

3x > 6

3x ( ÷ 3 ) > 6 ( ÷ 3 )

x > 2

Answer:

yes.

Step-by-step explanation:

cause yes.

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 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%.

To know more about   probability here

https://brainly.com/question/24756209

#SPJ1

Answer:

The answer is 15 nickels and 13 quarters\

Step-by-step explanation:

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

To learn more about the rectangular from the given link

https://brainly.com/question/31186475

#SPJ1

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

Read more about area at

https://brainly.com/question/24487155

#SPJ1

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].

Learn more about tangents at https://brainly.com/question/12465859

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:

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

76°c

Step-by-step explanation: