please help fast.Jack and Judy want to buy a new home. They will need to borrow $107,500 and the bank is offering two mortgage options. Mortgage A: 30 years at 6% with monthly payments of $644.52 and a total payback of $232,027.20 Mortgage B: 20 years at 5.75% with monthly payments of $754.74 and a total payback of $181,137.60 How much more interest will Jack and Judy pay with mortgage A?

If there exists an odd prime number p and a number x that is co-prime to p, such that x^2 ≡ a (mod p), then Fermat's Little Theorem can be used to show that a^(p-1)/2 ≡ 1 (mod p).


1. Assume that there exists an odd prime number p and a number x that is co-prime to p, such that x^2 ≡ a (mod p).

2. According to Fermat's Little Theorem, if p is a prime number and a is an integer not divisible by p, then a^(p-1) ≡ 1 (mod p).

3. Since x is co-prime to p, x is not divisible by p, and thus a^((p-1)/2) ≡ 1 (mod p) for x^2 ≡ a (mod p).

4. Therefore, a^(p-1)/2 ≡ 1 (mod p) is proven using Fermat's Little Theorem.

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a) The equation of the perpendicular bisector of the line segment AB is equal to y = - (2 / 3) · x + 3.

b) The equation of the circle is (x - 6)² + (y - 1)² = 34.

a) First, determine the midpoint of the line segment by using the midpoint formula:

M(x, y) = 0.5 · A(x, y) + 0.5 · B(x, y)

M(x, y) = 0.5 · (1, - 2) + 0.5 · (5, 4)

M(x, y) = (3, 1)

Second, calculate the slope of the line segment:

m = [4 - (-2)] / (5 - 1)

m = 6 / 4

m = 3 / 2

Third, derive the expression of the perpendicular bisector:

m' = - 1 / (3 / 2)

m' = - 2 / 3

b = y - m' · x

b = 1 - (- 2 / 3) · 3

b = 1 + 2

b = 3

The equation of the perpendicular bisector of the line segment AB is equal to y = - (2 / 3) · x + 3.

b) If the points A and B lie on a circle, then the following formula is constructed from Pythagorean Theorem:

(1 - 6)² + (- 2 - p)² = (5 - 6)² + (4 - p)²

25 + (4 + 4 · p + p²) = 1 + (16 - 8 · p + p²)

29 + 4 · p = 17 - 8 · p

12 · p = 12

p = 1

The coordinates of the center of the circle is (h, k) = (6, 1).

The radius of the circle is:

r = √[(1 - 6)² + (- 2 - 1)²]

r = √[(- 5)² + (- 3)²]

r = √34

The equation of the circle is (x - 6)² + (y - 1)² = 34.

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The required selling price of the game after two discount is $63.75.

The term discount designates a sum or a percentage subtracted from an item's usual selling price. Make sure you need all of the items you plan to get before waiting until after the holiday to make a purchase. A discount is a decrease in the cost of a commodity or service. ​

Discount = 25%,

Additional discount = 15%

Then,

Net discount is = 36.25%

We have,

Price of game = $100

Discount amount = 36.25% of $100 = $36.25

So, the selling price of the game after discount = $100 - $36.25

= $63.75

Thus, required selling price of the game is $63.75.

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Let E be the number of erasers and P the number of pencils, the cost for the erasers will be 1*E and the cost of the pencils will be 0.25*P.

Then, adding both, we get the total cost:

If the total cost has to be 5 or less, we can define the inequality as:

We can solve this inequality as E in function of P or viceversa.

Answer:

Being E the number of erasers and P the number of pencils, the inequality is E+0.25P<=5.

The inequality for E in function of P is E<=5-0.25P.

The inequality for P in function of E is P<=20-4E.

Answer: The answer is 1.25

Step-by-step explanation: 1+0.25=1.25

a)  The random variable is the number of fish caught in a single cast of the line and and the parameter is the probability of hooking a fish with a single hook

b) The probability of catching 9 fish on a single cast of the line is:

P(X=9) = p^9

c) The probability of catching at least one fish on a single cast is

1 - (1-p)^9

d) The smallest value of n that satisfies this condition is:

n = ⌈log(1-p/100) / log(1-p)⌉

(a) The random variable is the number of fish caught in a single cast of the line, denoted by X. This is a discrete random variable, taking values from 0 to 9. The parameter is the probability of hooking a fish with a single hook, denoted by p.

(b) The probability of catching 9 fish on a single cast of the line is:

P(X=9) = p^9

This is because each hook has a probability of p of catching a fish, and since the events are independent, the probability of all 9 hooks catching a fish is simply the product of their individual probabilities.

(c) The probability of catching at least one fish on a single cast is:

P(X ≥ 1) = 1 - P(X=0)

= 1 - (1-p)^9

This is because the probability of not catching any fish is the complement of the probability of catching at least one fish.

(d) Let n be the number of hooks needed to catch at least one fish on each cast with probability p%. We need to find the smallest value of n that satisfies this condition. Using the complement rule, the probability of not catching a fish with n hooks is:

P(X = 0) = (1-p)^n

So the probability of catching at least one fish is:

P(X ≥ 1) = 1 - (1-p)^n

We want this probability to be at least p%, so we have:

1 - (1-p)^n ≥ p/100

Simplifying, we get:

(1-p)^n ≤ 1 - p/100

Taking the logarithm of both sides, we get:

n log(1-p) ≤ log(1-p/100)

Dividing by log(1-p), we get:

n ≥ log(1-p/100) / log(1-p)

So the smallest value of n that satisfies this condition is:

n = ⌈log(1-p/100) / log(1-p)⌉, where ⌈x⌉ denotes the smallest integer greater than or equal to x.

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Answer:

Slope is -8

Step-by-step explanation:

The slope is the number in front of x.

The goal of maximizing sustainable yield is to find the optimum population size and yield that can be sustained over time. In this case, given the information provided, the optimum population size and yield is 8,750 fish and 750 fish per year respectively.

The first step in finding the optimum population size and yield is to calculate the maximum sustainable yield (MSY) using the equation MSY = 0.5 x R. In this case, R = 10,500 fish, and thus MSY = 0.5 x 10,500 = 5,250 fish. This is the number of fish that can be harvested each year without over-harvesting the population and depleting its resources.
However, the current population size of 3,250 fish is below the MSY, indicating that the population size is already being over-harvested. Therefore, to increase the sustainable yield and maximize it, the population size must be increased.

The optimum population size to maximize the sustainable yield is calculated using the equation:
Optimum Population Size = MSY + (1 - harvest rate) x (current population size - MSY).
In this case, the harvest rate is 1300/10,500 = 0.124. Therefore, the optimum population size is 8,750 fish.
Once the optimum population size is known, the yield can be determined by subtracting the current population size from the optimum population size. In this case, the optimum yield is 8,750 - 3,250 = 5,500 fish. However, since the harvest rate is uniformly distributed throughout the year, the sustainable yield is 750 fish per year (5,500/7.5).

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Answer:

1/12

Step-by-step explanation:

1) covert 1/2 to 3/6 to subtract 2/6 from it. this gets 2/6, which simplifies to 1/3

2) divide 1/3 by 4 which gets 1/12

Yes, two numbers could have 16 as their HCF and 380 as their LCM.

Right here's how you could find those two numbers:

Step 1: prime Factorization

Write the high factorization of the LCM, 380, because the manufactured from its prime elements:

\(380 = 2^2 * 5 * 19\)

Step 2: HCF Calculation

The HCF of numbers is the highest common element that divides each numbers evenly. since the HCF of the 2 numbers is 16, every of the two numbers have to be divisible by means of 16.

Step 3: number Formation

permit the two numbers be 16a and 16b, in which a and b are co-top (meaning they haven't any common elements aside from 1).

Step 4: LCM Calculation

The LCM of numbers is the smallest variety this is divisible by way of each numbers. because the LCM of the two numbers is 380, we will write:

LCM(16a, 16b) =\(2^2 * 5 * 19\)

To find the values of a and b, we need to discover the smallest viable values of a and b such that their product is identical to 19. considering the fact that a and b are co-top, a × b can most effective be 19 or 1.

Case 1: a × b = 1

If a × b = 1, then a = 1 and b = 1, which means that that the 2 numbers are 16 and 16. but, 16 isn't divisible by using 19, so this case is not valid.

Case 2: a × b = 19

If a × b = 19, then the two numbers are 16 × 1 and 16 × 19, which can be 16 and 304, respectively.

Step 5: Verification

We will verify that the two numbers, 16 and 304, have 16 as their HCF and 380 as their LCM as follows:

HCF(16, 304) = 16 (seeing that 16 is the largest common factor of 16 and 304)

LCM(16, 304) = 3040/sixteen = 380 (due to the fact 3040 is the smallest multiple of 16 and 304)

Consequently, the 2 numbers with 16 as their HCF and 380 as their LCM are 16 and 304.

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The final factorized form of the expression (a - b) (x²-7) - (a - b) (4x + 5) is:(a - b) (x - 6) (x + 2)

Given expression is (a - b) (x²-7) - (a - b) (4x + 5).Here, (a - b) is a common factor in the given expression. We can factor out (a - b) as follows:(a - b) (x²-7) - (a - b) (4x + 5) = (a - b) [(x² - 7) - (4x + 5)] = (a - b) (x² - 4x - 12)Next, we can further factorize the expression x² - 4x - 12 as follows:x² - 4x - 12 = (x - 6) (x + 2)

Therefore, the final factorized form of the expression (a - b) (x²-7) - (a - b) (4x + 5) is:(a - b) (x - 6) (x + 2) - this is the answer.

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Answer:

12 apples in the bowl

Step-by-step explanation:

in the ratio 3 is the apples and 4 is the oranges. it says there are 16 oranges meaning the 3:4 ratio was simplified. to simplify it they divided each fruit by 4. meaning there are 12 apples

Answer:

12 apples.

Step-by-step explanation:

4 = 16 oranges.

3 = ? apples.

¾ = ?/16

16 ÷ 4 = 4

4 × 3 = 12 apples.

The empirical probability density function (pdf) and cumulative distribution function (cdf) can be constructed based on the data on movie downloads on Netflix.

To construct the pdf, we would calculate the fraction of times each movie was downloaded, and then plot this fraction as a function of the movie title. The cdf would then be calculated by summing the probabilities of downloading each movie, starting from the movie with the lowest probability and proceeding to the movie with the highest probability.

It's important to note that the pdf and cdf for which movie a random person will download will depend on the popularity of movies on Netflix at the time of download. This may change over time as new movies are added and old movies are removed and as the popularity of movies changes.

From the pdf and cdf, we can observe that the most popular movies have a higher probability of being selected, while less popular movies have a lower probability.

This is because the more times a movie is downloaded, the higher its probability of being selected by a random person. Additionally, we can see from the cdf that there is a certain probability that a person will select a movie that is less popular. This is because, while the most popular movies have a higher probability of being selected, there is always a chance that a person will select a less popular movie.

Overall, the pdf and cdf provide a way to understand the probability of a random person selecting a particular movie on Netflix, based on the popularity of movies at the time of download.

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The left cosets of {1, 11} in u(30) are: {{1, 11}, {7, 17}, {13, 23}, {19, 29}}.

Here, u(30) represents the group of integers that are relatively prime to 30 under multiplication.

To find the left cosets of {1, 11} in u(30), we need to find all the possible subsets of u(30) that are of the form a(1,11) = {a, a*11} for some integer a.

First, we can list the elements of u(30), which are: {1, 7, 11, 13, 17, 19, 23, 29}.

Next, we can choose an integer a that is relatively prime to 30 and form the subset a(1,11) as follows:

If a = 1, then a(1,11) = {1, 11}.

If a = 7, then a(1,11) = {7, 17}.

If a = 11, then a(1,11) = {11, 1}.

If a = 13, then a(1,11) = {13, 23}.

If a = 17, then a(1,11) = {17, 7}.

If a = 19, then a(1,11) = {19, 29}.

If a = 23, then a(1,11) = {23, 13}.

If a = 29, then a(1,11) = {29, 19}.

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Answer:

b

Step-by-step explanation:

we require 2 equations with the repeated 2 placed after the decimal point.

let

x = 0.22... → (1) ← multiply both sides by 10

10x = 2.22... → (2)

subtract (1) from (2) thus eliminating the repeated 2

9x = 2 ( divide both sides by 9 )

x = \(\frac{2}{9}\)

Answer:

44 <= x

Step-by-step explanation:

The given terms in the question can be added by collecting like terms. So that the addition is:

8\(x^{2}\) + 5xy -5\(y^{2}\)

The addition of terms involve adding like terms so as to simplify the final expression.

Thus considering the terms in the given question, the addition would give:

(8\(x^{2}\) + 7xy - 6\(y^{2}\)) + (4

open the brackets and collect like terms to have;

8\(x^{2}\) + 7xy - 6\(y^{2}\) + 4

8\(x^{2}\) + 4

8\(x^{2}\) + 5xy -5\(y^{2}\)

Therefore on adding the given terms, the expression derived as answer is:

8\(x^{2}\) + 5xy -5\(y^{2}\)

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According to the given information for the normal distribution, we can expect that the corresponding z-score may be negative.

It is given to us that -

The normal distribution has mean 11

The normal distribution has data value of 9

We have to find out if the corresponding z-score is to be positive or negative.

The z-score calculates the number of standard deviations higher or lower the mean, the data point is present.

The formula for z-score is given by -

\(z=\frac{data point - mean}{standard deviation}\)

Although we do not have the value of standard deviation in the question, we can see that the value of

data point - mean = 9-11 = -2

So, we can see that the numerator of the z-value formula is already negative.

Thus, we can expect that the corresponding z-score may be negative according to the information provided for the normal distribution.

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To find the point P on the line y=3x that is closest to the point (50,0), we need to use the formula for the distance between a point and a line. This formula is:

distance = |Ax + By + C| / sqrt(A^2 + B^2)

Where A, B, and C are the coefficients of the equation of the line in the form Ax + By + C = 0.

In this case, the equation of the line y=3x can be written as -3x + y = 0. So we have:

A = -3, B = 1, C = 0

Now we can plug in the values of (50,0) and solve for x and y to find the point P that is closest to it:

distance = |-3x + y| / sqrt((-3)^2 + 1^2)
distance = |-3x| / sqrt(10)

To minimize the distance, we need to minimize |-3x|. This occurs when x = 0. So the point P that is closest to (50,0) is the point (0,0).
To find the point P on the line y = 3x that is closest to the point (50, 0), we need to minimize the distance between P and (50, 0).

Let P(x, y) be a point on the line y = 3x. The distance between P and (50, 0) is given by the distance formula:

D = sqrt((x - 50)^2 + (y - 0)^2)

Since y = 3x, we can rewrite the distance formula as:

D = sqrt((x - 50)^2 + (3x)^2)

To minimize the distance, we can minimize the square of the distance (D^2), as it avoids dealing with the square root:

D^2 = (x - 50)^2 + (3x)^2

Now, we can find the derivative of D^2 with respect to x and set it to 0 to find the critical points:

d(D^2)/dx = 2(x - 50) + 2(3x)^2 * 6x = 0

Solve this equation to find the x-coordinate of the point P. Then, use y = 3x to find the corresponding y-coordinate. Finally, you'll have the coordinates of the point P that is closest to (50, 0).

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The double asterisk operator (**) is used to raise the radius to the power of 3, which represents r³ in the formula.

To compute the volume of a sphere, the given formula is used. It is: volume of a sphere = (4.0 / 3.0) πr³ where r is the radius of the sphere.

Therefore, to find the volume of the sphere given the sphere_radius and pi, the formula above is used, as shown below: sphere_volume = (4.0 / 3.0) * pi * sphere_radius**3

where sphere_radius is the given radius of the sphere and pi is the constant pi.

The double asterisk operator (**) is used to raise the radius to the power of 3, which represents r³ in the formula.

Pi (π) is a mathematical constant that represents the ratio of the circumference of a circle to its diameter. It is an irrational number, which means it cannot be expressed as a simple fraction or as a finite decimal. The decimal representation of pi goes on infinitely without repeating.

The value of pi is approximately 3.14159, but it is typically rounded to 3.14 for simplicity in calculations. However, to maintain accuracy, mathematicians and scientists often use more decimal places, such as 3.14159265359, depending on the level of precision required for their calculations.

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Full question:

Given sphere_radius and pi, compute the volume of a sphere and assign to sphere_volume. Volume of sphere = (4.0 / 3.0) π r3

The length of the arc of the circle is 19.6 inches

We can find the length of the arc using the formula;

where ;

Substituting these values;

Answer:

Since there is one value of y for every x in the relation above, yes this relation is a function.

I hope this helped you get to your answer, or at least help you out a bit. Thanks and have a great day of learning!

The number stored in N7:3 that will cause output PL1 to be energized is 170 (option c).

To determine which of the numbers stored in N7:3 will cause output PL1 to be energized when subtracting 2 from each number, we need to perform the subtraction and check the result.

Let's subtract 2 from each number:

a) 048 - 2 = 046

b) 124 - 2 = 122

c) 172 - 2 = 170

d) 325 - 2 = 323

Based on the subtraction, the result that matches "2-2" is 170. Therefore, the number stored in N7:3 that will cause output PL1 to be energized is 170 (option c).

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\(f(s) = ℒ{(4t+1)(t-1)} = 8/s^3 - 3/s^2 - 1/s\) by using laplace transform for the function.

A mathematical technique for studying linear, time-invariant systems is the Laplace transform. It transforms a time-domain function into a frequency-domain function, making it simpler to analyse the behaviour of the system.

The transform is defined as the function's integral times an exponential that decays, with the integral being assessed from 0 to infinity. The Laplace transform can convert a differential equation into an algebraic equation, which can subsequently be solved more quickly, making it useful for solving differential equations.

Along with other disciplines, the Laplace transform has uses in signal processing, electrical engineering, and control theory. It is an effective technique for deconstructing complicated systems and comprehending their evolution.

To find f(s), we first need to apply the Laplace transform to the function (4t+1)(t-1), denoted by ℒ{(4t+1)(t-1)}. Using the linearity property of the Laplace transform, we can split this into two separate transforms:

\(ℒ{(4t+1)(t-1)} = ℒ{4t^2 - 3t - 1} = 4ℒ{t^2} - 3ℒ{t} - ℒ{1}\)

Next, we use the table of Laplace transforms to find the transforms of each term:

\(ℒ{t^n} = n!/s^(n+1)\) for any integer n greater than or equal to 0
\(ℒ{1} = 1/s\)

Using these formulas, we get:

\(ℒ{t^2} = 2!/s^3 = 2/(s^3)\\ℒ{t} = 1!/s^2 = 1/s^2\\ℒ{1} = 1/s\)

Substituting these results back into the original equation, we get:

\(ℒ{(4t+1)(t-1)} = 4(2/(s^3)) - 3(1/s^2) - (1/s)\\= 8/s^3 - 3/s^2 - 1/s\)

Therefore, \(f(s) = ℒ{(4t+1)(t-1)} = 8/s^3 - 3/s^2 - 1/s\).

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The surface area of a square pyramid net can be found by adding the areas of the base and the four triangular faces.

First, find the area of the base by multiplying the length and width of the square base.

Area of base = length × width

Next, find the area of one of the triangular faces by using the formula:

Area of triangle = 0.5 × base × height

Since there are four triangular faces in a square pyramid, multiply the area of one triangle by four to find the total area of the triangular faces.

Area of triangular faces = 4 × (0.5 × base × height)

Finally, add the area of the base and the area of the triangular faces to find the total surface area of the square pyramid net.

Surface area = Area of base + Area of triangular faces

= (length × width) + (4 × (0.5 × base × height))

= (length × width) + (2 × base × height)

So, the surface area of a square pyramid net is the sum of the area of the base and the area of the four triangular faces.

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Answer:

Answer: 3) Table B, because the given condition is that the person has a flower garden.

Step-by-step explanation:

In Table A, there are frequencies by column .

The columns of the table are "Vegetable Garden" and "No Vegetable Garden".

It means if we assume some one as a Vegetable Garden or No Vegetable Garden as initial condition, then Table A works.

In Table B, there are frequencies by rows .

The columns of the table are "Flower Garden" and "No Flower Garden".

It means if we assume some one as a Flower Garden or No Flower Garden as initial condition ,then Table B works.

Hence, Table B could be used to answer the given question  because the given condition is that the person has a flower garden.

Step-by-step explanation:

Answer:

c

Step-by-step explanation:

Answer:

No it is not possible

Step-by-step explanation:

The formula is (360/n)

and 360 is not completely divisible by 25.