A deli sells 640 sandwiches per day at a price of $8 A market survey shows that every$0.10 reduction in the price ,20 more sandwiches will be sold. How much should the deli charge in order to maximize revenue? The deli should charge $ for a sandwich revenue. (Round to the nearest cent as needed.) A different market survey shows that for every$0.20 reduction in the original $8 price,5 more sandwiches will be sold Now how much should the deli charge in order to maximize revenue? Now the deli should charge $ for a sandwich to maximize revenue. (Round the nearest cent as needed.)

An equation for the translation of y = 6/x that has the asymptotes x =4 and y = 5 is y = 6/(x - 4) + 5.

Given,

Vertical asymptotes x = 4

Horizontal asymptotes y = 5

The graph of y = 6/x has vertical asymptote x = 0 (the y-axis) and horizontal asymptote y = 0 (the x-axis).

An asymptote is a line to which the curve converges.

When the resulting independent variable is zero, the vertical asymptote occurs.

The values of the dependent variable coincide with the horizontal asymptote when the independent variable diverges to plus or minus infinity.

The translation formula is y = 6/(x - 4) + 5

Therefore, the equation is y = 6/(x - 4) + 5.

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The number of triangles that can be drawn with the given side lengths is equal to three.

In Mathematics, a triangle is a two-dimensional (2D) geometric shape that comprises three (3) sides, three (3) vertices and three (3) angles only.

In order to determine the magnitude of angle A (m∠A), angle B (m∠B), and angle C (m∠C) in this triangle with the adjacent, opposite and hypotenuse side lengths given, we would have to apply the law of cosine:

C² = A² + B² - 2(A)(B)cosθ

Where:

A, B, and C represent the side lengths of a triangle.

In this context, we can reasonably infer and logically deduce that three (3) triangles can be drawn based on the given side lengths because they would each have three different angle measures.

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The equation for the circle described is:

(x + 2)² + (y - 10)² = 169

For a circle whose center is at (a, b) and has a radius of R units, the equation can be written as:

(x - a)² + (y - b)² = R²

Here we know that the center is at (-2, 10), and the circle contains the point (10, 5), then the radius of the circle is:

R = √( (-2 - 10)² + (10 - 5)²)

R = √( 144 + 25)

R= √169

Then the equation for the circle is:

(x + 2)² + (y - 10)² = 169

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Answer: $8.75

Step-by-step explanation:

An 8% increase means that the Lasagne's cost has increased to 108% of the original amount.

We can use the inverse operation.

To work out an increase of 8%, you multiply by 1.08

The inverse of multiplying by 1.08 is dividing by 1.08

Original price = Price after increase ÷ 1.08

                       = $9.45 ÷ 1.08

                       =  $8.75

Therefore, the original price of the Lasagne is $8.75.

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(a) The work done by gravity is: -1177 J.

(b) The watermelon's kinetic energy is 1177 J and the speed of the is approximately 12.1 m/s.

(c) The answer to part (b) would be different if there were appreciable air resistance.

(a) The work done by gravity on the watermelon as it falls from the roof to the ground is equal to the change in the potential energy of the watermelon. The potential energy of the watermelon when it is at rest on the roof is:

U = mgh = (4.21 kg)(9.81 m/s^2)(27.8 m) = 1177 J

The potential energy of the watermelon when it hits the ground is:

U' = mgh = (4.21 kg)(9.81 m/s^2)(0 m) = 0 J

Therefore, the work done by gravity is:

W = U' - U = -1177 J

The negative sign indicates that the work done by gravity is done on the watermelon, decreasing its potential energy.

(b) (i) Just before it strikes the ground, the watermelon's kinetic energy is equal to the amount of potential energy it had when it was on the roof, since all of its potential energy has been converted to kinetic energy during the fall:

K = U = 1177 J

(ii) The kinetic energy can be expressed in terms of the velocity of the watermelon just before it hits the ground:

K = (1/2)mv^2

Solving for v, we get:

v = sqrt(2K/m) = sqrt(2(1177 J)/(4.21 kg)) = 12.1 m/s

Therefore, the speed of the watermelon just before it hits the ground is approximately 12.1 m/s.

(c) The answer to part (b) would be different if there were appreciable air resistance. Air resistance would cause the watermelon to experience a force opposite to its direction of motion, which would cause a loss of kinetic energy and a decrease in speed.

It would also cause the work done by gravity to be less than the change in potential energy, since some of the potential energy would be converted to heat due to air resistance.

The exact values of the work done and the final kinetic energy and speed would depend on the amount of air resistance and the exact nature of its effects.

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Answer: 140 units

Step-by-step explanation:

     To find the perimeter, we will add all four side lengths together. However, you likely notice we are missing one. We will use the Pythagorean theorem to solve for this missing side length. I have attached a picture of where we got this right triangle from. We find this side length is 39 units.

           a² + b² = c²

           36² + (40 - 25)² = c²

           36² + 15² = c²

           1,296 + 225 = c²

           1,521 = c²

           c = [tex]\sqrt{1,521}[/tex] = 39

     Lastly, we can add all of the side lengths together.

           36 units + 25 units + 40 units + 39 units = 140 units

The shortest path connecting two points on a sphere is called a great circle, which is the circle on the surface of the sphere that has the same radius as the sphere. Therefore, Length of shortest path = θ⋅ radius

= 3π​ ⋅1 = 3π

​

Since the two points are on a unit sphere, the radius of the sphere is 1. Therefore, the length of the shortest path between $a$ and $b$ is equal to the angle between them, measured in radians, multiplied by the radius of the sphere.

To find the angle between $a$ and $b$, we can use the dot product formula for the angle between two vectors:

cosθ = a× b/ ∥a∥∥b∥,

​where θ is the angle between the two vectors, $a\cdot b$ is the dot product of the two vectors, and $\lVert a \rVert$ and $\lVert b \rVert$ are the magnitudes of the two vectors. Since the points $a$ and $b$ are on the surface of the unit sphere, their magnitudes are both equal to 1. Therefore, the dot product of $a$ and $b$ is equal to the cosine of the angle between them: cos a⋅ b = cosθ.

Since we know the distance between $a$ and $b$ is $\sqrt 2$, we can write $b$ as $a+d$, where $d$ is the displacement vector from $a$ to $b$. Therefore,

∥a+ d∥² =∥a∥² +∥d∥² + 2a⋅d =2,

where the last equality follows from the fact that $a\cdot d = \cos\theta$ and $\lVert a \rVert = \lVert d \rVert = 1$. Solving for $\cos\theta$, we get

cosθ =a⋅ d = 1/2,

which implies that $\theta = \frac{\pi}{3}$ (since $\cos^{-1}\frac{1}{2} = \frac{\pi}{3}$). Therefore, the length of the shortest path on the sphere that connects $a$ to $b$ is

Length of shortest path = θ⋅radius = 3π​ ⋅1 = 3π

​

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If t is the expected completion time for a given activity, then the correct formula for calculating the early finish time (EF) is EF = ES + t.

This formula adds the expected completion time to the early start time (ES) to determine when the activity is expected to finish.

The other options listed, LF = LS - t and EF = LS - t, involve subtracting the expected completion time from the late start or late finish times, which would not accurately reflect the expected completion time of the activity.

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The calculated area of the shape if it is a circle with a radius of 1 1/9 inches is 2200/567 square inches

The dimension given from the question is

Dimension = 1 1/9 inches

This is a single dimension, and as such the shape whose area is to be calculated is a circle

When the dimension is rewritten as radius, we have

Radius, r = 1 1/9 inches

Calculating the area, we make use of the following formula

Area = πr²

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

Area = π * (1 1/9)²

This gives

Area = 22/7 * (10/9)²

So, we have

Area = 22/7 * 100/81

Evaluate the product

Area = 2200/567

Hence,  the area is 2200/567 square inches

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The length of the other ribbon is 10 1/2 yards.

What is a fraction?

A fraction is a mathematical expression that represents a part of a whole or a division of one quantity by another. It consists of two numbers separated by a horizontal line called a fraction bar or a vinculum.

The length of each of Victoria's blue ribbons is 1 3/4 yd. We can represent this length in terms of yards as:

1 3/4 = 7/4 yards

If another ribbon is 6 times as long as one of Victoria's ribbons, then its length would be:

6 * (7/4) yards = 42/4 yards = 10 1/2 yards

Therefore, the length of the other ribbon is 10 1/2 yards.

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Answer: When graphed, the line will cross through the letter "Y" at the point (−3, 0).

Step-by-step explanation: To graph the equation y = x + 3, we can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is 1 and the y-intercept is 3.

We can start by plotting the y-intercept at the point (0, 3). Then, since the slope is 1, we can move up 1 unit and to the right 1 unit to find another point on the line. If we repeat this process, we can find more points on the line and connect them to create the graph of the equation.

When graphed, the line will cross through the letter "Y" at the point (−3, 0).

Answer:

[tex]\large\boxed{\tt m \angle A = 40^{\circ}.}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked to identify} \ \tt m \angle A. \ \textsf{We are given a diagram which tells us the shape}[/tex]

[tex]\textsf{is a Triangle.}[/tex]

[tex]\large\underline{\textsf{What is a Triangle?}}[/tex]

[tex]\textsf{A Triangle is a 2D Shape that has 3 sides and 3 angles. We know this since}[/tex]

[tex]\textsf{Triangle begins with "Tri" which means 3, hence there's 3 sides that make up 3}[/tex]

[tex]\textsf{angles. The Sum of the 3 angles is equal to 180}^{\circ}. \ \textsf{This means that Triangle can}[/tex]

[tex]\textsf{have many unique differences with its angles, making it an entirely different}[/tex]

[tex]\textsf{Triangle.}[/tex]

[tex]\textsf{We know that the angles' measures add up to 180}^{\circ}. \ \textsf{We can now form an equation.}[/tex]

[tex]\tt 180^{\circ} = m \angle A + m \angle B + m \angle C[/tex]

[tex]\underline{\textsf{Substitute what's given;}}[/tex]

[tex]\tt 180^{\circ} = 80^{\circ} + 7 + 3x + 6x - 6.[/tex]

[tex]\large\underline{\textsf{Solving;}}[/tex]

[tex]\textsf{We identified our equation, now we are ready to solve for x. Remember that x}[/tex]

[tex]\textsf{isn't given to us, which means that we have to identify x, then substitute the}[/tex]

[tex]\textsf{value in to find m} \tt \angle A.[/tex]

[tex]\underline{\textsf{Combine All Like Terms;}}[/tex]

[tex]\tt 180^{\circ} = \boxed{\tt 80^{\circ}} + \boxed{\tt 7} + 3x + 6x \boxed{\tt- 6}[/tex]

[tex]\tt 180^{\circ} = 81^{\circ} + \boxed{\tt 3x} + \boxed{\tt6x}[/tex]

[tex]\tt 180^{\circ} = 81^{\circ} + 9x[/tex]

[tex]\textsf{Now, let's use the Subtraction Property of Equality which states that whenever}[/tex]

[tex]\textsf{2 equal expressions are subtracted by the same term, they're still equal.}[/tex]

[tex]\underline{\textsf{Subtract 81 from both sides using the Subtraction Property of Equality;}}[/tex]

[tex]\tt 180^{\circ} - 81^{\circ} = 81^{\circ} - 81^{\circ}+ 9x[/tex]

[tex]\tt 99^{\circ} = 9x[/tex]

[tex]\textsf{Let's now use the Division Property of Equality, but for dividing by the same term.}[/tex]

[tex]\underline{\textsf{Divide 9 from both sides using the Division Property of Equality;}}[/tex]

[tex]\tt \frac{99^{\circ}}{9} = \frac{9x}{9}[/tex]

[tex]\large\boxed{\tt x = 11}[/tex]

[tex]\textsf{Now that we know the value of x, we are able to find m} \tt \angle A.[/tex]

[tex]\underline{\textsf{Substitute;}}[/tex]

[tex]\tt m \angle A = 7+3(11)[/tex]

[tex]\underline{\textsf{Evaluate;}}[/tex]

[tex]\large\boxed{\tt m \angle A = 40^{\circ}.}[/tex]

These flaws make it difficult to draw valid conclusions from the survey results.

There are several major flaws in this survey:

Self-selection bias: The survey relied on readers to call in their responses, which means that only people who were motivated enough or had strong opinions on the issue would participate. This could lead to an over-representation of individuals who are strongly against atomic weapons, and the results may not be representative of the general population.

Sampling bias: The survey was conducted using a non-random sampling method, as only readers of the Oakland Tribune were surveyed. This means that the sample may not be representative of the entire population, and the results may not be generalizable to the wider population.

Loaded question: The question itself is biased and may influence respondents' answers. The question includes the statement "that could kill millions of innocent people," which could influence respondents to answer in a certain way, regardless of their true opinions on atomic weapons.

Lack of control: There is no information provided on how the survey was administered, how responses were recorded, or how the data was analyzed. This lack of control means that the survey results may not be accurate or reliable.

In summary, the major flaws in this survey include self-selection bias, sampling bias, a loaded question, and lack of control. These flaws make it difficult to draw valid conclusions from the survey results.

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49 is the amount that would need to be added in order to "complete the square".

Given the quadratic equation below:

x² + 14x + 3=0

In order to complete the square, we will add the square of the half of the coefficient of x to both sides.

Coefficient of x = 14

Half of coefficient of x = 14/2 = 7

Square of the half of coefficient of x = 7²

Square of the half of coefficient of x = 49

Hence the number that would have to be added to "complete the square" is 49.

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The intercepts are at the origin (0, 0, 0) and the graph is an elliptical paraboloid opening upwards with its vertex at the origin (0, 0, 0).

a. To find the intercepts with the three coordinate axes, we can set the other two variables to zero in the given equation.

x-intercepts:
Set y = 0 and z = 0:
[tex]0 = x^2 + 289(0)[/tex]
x^2 = 0
x = 0

y-intercepts:
Set x = 0 and z = 0:
[tex]0 = (0)^2 + 289y^2[/tex]
[tex]y^2 = 0[/tex]
y = 0

z-intercepts:
Set x = 0 and y = 0:
[tex]z = (0)^2 + 289(0)[/tex]
z = 0

The intercepts are at the origin (0, 0, 0).

b. To find the equations of the xy-, xz-, and yz-traces, we can set one of the variables to zero in the given equation.

xy-trace (z = 0):
[tex]0 = x^2 + 289y^2[/tex]

xz-trace (y = 0):
[tex]z = x^2 + 289(0)[/tex]
[tex]z = x^2[/tex]

yz-trace (x = 0):
[tex]z = (0)^2 + 289y^2[/tex]
[tex]z = 289y^2[/tex]

c. To sketch a graph of the surface, note that the given equation is a quadric surface of the form z = x^2 + 289y^2, which represents an elliptical paraboloid. Since the coefficients of x^2 and y^2 are both positive, the paraboloid opens upwards.

In summary:
a. The x-intercept is located at x = 0.
b. The equations of the traces are:
[tex]xy-trace: 0 = x^2 + 289y^2[/tex]
  - xz-trace[tex]: z = x^2[/tex]
  - yz-trace:[tex]z = 289y^2[/tex]
c. The graph is an elliptical paraboloid opening upwards with its vertex at the origin (0, 0, 0).

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The is_number() function tests whether a variable or cell contains a numeric data type. So, correct option is A.

This function returns TRUE if the argument is a number and FALSE otherwise.

The is_numeric() function is not a standard function in most programming languages. However, some programming languages may have similar functions or methods to test for numeric data types.

On the other hand, the is_digit() function tests whether a character is a digit or not. This function returns TRUE if the character is a digit and FALSE otherwise.

This function can be useful in scenarios where you need to check if a character is a number, such as validating user input in a form. However, it only works for individual characters and does not test for numeric data types of variables or cells.

So, correct option is A.

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Answer:

Mean is the average of all of the numbers. Median is the middle number, when in order.

By answering the presented question, we may conclude that P(990< X <1150) = normal distribution  P[(990-1000)/100 < Z < (1150-1000)/100] = P(-1.0 < Z < 1.5) = 0.7745

The normal distribution is an example of a continuous probability distribution, in which the majority of data points cluster at the middle of the range and the remaining ones drop symmetrically towards one of the extremes. The mean of the distribution is another term for the range's centre. According to the normal distribution, also known as the Gaussian distribution, which is symmetrical around the mean, data that are close to the mean are more frequent than data that are far from the mean. I'm utilising heuristics in normal distribution to obtain a student's SAT scores from a new exam prep course. The data exhibit a normal distribution with a mean (M) of 1150 and a standard deviation (SD) of 150.

Question 1:

a) P(Z<1.72) = 0.9582 (using standard normal distribution table or calculator)

b) P(Z>1.1) = 0.1357

c) P(1.1<Z<1.70) = 0.0808

Question 2:

a) P(X<1070) = P(Z<(1070-1000)/100) = P(Z<0.7) = 0.7580

b) P(990< X <1150) = P[(990-1000)/100 < Z < (1150-1000)/100] = P(-1.0 < Z < 1.5) = 0.7745

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Long division is a method of dividing two numbers that involves writing out the division problem in a step-by-step process, using repeated subtraction and multiplication to find the quotient and remainder.

To perform long division, the dividend (the number being divided) is written on top of the division symbol, with the divisor (the number doing the dividing) written below it. The goal is to divide the dividend by the divisor, and write the quotient (the answer) above the division symbol, with any remainder written to the right of the quotient.

The steps involved in long division include:

Divide: Determine how many times the divisor can be subtracted from the first digit (or digits) of the dividend, and write this above the division symbol as the first digit(s) of the quotient.

Multiply: Multiply the quotient digit(s) by the divisor, and write the result below the corresponding digits of the dividend.

Subtract: Subtract the product from the previous step from the dividend, and write the result below the line.

Bring down: Bring down the next digit of the dividend (if any) and write it next to the result from the previous step.

Repeat: Repeat the process until there are no more digits to bring down, and the remainder (if any) is less than the divisor.

Long division can be used to divide any two numbers, including decimals and fractions. It is an important skill in mathematics, and is often used in algebra and other advanced math courses.

Brainliest?

Answer:

when normaldivision doesnt work do this

Step-by-step explanation:

truse me it works everytime

The larger hose will take 23.33 minutes to fill the pool by itself. This can be determined by solving a system of equations.

It has two expressions of numbers, variables, and operators (such as +, -, x, /) are equal to each other. An equation can be written in a variety of forms, including algebraic, graphical and verbal.

Let x be the time it takes for the larger hose to fill the pool, and let y be the time it takes for the smaller hose to fill the pool. We can set up two equations to solve for x and y.

x + y = 35

(2)(y) = x

y = (1/2)x

We can then substitute this value of y into the first equation to solve for x.

x + (1/2)x = 35

(3/2)x = 35

x = (35)(2/3)

x = 23.33

Since x is the time it takes for the larger hose to fill the pool by itself, we can conclude that it will take 23.33 minutes for the larger hose to fill the pool by itself.

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The equation of the tangent plane to the given surface at the point (1,2,-4) is 16x + 3y - z = 27.

To find the equation of the tangent plane to the given surface at the point (1,2,-4), we first need to find the partial derivatives of the surface equation with respect to x and y:
∂z/∂x = 4xy^2
∂z/∂y = 4x^2y - 5

Then, we can evaluate these partial derivatives at the point (1,2,-4):
∂z/∂x(1,2) = 4(1)(2^2) = 16
∂z/∂y(1,2) = 4(1^2)(2) - 5 = 3

Next, we can use these values to find the equation of the tangent plane using the point-normal form:
z - z0 = ∂z/∂x(x0,y0)(x - x0) + ∂z/∂y(x0,y0)(y - y0)

Plugging in the values we found, we get:
z - (-4) = 16(x - 1) + 3(y - 2)

Simplifying, we get the equation of the tangent plane:
16x + 3y - z = 27

Therefore, the equation of the tangent plane to the given surface at the point (1,2,-4) is 16x + 3y - z = 27.

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To investigate the relationship between the amount of kinetic energy in the puck-earth system and the velocity of the puck, we need to collect data for the puck mass. The mathematical model for this relationship can be derived using the equation for kinetic energy, which is KE=1/2mv^2, where KE is the kinetic energy, m is the mass of the puck, and v is the velocity of the puck.

To collect data, we can conduct experiments where we launch the puck at different velocities and measure the kinetic energy using a device such as a force sensor or accelerometer. We can then plot the data on a graph, with velocity on the x-axis and kinetic energy on the y-axis.

By analyzing the data, we can derive a mathematical model that describes the relationship between kinetic energy and velocity. The model is expected to show that the kinetic energy increases with the square of the velocity, meaning that a small increase in velocity can result in a large increase in kinetic energy. This model can be useful in predicting the kinetic energy of a moving object, and in designing systems that require the transfer of kinetic energy.

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Answer:

35% when awake 65% when asleep

Step-by-step explanation:

Answer:

59%

Step-by-step explanation:

divide sleep hours by total hours

295 ÷ 500= .59 = 59%

A

I first divided the numbers and got 61.

On the last one for 40-39= 1

That 1 will be the numerator and the 39, which is what you divided the 2,380 from goes underneath, which is the denominator.

If that doesn't make sense, here is a better explanation:

39 divided by 2380 = 61

The 61 will go to the right and keep it where it is.

That 1 after subtracting your numbers will go on top which is the numerator

The 39 divided from 2380 goes at the bottom as the denominator.

[Note: The numerator can never be bigger than the denominator!]

I hope that helps!

Sure! Since x is rational and x = -33 + 234, we can write x as a fraction in simplest form:

x = (201)/1

To find the minimum value of x, we want to make the denominator as small as possible. One way to do this is to repeatedly subtract 33 from the numerator until we get to a multiple of 33.

(201)/1 = (168 + 33)/1 = (135 + 33 + 33)/1 = (102 + 33 + 33 + 33)/1 = ...

Continuing this pattern, we see that x can be written as:

x = (33n + 201)/1

where n is a non-negative integer. To minimize x, we want to choose the smallest possible value of n, which is 6.

Therefore, the minimum value of x is:

x = (33(6) + 201)/1 = 399

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S and X are the poair at the same structural organization

Structurаl orgаnizаtion in humаns аnd plаnts refers tо thе hierаrсhy оf biologicаl structures аnd systems thаt mаkе up thеse orgаnisms. Bоth humаns аnd plаnts еxhibit levels оf orgаnizаtion, stаrting frоm thе bаsic building blocks (аtоms аnd mоlecules) tо thе complеx orgаn systems in humаns аnd рlаnt structures. Нere's а brief compаrison оf structurаl orgаnizаtion in humаns аnd plаnts:

Atоms аnd mоlecules: Тhe bаsic building blocks оf аll living orgаnisms, аtоms сombine tо form mоlecules, whiсh furthеr сombine tо form more complеx structures.

Cells: Тhe fundаmentаl unit оf life. In both humаns аnd plаnts, сells cаrry оut essentiаl functions аnd cоme in diffеrеnt typеs, eаch with speсifiс roles.

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To convert the binary numbers to decimal:
a. 11111 (binary) = 1(2⁴) + 1(2³) + 1(2²) + 1(2¹) + 1(2⁰) = 31 (decimal)
b. 101010101 (binary) = 341 (decimal)
c. 100000001 (binary) = 257 (decimal)

To convert the hexadecimal numbers to binary:
a. 80E (hex) = 1000 0000 1110 (binary)
b. ABBA (hex) = 1010 1011 1011 1010 (binary)
c. 135AB (hex) = 0001 0011 0101 1010 1011 (binary)

To convert 7345321 (decimal) to binary, and 101011011 (binary) to octal:
(7345321)10 = 11100001011011001001 (binary)
(101011011)2 = 2533 (octal)

Using the Euclidean algorithm to find the GCD:
a. GCD(12, 18) = 6
b. GCD(1001, 1331) = 7
c. GCD(1000, 5040) = 40
d. GCD(111, 201) = 3
e. GCD(12345, 54321) = 3

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a) To find the number of customers who have both checking and money market savings accounts, we need to use the principle of inclusion and exclusion.

Let A be the set of customers with checking accounts, B be the set of customers with money market savings accounts, and C be the set of customers with both checking and money market savings accounts. Then we have:

|A ∪ B| = |A| + |B| - |A ∩ B|

We know that |A| = 168, |B| = 120, and |A ∩ B| = ? (what we need to find)

We can also use the fact that the total number of customers is 250, so:

|A ∪ B| = 250 - |A' ∩ B'|

where A' and B' are the complements of A and B, respectively. Since no customer is allowed to have both regular savings and money market savings accounts, we know that:

|B ∩ C| = 0

So we can use the principle of inclusion and exclusion again to find:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |C ∩ A| + |A ∩ B ∩ C|

We know that |A ∪ B ∪ C| = 250 and |A|, |B|, and |C| are given. We also know that |B ∩ C| = 0 and we can use the fact that:

|A' ∩ B' ∩ C'| = |(A ∪ B)' ∩ (B ∪ C)' ∩ (C ∪ A)'|

= |(A' ∩ B') ∪ (B' ∩ C') ∪ (C' ∩ A')|

= |A' ∩ B'| + |B' ∩ C'| + |C' ∩ A'| - |A' ∩ B' ∩ C'|

We know that |A' ∩ B'| = |(A ∪ B)'| = 250 - |A ∪ B| and |C' ∩ A'| = |(C ∪ A)'| = 250 - |C ∪ A|. Since we already know |A ∪ B| and |C ∪ A| from the previous calculations, we can find |A' ∩ B' ∩ C'|. Then we can use the principle of inclusion and exclusion to find |A ∩ B|:

|A ∩ B| = |A| + |B| + |C| - |A ∪ B ∪ C| + |A' ∩ B' ∩ C'|

Substituting the given values, we get:

|A ∩ B| = 168 + 120 + 0 - 250 + (250 - 168 - 75 - 120 + |A ∩ B|)

Solving for |A ∩ B|, we get:

|A ∩ B| = 26

Therefore, there are 26 customers who have both checking and money market savings accounts.

b) To find the number of customers who have a checking account but no savings account, we can use the same principle of inclusion and exclusion. Let D be the set of customers with no savings account. Then we have:

|A ∩ D| = |A| - |A ∩ B ∪ A ∩ C|

We know that |A| = 168 and we just found |A ∩ B| = 26. To find |A ∩ C

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a) The 2x2 matrices A≠I such that A^2 = I is:

b) The determinant of A must be ±1, and all such determinants are possible.

(a) To find three 2x2 matrices A≠I such that A^2 = I, we can start with the matrix:

A = [0 1

1 0]

which is known as the permutation matrix. We can verify that A^2 = I:

A^2 = [0 1] [0 1]

[1 0] * [1 0]

= [1 0] [0 1]

[0 1] [1 0]

= I

Another example is:

B = [1 1

1 -1]

which also satisfies B^2 = I:

B^2 = [1 1] [1 1]

[1 -1] * [1 -1]

= [2 0] [1 -1]

[0 2] [1 -1]

= I

A third example is:

C = [-1 0

0 1]

which satisfies C^2 = I:

C^2 = [-1 0] [-1 0]

[ 0 1] * [ 0 1]

= [ 1 0] [-1 0]

[ 0 1] [ 0 1]

= I

(b) To characterize exactly which determinants are possible for such a matrix, we can observe that for a 2x2 matrix A, we have:

det(A^2) = det(A)^2

Since A^2 = I, we have det(A^2) = det(I) = 1, and so det(A)^2 = 1. Therefore, det(A) = ±1.

To prove that this conjecture is correct, we can use the fact that the determinant of a matrix is equal to the product of its eigenvalues. For a 2x2 matrix A, the eigenvalues λ1 and λ2 satisfy:

λ1 + λ2 = tr(A) = a11 + a22

λ1 λ2 = det(A) = a11 a22 - a12 a21

Since A is its own inverse, we have A^2 = I, which implies that A has eigenvalues ±1. Therefore, we have two cases:

Case 1: λ1 = 1, λ2 = -1

In this case, we have:

a11 + a22 = 1 - 1 = 0

a11 a22 - a12 a21 = 1

Solving these equations, we obtain:

a11 = a22 = 0, a12 a21 = -1

Therefore, the determinant of A is det(A) = 0 or ±1.

Case 2: λ1 = λ2 = 1

In this case, we have:

a11 + a22 = 2

a11 a22 - a12 a21 = 1

Solving these equations, we obtain:

a11 = a22 = ±1, a12 a21 = 0

Therefore, the determinant of A is det(A) = ±1.

Thus, we have shown that for a matrix A that is its own inverse, the determinant of A must be ±1, and that all such determinants are possible.

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The value of b include the following: C. 2√21.

In order to determine the value of b, we would apply Cathetus theorem (leg rule), which states that each leg of a right-angled triangle is the geometric mean that's directly proportional between the hypotenuse and the part of the hypotenuse that's directly below the leg.

By applying Cathetus theorem (leg rule) to this right-angled triangle, we have the following:

b² = m × a

b² = (8 + 6) × 6

b² = 14 × 6

b² = 84

b = √84

b = 2√21 units.

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