Solve for a and b: 2a - 3b=5 -3a+b=3​

the expected value of the sum of two fair 6-sided dice is 7.

The possible values of the sum of two fair 6-sided dice range from 2 (when both dice show a 1) to 12 (when both dice show a 6). The probability of each possible sum can be calculated using a probability distribution table or by listing all the possible outcomes and counting them.

Here is the probability distribution table:

Sum Number of Outcomes Probability

2 1 1/36

3 2 2/36

4 3 3/36

5 4 4/36

6 5 5/36

7 6 6/36

8 5 5/36

9 4 4/36

10 3 3/36

11 2 2/36

12 1 1/36

The expected value of the sum, denoted E(X), is the sum of each possible value of X multiplied by its probability, or:

E(X) = 2*(1/36) + 3*(2/36) + 4*(3/36) + 5*(4/36) + 6*(5/36) + 7*(6/36) + 8*(5/36) + 9*(4/36) + 10*(3/36) + 11*(2/36) + 12*(1/36)

E(X) = 7

what is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain. A probability of 0.5 (or 50%) means the event is just as likely to occur as it is not to occur.

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The type of error that could have occurred in this scenario is a Type II error. A Type II error occurs when the null hypothesis is not rejected, even though it is false.

In this case, the null hypothesis is that the proportion of defective bulbs in the shipment is greater than or equal to 5%. By not rejecting this null hypothesis, the company is accepting the possibility that the shipment has a high proportion of defective bulbs, when in reality it does not. This can lead to significant costs for the company, such as the cost of returning the shipment or the cost of replacing defective bulbs in the future.

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So, Lena sent 24 messages, Alan sent 32 messages, and Bill sent 48 messages.

Let's denote the number of messages Lena, Alan, and Bill sent as L, A, and B, respectively. We are given the following information:

L + A + B = 104 (Total messages)

L = A - 8 (Lena sent 8 fewer messages than Alan)

B = 2L (Bill sent 2 times as many messages as Lena)

Now, we'll use the second equation to express A in terms of L:

A = L + 8

Next, substitute the expressions for A and B from equations 2 and 3 into equation 1:

L + (L + 8) + 2L = 104

Combine like terms:

4L + 8 = 104

Subtract 8 from both sides:

4L = 96

Divide by 4:

L = 24

Now that we have the number of messages Lena sent, we can find the number of messages Alan and Bill sent:

A = L + 8 = 24 + 8 = 32

B = 2L = 2 * 24 = 48

Given the inverse demand function p = 100 - 20q for toasters and a price of $35, we can find the consumer surplus.

First, we'll find the quantity demanded at the given price:

35 = 100 - 20q
20q = 100 - 35
q = (100 - 35) / 20
q = 65 / 20
q = 3.25

Now, to find the consumer surplus, we'll use the formula:

Consumer Surplus = (1/2) × Base × Height

The base represents the quantity (q = 3.25) and the height is the difference between the maximum willingness to pay (p = 100) and the actual price (p = 35).

Consumer Surplus = (1/2) × 3.25 × (100 - 35)
Consumer Surplus = 0.5 × 3.25 × 65
Consumer Surplus = 105.625

So, the consumer surplus is $105.63 when rounded to two decimal places.

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Answer: y = -5

Step-by-step explanation:

We move the constants to one side to get y^3 = -125.

Then, we cube root both sides to get y = -5.

Answer:

5x - 15

Step-by-step explanation:

5(x - 3) ← multiply each term in the parenthesis by the 5 outside

= 5x - 15

Explanation:

The w/4 term is the same as (1/4)w or 0.25w since 1/4 = 0.25

Because a variable is attached to this term, it is not constant. The same can be said about the -7z term as well.

The 12.5 is constant. It never changes. It will always be 12.5 no matter what the other variable terms change to.

For example, if w = 12, then the term w/4 becomes 12/4 = 3. Or if w = 24, then w/4 = 24/4 = 6 is the new value. This shows that term changing depending on the input variable.

BMED is a trapezoid and OD is equals to BD/2. Proof of these is given below.

It should be noted that to prove that BMED is a trapezoid, we need to show that either BM || ED or BM || DE. We will show that BM || DE.

Since O is the midpoint of AM, we have OA = OM, and thus triangle ODM is isosceles. Therefore, OD = DM.

Also, since E is the midpoint of DC, we have DE || AB (because AB is also a mid-segment of triangle ADC). Therefore, angle ADE = angle ABC.

Using the fact that angles in a triangle add up to 180 degrees, we have:

angle EDC + angle ADE + angle CED = 180 degrees

Substituting angle ADE = angle ABC, we get:

angle EDC + angle ABC + angle CED = 180 degrees

But angle CED = angle BCD (because they are alternate interior angles formed by transversal BD cutting parallel lines DC and AB). Therefore:

angle EDC + angle ABC + angle BCD = 180 degrees

This means that angles BCD, ABC, and EDC are all on the same line, and thus angle BCD + angle ABC = 180 degrees.

Now, using the fact that angles in a triangle add up to 180 degrees, we have:

angle BAC + angle ABC + angle ACB = 180 degrees

Substituting angle BCD + angle ABC = 180 degrees, we get:

angle BAC + angle BCD + angle ACB = 180 degrees

Since triangle BCD is isosceles (because BD is an angle bisector), we have angle BCD = angle CBD. Therefore:

angle BAC + angle CBD + angle ACB = 180 degrees

But angle ACD = angle ACB + angle CBD, so:

angle BAC + angle ACD = 180 degrees

This means that triangle ABC is similar to triangle ACD.

Now, since BM is a median of triangle ABC, we have:

BM = 1/2 AC

But triangle ADC is similar to triangle ABC, so:

AC = 2 AD

Therefore:

BM = AC/2 = AD

So we have BM || DE, and BM = DE, which means that BMED is a trapezoid.

2. To prove that D is the midpoint of AE, we will use the fact that BMED is a trapezoid. Since BM || DE, we have:

BD/DM = BE/EM

But DM = OD (since triangle ODM is isosceles) and EM = DC/2 = AC/4 (since E is the midpoint of DC and AC is a mid-segment of triangle ADC).

Therefore:

BD/OD = BE/(AC/4)

But we also have:

BE = BD + DE = BD + BM = 2BD

Substituting this into the previous equation, we get:

BD/OD = 2BD/(AC/4)

Simplifying:

BD/OD = 8BD/AC

This means that:

OD = AC/8

But we also know that AC = 2AD, so:

OD = AD/4

Since O is the midpoint of AM, we also have:

OD = OM = AM/2

Therefore:

AD/4 = AM/2

This means that:

AD = 2AM/4 = AM/2 = OD

So D is the midpoint of AE, and CD = 2AD.

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In a triangle ABC. [AM] is the median relative to [BC] and O is the midpoint of [AM]. (BO) cuts [AC] at D. Let E be the midpoint of [DC]. 1) Prove that BMED is a trapezoid. 2) Prove that D is the midpoint of [AE); deduce that CD = 2AD. 3) Prove that OD == BD.

This means that c must be 1/2, since it is the only value in the interval (0,2) where f'(c) = -1/2, which satisfies the equation given by the Mean Value Theorem.

To apply the Mean Value Theorem, we need to check if f(x) is continuous on [0,2] and differentiable on (0,2).

[tex]f(x) = x^3 - 2x^2[/tex] is a polynomial function, so it is continuous and differentiable everywhere.

Now, we can use the Mean Value Theorem, which states that there exists a number c in (0,2) such that:
[tex]f'(c) = (f(2) - f(0))/(2-0)[/tex]

First, let's find f'(x):

[tex]f'(x) = 3x^2 - 4x[/tex]

Now, we can solve for c:

f'(c) = 3c^2 - 4c

f(2) = 2^3 - 2(2^2) = -4

f(0) = 0^3 - 2(0^2) = 0

(f(2) - f(0))/(2-0) = -4/2 = -2

So, we need to solve the equation 3c^2 - 4c = -2 for c.

Rearranging, we get:

3c^2 - 4c + 2 = 0

Using the quadratic formula, we get:

c = (4 ± sqrt(16 - 24))/6

c = (4 ± 2i)/6

Since the interval is [0,2], we only need to consider real solutions.
[tex]f'(c) = 3c^2 - 4c\\f(2) = 2^3 - 2(2^2) = -4\\f(0) = 0^3 - 2(0^2) = 0\\(f(2) - f(0))/(2-0) = -4/2 = -2\\[/tex]
Therefore, c = 4/3 is not a valid solution.

To choose between the remaining options, we can test if f'(c) is positive or negative.

For c = 0, f'(0) = 0 - 0 = 0.

For [tex]c = 1/2, f'(1/2) = 3(1/2)^2 - 4(1/2) = -1/2.[/tex]

For[tex]c = 1, f'(1) = 3(1)^2 - 4(1) = -1.[/tex]
Therefore, f'(c) is negative on the interval [0,1/2] and positive on the interval [1/2,2].

This means that c must be 1/2, since it is the only value in the interval (0,2) where f'(c) = -1/2, which satisfies the equation given by the Mean Value Theorem.

Therefore, the answer is (b) 1/2.

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Answer:

a) m || n

b) p || q, and m must be perpendicular to both p and q.

c) p || q

d) p || q

e) m || n and p || q

Answer: 3/8 >2/6

Step-by-step explanation:

The value of k is 20.

An equation is a mathematical statement that shows that two expressions are equal. It contains one or more variables and may involve arithmetic operations such as addition, subtraction, multiplication, division, or exponents. The goal of solving an equation is to determine the value of the variable that makes the equation true. Equations are commonly used in algebra and other mathematical fields to model real-world situations and solve problems. They can be written in various forms, including standard form, slope-intercept form, point-slope form, and general form, among others.

Since the sum of two angles on a straight line is always 180 degrees, we can set up an equation:

First angle + Second angle = 180

112 + 3k + 8 = 180

Simplifying the equation, we get:

3k + 120 = 180

Subtracting 120 from both sides, we get:

3k = 60

Dividing both sides by 3, we get:

k = 20

Therefore, the value of k is 20.

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Answer:

2(1/2)(8)(15) + 17(2) + 8(2) + 15(2)

= 200 cm^2

Answer:

Yes

Step-by-step explanation:

The definition of dimension is Length, Width, and Height so if he doubles the dimensions it will be double. The volume is 630.

Answer:

H = 45 ½ ÷ 6 ½

Step-by-step explanation:

The formula  for the area of the base of the box is L * W.

The formula for the volume of the box is L * W * H

Therefore dividing 45 ½ (volume of box) by 6 ½ (area of base) will give you the height of the box

the nicknames of the four friends are:

Andre: Boss

Bilal: Buzz

Glenna: Cosmo

Juanita: Tiger

From clue 1, we realize that Juanita's moniker isn't Tiger or Buzz, and Tiger's epithet isn't Chief or Cosmo. Along these lines, Juanita's epithet should be Cosmo or Chief, and Tiger's moniker should be Buzz or Tiger.

From clue 2, we realize that Glenna's moniker isn't Chief or Buzz, and Tiger's epithet isn't Cosmo or Tiger (currently disposed of in the past step). In this way, Glenna's moniker should be Cosmo, and Tiger's epithet should be Buzz.

From clue 3, Cosmo's nickname has already been determined to be Glenna, and we are aware that Andre's nickname cannot be Boss, Buzz, or Tiger (all of which have been eliminated). Boss has to be Andre's nickname.

From clue 4, Cosmo's nickname has already been identified as Glenna, and we are aware that Juanita's nickname cannot be Buzz or Tiger (both of which have already been eliminated). Therefore, Juanita must go by the name Boss.

So, the nicknames of the four friends are:

Andre: Boss

Bilal: Buzz

Glenna: Cosmo

Juanita: Tiger

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Answer: 5 (first round) + 25 (second round) + 125 (third round) + 625 (fourth round) = 780

So, at the end of 4 rounds, including Jacob himself, 780 people will have gotten fliers.

Step-by-step explanation: In the first round, Jacob hands out fliers to 5 people.

In the second round, each of the 5 people from the first round hands out 5 fliers to 5 other people. So, there are now 5 x 5 = 25 people with fliers (including the original 5).

In the third round, each of the 25 people from the second round hands out 5 fliers to 5 other people. So, there are now 25 x 5 = 125 people with fliers (including the original 5 and the 25 from the second round).

In the fourth round, each of the 125 people from the third round hands out 5 fliers to 5 other people. So, there are now 125 x 5 = 625 people with fliers (including the original 5, the 25 from the second round, and the 125 from the third round).

The LCM of 63 and 42 is 126 where 63 is the required number lying in between 60 and 70.

Let the missing number be x.

The number x lies in between 60 and 70.

The LCM ( Least Common Multiple) refers to the least value that is divisible by any two (or more) numbers.

Here LCM of x and 42 is 126.

Simplifying 126 we can write it as,

126 = 2*3*21

Thus, one of the number having 126 is as multiple is 42  (as already given).

The other number, that is x, having 126 as multiple lying between 60 and 70 is,

x = 3*21 = 63

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Given the above conditions in the word problem above, x is also equal to 6.5°.

The angle x is the angle between this perpendicular line and the Earth's surface. It is also equal to the angle between the tangent line and the horizontal axis.

Since the reentry angle is 6.5°, we know that the tangent line makes an angle of 6.5° with the horizontal axis. We also know that the tangent line and the perpendicular line form a right angle, so the angle between the perpendicular line and the horizontal axis is 90° - 6.5° = 83.5°.

Finally, we can subtract this angle from 90° to find x:

x = 90° - 83.5° = 6.5°

Therefore, x is also equal to 6.5°.

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The slope and derivative of a vertical tangent are both undefined due to the vertical nature of the tangent line.

The slope/derivative of a vertical tangent.
A vertical tangent occurs when a curve has a point where its tangent line is vertical.

In terms of the slope and derivative, the following is true about the vertical tangent:
Slope:

The slope of a vertical tangent is undefined because the vertical line has no run (change in x). Slope is calculated as the rise (change in y) divided by the run (change in x), and division by zero is not possible.
Derivative:

The derivative of a function at a point represents the slope of the tangent line to the curve at that point.

When the tangent is vertical, the derivative is also undefined because the slope is undefined.

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In the above isosceles trapezoid,

Since R is the midpoint of HN, HR = RN. Similarly, since T is the midpoint of KJ, KT = TJ.

Let's use these properties to write expressions for HJ and NK in terms of x:

HJ = 5x + 9

NK = 3x - 2

Since NKJH is an isosceles trapezoid, we know that HJ = NK + 2RT. Substituting the expressions we found earlier, we get:

5x + 9 = (3x - 2) + 2(3x + 6)

Simplifying this equation gives:

5x + 9 = 9x + 10

Subtracting 5x from both sides gives:

4 = 4x

Dividing both sides by 4 gives:

x = 1

Now that we know x, we can find the values of HJ, NK, and RT:

HJ = 5x + 9 = 14 cm

NK = 3x - 2 = 1 cm

RT = 3x + 6 = 9 cm

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The circle graph that correctly represents the data in the table is that with:  A. four sections, labeled turkey 30 percent, ham 20 percent, chicken 35 percent, and vegetarian 15 percent correctly represents the data in the table.

To find the percentage for each type of sandwich, we can divide the number of customers who ordered that sandwich by the total number of customers (100) and multiply by 100.

Using the data given in the table, we get:

Turkey: 30/100 x 100% = 30%

Ham: 20/100 x 100% = 20%

Chicken: 35/100 x 100% = 35%

Vegetarian: 15/100 x 100% = 15%

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A bank's loan officer rates applicants for credit. the ratings are normally distributed with a mean of 200 and a standard deviation of 50.

We will find the score that separates the lower 60% of ratings from the top 40% of ratings.

Let x be the score that separates these two portions. We can find the corresponding z-scores using the standard normal distribution, since the ratings are normally distributed with mean 200 and standard deviation 50.

The z-score corresponding to the 60th percentile is the value z such that

P(Z ≤ z) = 0.6

With the help of standard normal tables, we can find that z = 0.25. Therefore, we have

(x - 200)/50 = 0.25

For x, we get

x - 200 = 0.25 * 50

x - 200 = 12.5

x = 212.5

Similarly, the z-score corresponding to the 40th percentile is the value z such that

P(Z ≤ z) = 0.4

With the help of standard normal tables, we can find that z = -0.25. Therefore, we have

(x - 200)/50 = -0.25

For x, we get

x - 200 = -0.25 * 50

x - 200 = -12.5

x = 187.5

Hence, the score that separates the lower 60% from the top 40% is between 187.5 and 212.5.

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The linear approximation of [tex]f(x) = x^2 at x = 3.5 is f(3.5)[/tex]  ≈ 9.5.

To write a linear approximation (equation of the tangent) and use it to approximate the value of a function, we can follow these steps:

Choose a point on the function where you want to approximate the value.

Let's call this point (a, f(a)).

Find the slope of the tangent line at that point using the derivative of the function at that point.

The slope of the tangent line is equal to the derivative of the function at that point, which is denoted by f'(a).

Write the equation of the tangent line using the point-slope form, which is given by: y - f(a) = f'(a)(x - a)

Use this equation to approximate the value of the function at a nearby point.

Plug in the value of x that is close to a into the equation and solve for y. The resulting y-value will be an approximation of the value of the function at the nearby point.

For example, suppose we want to approximate the value of [tex]f(x) = x^2 at x = 3.[/tex]

We can follow these steps:

Choose the point (3, f(3)) = (3, 9) on the function.

Find the slope of the tangent line at x = 3 using the derivative: f'(x) = 2x, so f'(3) = 6.

Write the equation of the tangent line using the point-slope form: y - 9 = 6(x - 3), which simplifies to y = 6x - 9.

Use this equation to approximate the value of f(x) at x = 3.5: f(3.5) ≈ 6(3.5) - 9 = 9.5.

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a) The appropriate hypothesis testing are states as [tex]H_0 : p = 0.50[/tex]

[tex]H_a : p > 0. 50 [/tex]

b) The p-value for z-test is equals to the 0.000. Also, p-value < α.

c) Conclusion: Null hypothesis is rejected and there is evidence to support that more than 50% do not belief that such software unfairly targets students.

We have a sample of at an australian university that introduced the use of plagiarism-detection software.

Sample size = 178

Number of students who belief that such software unfairly targets student = 53

a) Null and alternative hypothesis are stated here as for right tailed t test.

[tex]H_0 : p = 0.05 [/tex]

[tex]H_a : p > 0.05[/tex]

sample proportion for students who does not belief that such software unfairly targets student [tex]\hat p = \frac {178- 53 }{178} [/tex] = 0.71

Now, test-statistic : [tex]t = \frac { \hat p - p} {\sqrt{ \frac{ p( 1- p)}{n}}}[/tex]

Substitute all the known values in above formula, [tex]= \frac {0.71 - 0.05} {\sqrt{ \frac{ 0.05( 1- 0.05)}{178}}}[/tex]

=> t = 41.25

b) Now, using Excel, the p-value for z

= 41.25 is 0.000. So, p-value < 0.05

Therefore, the Null hypothesis is rejected here.

c) Conclusion: there is evidence to support the claim that more than 50% do not belief that such software unfairly targets students.

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Complete question:

in a sample of 178 students at an australian university that introduced the use of plagiarism-detection software in a number of courses, 53 students indicated a belief that such software unfairly targets students. does this suggest that a majority of students at the university do not share this belief? test appropriate hypotheses at level 0.05.

a) State appropriate hypothesis

b) find p-value

c) draw conclusion

Answer:

c = 56°

Step-by-step explanation:

the exterior angle of a triangle is equal to the sum of the 2 opposite interior angles.

120° is an exterior angle of the triangle , then

c + 64° = 120° ( subtract 64° from both sides )

c = 56°

The opposite side of the right triangle (x) is approximately 6.888 inches long.

To solve this problem, we can use the trigonometric ratio for the sine function, which is opposite/hypotenuse. We have the hypotenuse as 12 inches and the opposite angle as 35 degrees, so we can set up the equation:

sin(35) = opposite/12

To solve for the opposite, we can multiply both sides by 12:

opposite = 12 * sin(35)

sin(35) as approximately 0.574:

opposite = 12 * 0.574

opposite ≈ 6.888

Therefore, the opposite side of the right triangle (x) is approximately 6.888 inches long.

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If the base of an isosceles triangle is 50 cm and the length of one of its legs is 65 cm, the height of the isosceles triangle is 60 cm.

To find the height of the isosceles triangle, we need to use the Pythagorean theorem and the properties of isosceles triangles.

First, we draw a diagram of the triangle and label the known values. We know that the base is 50 cm and one of the legs is 65 cm. Since the triangle is isosceles, the other leg is also 65 cm.

Next, we can draw a perpendicular line from the top vertex to the base, which represents the height of the triangle. We can label this height as "h".

Now, we have a right triangle with a base of 50 cm, a hypotenuse of 65 cm, and a height of "h". We can use the Pythagorean theorem to find the value of "h":

h² + 25² = 65²

h² + 625 = 4225

h² = 3600

h = 60

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The values of x that makes line segment m parallel to n (m ║ n) are as follows;

13. x = 107°.

14. x = 133°.

15. x = 20°.

16. x = 23°

The outer strings of the oud are parallel because they are all cut or intersected by a transversal.

In Mathematics and Geometry, corresponding angles can be defined as a postulate (theorem) which states that corresponding angles are always congruent when the transversal intersects two (2) parallel lines. This ultimately implies that, the corresponding angles will be always equal (congruent) when a transversal intersects two (2) parallel lines.

By applying corresponding angles theorem, we have the following:

x + 73 = 180° (sum of all interior angles)

x = 180 - 73

x = 107°.

x + 14 = 147 (vertical angles theorem).

x = 147 - 14

x = 133°.

3x + 2x + 20 = 180° (sum of all interior angles)

5x = 180 - 20

x = 60/3

x = 20°.

7x - 11 = 4x + 58 (corresponding angles theorem)

7x - 4x = 58 + 11

3x = 69

x = 23°

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