For the piecewise function, find the values h(- 8), h(-3), h(2), and h(5). -2x 10, for x< -7 2. x+3, forx22 h(x) = for 7sx<2 h(-8) = h(-3) = h(2) = h(5)

The car traveled for 6 hours at 70 mph and 4 hours at 80 mph.

To find out how many hours the car traveled at each speed, we can use a system of equations. Let x be the number of hours the car traveled at 70 mph and y be the number of hours the car traveled at 80 mph. We can set up the following equations:

Distance equation: 740 = 70x + 80y
Time equation: 10 = x + y

We can rearrange the time equation to solve for one of the variables in terms of the other:

y = 10 - x

Now we can substitute this into the distance equation:

740 = 70x + 80(10 - x)
740 = 70x + 800 - 80x
10x = 60
x = 6

So the car traveled for 6 hours at 70 mph. We can use this to find the number of hours the car traveled at 80 mph:

y = 10 - x = 10 - 6 = 4

So the car traveled for 4 hours at 80 mph.

Therefore, the car traveled for 6 hours at 70 mph and 4 hours at 80 mph.

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Answer:

Bears: 12

Tigers: 5

Step-by-step explanation:

9:15

the common factor of 9 and 15 is 3. so we divide both sides by 3.

3:5

12:20

The function f(x) = 3x² + 24x + 48 has a zero with a multiplicity of 2.

To determine which function has a zero with a multiplicity of 2, we need to find the discriminant of each quadratic function and check if it is equal to zero.

The discriminant is given by the formula: b² - 4ac, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.

Let's calculate the discriminant for each function:

1) f(x) = -3x² + 12x + 12

Discriminant: b² - 4ac = (12)² - 4(-3)(12) = 144 + 144 = 288 (not zero)

2) f(x) = 3x² + 24x + 48

Discriminant: b² - 4ac = (24)² - 4(3)(48) = 576 - 576 = 0 (zero)

3) f(x) = x² + 3x + 9

Discriminant: b² - 4ac = (3)² - 4(1)(9) = 9 - 36 = -27 (not zero)

4) f(x) = x² - 2x - 1

Discriminant: b² - 4ac = (-2)² - 4(1)(-1) = 4 + 4 = 8 (not zero)

Based on the calculations, the function f(x) = 3x² + 24x + 48 has a zero with a multiplicity of 2.

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We can use LCM to find a common denominator for these fractions. The LCM of 70, 2, and 5 is 70 x 2 x 5 = 700. The result of the expression is approximately 29.3885.

To solve this problem using LCM, we first need to convert all the mixed numbers to improper fractions:

Three and three seventieths = (3 x 70 + 3) / 70 = 213 / 70

Six and a half = (6 x 2 + 1) / 2 = 13 / 2

Nine and three fifths = (9 x 5 + 3) / 5 = 48 / 5

Now we can use LCM to find a common denominator for these fractions. The LCM of 70, 2, and 5 is 70 x 2 x 5 = 700.

We can then convert each fraction to an equivalent fraction with denominator 700:

213/70 = 3.042857... ≈ 3.043

13/2 = 455/70 = 6.5

48/5 = 192/20 = 96/10 = 480/50

Now we can substitute these equivalent fractions into the original expression and simplify:

3.043 × 6.5 + 480/50 = 19.7885... + 9.6 = 29.3885.

LCM stands for the "Least Common Multiple" and is a mathematical concept used to find the smallest multiple that two or more numbers have in common. In other words, it is the smallest positive integer that is divisible by all the given numbers.

To find the LCM of two or more numbers, we can start by finding their prime factorization. Then, we can take the highest power of each prime factor that appears in any of the factorizations and multiply them together to get the LCM. Taking the highest power of each prime factor (2^3 x 3^1), we get 24, which is the smallest multiple that both 6 and 8 have in common.

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The probability that Bob is the last customer to leave the post office comes out as (1 - e^(-λx))^2. The probability that Bob is the last customer to leave the post office can be calculated using the exponential distribution formula.

The exponential distribution is a continuous probability distribution used to model the time between events in a Poisson process. The formula for the exponential distribution is: P(X=x) = λe^(-λx)

Where X is the random variable representing the time between events, λ is the rate parameter, and e is the base of the natural logarithm.


P(Bob is the last customer to leave) = P(Bob's serving time > Alice's serving time) * P(Bob's serving time > Claire's serving time). Since the serving times follow an exponential distribution with parameter λ, we can use the formula to calculate the probabilities:

P(Bob's serving time > Alice's serving time) = ∫_0^∞ λe^(-λx) dx = 1 - e^(-λx)
P(Bob's serving time > Claire's serving time) = ∫_0^∞ λe^(-λx) dx = 1 - e^(-λx)

P(Bob is the last customer to leave) = (1 - e^(-λx))^2
Therefore, the probability that Bob is the last customer to leave the post office is (1 - e^(-λx))^2.

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If the balloon can hold 14 passengers at a time, then number of rounds to empty the que is 9.

In order to find out number of rounds it will take to empty the queue of 124 people, we divide the total number of people by the number of people that can be carried in each round,

⇒ Number of rounds = (Total number of people)/(Number of people per round);

⇒ Number of rounds = 124/14,

⇒ Number of rounds = 8.86 (rounded to two decimal places)

Since we can't have a fraction of a round, we round up to the nearest whole number.

Therefore, the balloon will need to make 9 rounds to empty the queue of 124 people.

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The given question is incomplete, the complete question is

There are 124 people waiting to take a ride in a hot air balloon. The balloon can hold 14 passengers at a time, how many rounds will it take to empty the que?

The solutions for the absolute value equation are x = 0 and x = -4.

To solve the absolute value equation |(5x + 10)/(2)| = 5, we need to remove the absolute value bars and create two separate equations, one positive and one negative. Then we can solve for x in each equation.

First, let's remove the absolute value bars and create two separate equations:

(5x + 10)/2 = 5 and (5x + 10)/2 = -5

Now we can solve for x in each equation:

(5x + 10)/2 = 5

5x + 10 = 10

5x = 0

x = 0

And:

(5x + 10)/2 = -5

5x + 10 = -10

5x = -20

x = -4

So the solutions to the equation |(5x + 10)/(2)| = 5 are x = 0 and x = -4.

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Answer:

To determine the location of point H, we need to first find the length of the three sides of the fenced area using the given points E, F, and G.

The length of EF is:

sqrt[(3-1)^2 + (5-5)^2] = 2 units

The length of FG is:

sqrt[(6-3)^2 + (1-5)^2] = sqrt(25) = 5 units

The length of EG is:

sqrt[(6-1)^2 + (1-5)^2] = sqrt(20) = 2sqrt(5) units

The total length of the three sides is:

2 + 5 + 2sqrt(5) = 7 + 2sqrt(5) units

Since Landon has 16 units of fencing, he needs to find a point H such that the length of the fourth side of the fenced area (EH) is:

16 - (7 + 2sqrt(5)) = 9 - 2sqrt(5) units

Let's assume that point H has coordinates (x, y). Then, we can use the distance formula to find the length of EH:

sqrt[(x-1)^2 + (y-5)^2] = 9 - 2sqrt(5)

Squaring both sides and simplifying, we get:

(x-1)^2 + (y-5)^2 = 81 - 36sqrt(5) + 20

(x-1)^2 + (y-5)^2 = 101 - 36sqrt(5)

Now, we can plug in the coordinates of each of the answer choices and see which one satisfies this equation:

For (0, 1):

(0-1)^2 + (1-5)^2 = 16, which is not equal to 101 - 36sqrt(5)

For (0, -2):

(0-1)^2 + (-2-5)^2 = 65, which is not equal to 101 - 36sqrt(5)

For (1, 1):

(1-1)^2 + (1-5)^2 = 16, which is not equal to 101 - 36sqrt(5)

For (1, -2):

(1-1)^2 + (-2-5)^2 = 65, which is equal to 101 - 36sqrt(5)

Therefore, the answer is (1, -2), and Landon can place point H at coordinates (1, -2) to fence in the dog park without having to buy more fencing.

The solution to the logarithmic equation log_(3x - 2)(125) = 3 is given as follows:

x = 7/3.

The logarithmic equation for this problem is defined as follows:

log_(3x - 2)(125) = 3

The definition of the logarithm can be applied, which means that if we elevate the base of 3x - 2 to the power of 3, we obtain the result of 125.

With the application of the definition, the expression is given as follows:

(3x - 2)³ = 125.

125 is the third power of 5, hence:

(3x - 2)³ = 5³.

The third power function is one-to-one function, meaning that each output is related to a single input, hence the value of x is obtained as follows:

3x - 2 = 5

3x = 7

x = 7/3.

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The distance between the hunter and the bird is 5.39

We know that the distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula below.

distance = √( (x₂ - x₁)² + (y₂ - y₁)²)

The hunter is at (0, 0) and shoots at (2, 5), so we can plug those values ​​into the formula for the distance:

distance = √( (2 - 9)² + (5 - 0)²)

distance = √( (2 )² + (5 )²)

distance = √( 4 + 25) = 5.39

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Your final map should include two streets, a highway, and labels for each angle with their corresponding measures. The key should clearly explain each type of angle and their relationships.

Maps are also used to help people understand the relationship between different places and their relative size and position. Maps can provide information about the landforms of an area, such as mountains, rivers, valleys, and deserts.

To begin, draw two parallel lines on the map, and then add a transversal crossing them. Label the parallel lines as “streets” and the transversal as “Highway.” Then, label each pair of angles with the corresponding names listed above. For example, the angles that form when the two parallel lines are crossed by the transversal would be labeled as “Alternate Interior Angles.”

Next, measure the angles using a protractor or a ruler. For example, if two parallel lines are crossed by a transversal, measure the four resulting angles. Each angle should be measured and labeled with its degrees.

Finally, add a key to your map. This should include an explanation of each type of angle and their relationships with one another. For example, the key should explain that complementary angles add up to 90 degrees, supplementary angles add up to 180 degrees, etc.

Your final map should include two streets, a highway, and labels for each angle with their corresponding measures. The key should clearly explain each type of angle and their relationships. By completing this assignment, you will have a visual understanding of the different types of angles and lines we have learned in class.

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The distance between the two chords is approximately 0.894 cm where |AB|=10cm and |CD|=6cm

Chord is the longest segment that can be drawn within the circle, connecting two points on the circle that are not necessarily opposite each other.

We can use the following formula to find the distance between two chords in a circle:

d = |(AB-CD)/(2√(r²-(AB/2-CD/2)²))|

where d is the distance between the chords, AB and CD are the lengths of the chords, and r is the radius of the circle.

By substituting, we get:

d = |(10-6)/(2√(6²-(10/2-6/2)²))|

d = |4/(2√(6²-(8/2)²))|

d = |4/(2√(6²-4²))|

d = |4/(2√(20))|

d = |4/(2×2√(5))|

d = |2/√(5)|

d ≈ 0.894 cm

Therefore, the distance between the two chords is approximately 0.894 cm.

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Using the triangle proportionality theorem, the length of CD is calculated as: 12 cm.

The triangle proportionality theorem states that if a line is parallel to one side of a triangle, it divides the other two sides proportionally.

To calculate the length of the line, recall the triangle proportionality theorem which states that:

Given the following:

AB = 10 cm

AC = 15 cm

BE = 8 cm

CD = ?

Based on the triangle proportionality theorem, we have:

AC/AB = CD/BE

Substitute:

15/10 = CD/8

Cross multiply:

10CD = 120

CD = 120/10

CD = 12 cm

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The natural logarithm function is ln(p) = ln(100) - 0.35t

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

The table of values

The function can be represented as

p = ae^kt

Using the points on the table, we have

ae^(k * 0) = 100

So, we have

a = 100

This gives

p = 100e^kt

Using another point, we have

70.5y = 100e^(k * 1)

70.5y = 100e^k

So, we have

e^k = 0.705

Take the natural logarithm of both sides

k = ln(0.705)

k = -0.35

The function becomes

p = 100e^(-0.35t)

Take the natural logarithm of both sides

ln(p) = ln(100) - 0.35t

Hence, the function is ln(p) = ln(100) - 0.35t

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The numbers for which the rational expression is undefined are x=0 and x=-4.

To find the numbers for which the rational expression (7x^(4)+8)/(5x^(2)+20x) is undefined, we need to find the values of x that make the denominator equal to zero. This is because division by zero is undefined.

So, we need to solve the equation 5x^(2)+20x=0 for x.

We can factor out a common factor of 5x from the equation:

5x(x+4)=0

Now, we can use the zero product property to set each factor equal to zero and solve for x:

5x=0 or x+4=0

x=0 or x=-4

So, the numbers for which the rational expression is undefined are x=0 and x=-4.

In summary, the rational expression (7x^(4)+8)/(5x^(2)+20x) is undefined for x=0 and x=-4.

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The complete equation using the numbers from1 to 9 will be:

2:2 = 3×3:9 = 4×4 =16.

A contrast between two numbers in arithmetic, this is referred to as a proportion. According to the law of proportion, two sets of given numbers are said to be directly proportional to one another if they grow or shrink in the same ratio.

In the question given:

By using the digits 1 to 9 at one time each, we have to form the equation to make the equality true.

Now,

All the numbers from 1 to 9 are = 1, 2 ,3, 4, 5, 6, 7, 8 and 9

Let the proportion be = 1

Hence, the equivalent ratios are:

1:1, 2:2. 3:3

Now, according to the question the required equation will be:

2:2 = 3×3:9 = 4×4 =16.

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Answer:

Step-by-step explanation:

Pages Alana writes = m+4

to fill in the blanks add 4 to 5, 7, and 9 because of the 4 extra pages Alana writes.

That will give you: 9, 11, and 13 as your answers

The five rules given in this chapter are important for algebraic manipulation. Firstly, the Commutative Rule states that when two numbers are added or multiplied, the order of the numbers does not matter.

Secondly, the Associative Rule states that when three or more numbers are added or multiplied, the order in which the operations are performed does not affect the result.

Thirdly, the Distributive Rule states that when a number is multiplied by a sum of two numbers, the number can be distributed to each of the two numbers.

Fourthly, the Additive Identity Rule states that when any number is added to zero, the result is the same number. Finally, the Multiplicative Inverse Rule states that if a number is multiplied by its reciprocal, the result is one.

Together, these five rules form the basis for many algebraic operations. They are important for understanding and manipulating equations, solving for unknowns, and more. By understanding and applying these five rules, students will be able to work more efficiently and accurately with algebraic expressions.

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a) The expression can be factored as (5xy+8z)(5xy-8z).

b) The expression can be factored as (2x+3y)(2x-3y).

The expressions that can be factored as a difference of squares are:

- 25x^(2)y^(2)-64z^(2)
- 4x^(2)-9y^(2)

A difference of squares is an expression in the form a^2 - b^2, which can be factored as (a+b)(a-b). In the first expression, 25x^(2)y^(2) can be rewritten as (5xy)^2 and 64z^(2) can be rewritten as (8z)^2. Therefore, the expression can be factored as (5xy+8z)(5xy-8z).

In the second expression, 4x^(2) can be rewritten as (2x)^2 and 9y^(2) can be rewritten as (3y)^2. Therefore, the expression can be factored as (2x+3y)(2x-3y).

The other two expressions, 16x^(2)+25y^(2) and x^(3)-125, cannot be factored as a difference of squares because they do not have the form a^2 - b^2.

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The distance between the points (-9, 4) and (3, 12) is approximately 14.42 units.

what exactly is distance?

Distance is the amount of space between two objects or locations. It is a measure of how far apart two things are from each other, typically expressed in units such as meters, kilometers, miles, feet, or yards. Distance can be measured in a straight line, known as the "Euclidean distance," or it can be measured along a path or route, known as the "distance traveled." Distance is a fundamental concept in many areas of science and mathematics, including physics, geography, and geometry

The formula for calculating the distance between two points is d = ((x2 - x1)2 + (y2 - y1)2).

Let's use this formula to find the distance between the two points (-9, 4) and (3, 12):

d = √((3 - (-9))² + (12 - 4)²)

= √(12² + 8²)

= √(144 + 64)

= √208

≈ 14.42

So the distance between the points (-9, 4) and (3, 12) is approximately 14.42 units.

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Answer: $170

Step-by-step explanation:

6x + 18(6) = 1128.

6x + 108 = 1128

6x = 1020

x = 170

The price of one ticket is 170

Answer:75

Step-by-step explanation:
7+5=12
15x5=75
75 is odd and I just went through the multiples of 15 haha

We will see that the measure of angle z is 70°

First we can get the angle in the right vertex in the triangle in the right side.

We know that the right angle and the 40° one are vertical angles, then the right angle also measures 40°

Also remember that the sum of the interior angles of any triangle is 180°, then:

105° + 40° +x = 180°

x = 180° - 40° - 105° = 35°

Then the right angle of the second triangle (the one that is below the line) also measures 35°

The bottom angle measures:

y + 85 = 180

y = 180 - 85 = 95

And if the last angle is k:

k + 95 + 35 = |80

k = 180 - 35 - 95 = 50

Then the right angle of the last triangle is also 50°, then we can write:

z + 60 + 50 = 180

z = 180 - 50 - 60 = 70°

That is the measure.

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The size of the angle that the line DE makes with the plane ABCD is 50.8 degrees

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

The figure

To start with, we calculate the measure of length BE using the following tangent ratio

tan(60) = BE/60

So, we have

BE = 60√3

Next, we calculate DB using the Pythagoras theorem

DB = √(60² + 60²)

Evaluate

DB = 60√2

The measure of the required angle is then calculated as

tan(Angle) = 60√3/60√2

Evaluate

tan(Angle) = 1.2247

Take the arc tan

Angle = 50.8 degrees

Hence, the angle is 50.8 degrees

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The volume of the right square pyramid is 320 in³.

Volume is a measure of the amount of space that an object takes up. It is usually measured in cubic units, such as cubic centimeters, cubic meters, or cubic feet. Volume is an important concept in mathematics, physics, and engineering, as it can be used to calculate the size and mass of an object. It can also be used to calculate the amount of liquid or gas that can fit into a container.

The volume of a right square pyramid can be calculated using the formula V = (1/3)lhs.

Therefore, the volume of the right square pyramid with the given dimensions is:

V = (1/3) x 10 in x 6 in x 16 in

V = 320 in³

Therefore, the volume of the right square pyramid is 320 in³.

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The volume of the pyramid is 320in cubic units

volume is a measure of space occupied by an object,

the formula for the volume of a pyramid is equal to one third of the product of the height and the base area of the pyramid

The pyramid that has a square base is known as Square pyramid.

square pyramid has 4 faces and 1 square base.

The volume of the square pyramid is given by the formula

V=(1/3)*(base area of a square)*(height of the square pyramid)cubic units

V=(1/3)AH cubic units

The volume of the pyramid is V= (base length*base width*height)/3

V=(1/3)l*s*h

V=(1/3)10*16*6

V=320 in

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Given that A = |b с| . |a d| and det(A) = -11, the following computations can be made:
(b) det(4A) det(A) = 4^2 * (-11) * (-11) = 1936
(c) det(2A - 1) = 2^2 * (-11) - 1 = -43
(d) det((24)^-1) = (1/24) ^2 = 1/576
(e) d a 9 det b h с e

The determinant of a matrix is defined as the sum of the products of the elements of the matrix multiplied by the corresponding cofactor. The cofactor is a signed minor of the matrix, which is obtained by deleting the row and column of the element for which the cofactor is being computed.
In the case of the matrix A = |b с| . |a d|, the determinant can be computed as follows:
|b с| . |a d| = (b*d) - (a*c)
Therefore, det(A) = (b*d) - (a*c) = -11.
To compute det(4A) det(A), first, find det(4A), which is equal to:
|4b 4c| . |4a 4d| = 4^2 * (b*d - a*c)
Thus, det(4A) = 16(b*d - a*c) = 16(-11) = -176.
Then, det(4A) det(A) = (-176) * (-11) = 1936.

For det(2A - 1), first, find 2A - 1, which is equal to:
|2b 2c| . |2a 2d| - |1 0| . |0 1|
= 2|b с| . |a d| - |1 0| . |0 1|
= 2A - |1 0| . |0 1|
= 2A - I
where I is the identity matrix.
Therefore, det(2A - 1) = det(2A - I) = det(2A) det(I^-1)
Since det(I) = 1, det(I^-1) = 1/det(I) = 1/1 = 1.
Therefore, det(2A - 1) = 2^2 * (-11) * 1 - 1 = -43.
Finally, to compute det((24)^-1), it is necessary to find the inverse of the matrix 24.
|a b| . |c d| = 24I
=> |a b|^-1 . |c d| = (1/24)I
=> (1/24) . |d -b| . |-c a| = (1/24)I
Therefore, |d -b| . |-c a| = I
Since the determinant of the identity matrix is 1, it follows that:
1 = det(I) = det(|d -b| . |-c a|) = (a*d) - (b*c)
Hence, det((24)^-1) = (1/24)^2 = 1/576.

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23) The equation for the hyperbola is (y-2)^2/9 - (x-1)^2/16 = 1.

24) The equation for the circle is (x+1)² + (y-3)² = 17.

23. The equation for a hyperbola with center (h,k) and vertices (h,k+a) and (h,k-a) is:

(y-k)²/a²- (x-h)²/b² = 1

In this case, the center is (1,2), so h = 1 and k = 2. The vertices are (1,5) and (1,-1), so a = 3. To find b, we can use the fact that c^2 = a^2 + b^2, where c is the distance from the center to the foci. The foci are (1,7) and (1,-3), so c = 5. Therefore:

5² = 3² + b²

b² = 25 - 9

b² = 16

b = 4

So the equation for the hyperbola is:

(y-2)^2/3^2 - (x-1)^2/4^2 = 1

(y-2)^2/9 - (x-1)^2/16 = 1

24. The equation for a circle with center (h,k) and radius r is:

(x-h)² + (y-k)² = r²

In this case, the center is (-1,3), so h = -1 and k = 3. To find the radius, we can use the distance formula with the center and the point (3,2):

r = sqrt((3-(-1))² + (2-3)²

r = sqrt(4² + (-1)²)

r = sqrt(16 + 1)

r = sqrt(17)

So the equation for the circle is:

(x-(-1))² + (y-3)² = (sqrt(17))²

(x+1)² + (y-3)² = 17

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1. a) The price is 5 dollars. b) The rate of change of demand with respect to price, p, when x = 4 is 5/6 thousands of units per dollar. c) The elasticity of demand when x = 4 is 25/24.

2. a) The price is 2*sqrt(3) dollars. b) The rate of change of demand with respect to price, p, when x = 10 is - (3 * sqrt(3))/(7) thousands of units per dollar. c) The elasticity of demand when x = 10 is -6/7.

3. Demand is increasing at -23.30 thousands of units per week.

4. If the daily production is 800 cases, then revenue is decreasing at a rate of -1200 dollars per day.

5. Supply is rising at a rate of 150.90 thousands of styluses per week.

1)
(a) Compute the price, p, when x = 4.
Price, p = −3(4)^2 − 4(4) + 69 = -48 -16 + 69 = 5 dollars
(b) Use implicit differentiation to compute the rate of change of demand with respect to price, p, when x = 4.
Rate of change of demand, x' = - (2 * -3 * x + 4)/(-3 * 2 * x) = -(-24 + 4)/(-24) = 20/24 = 5/6 thousands of units per dollar
(c) Compute the elasticity of demand when x = 4.
The elasticity of Demand = (x'/x) * (p) = (5/6)/(4) * (5) = 25/24

2)
(a) Compute the price, p, when x = 10.
Price, p = sqrt((216 - 10^2 - 8*10)/3) = sqrt((216 - 100 - 80)/3) = sqrt(12) = 2*sqrt(3) dollars
(b) Use implicit differentiation to compute the rate of change of demand with respect to price, p, when x = 10.
Rate of change of demand, x' = - (2 * 3 * p)/(2 * x + 8) = - (6 * 2 * sqrt(3))/(20 + 8) = - (12 * sqrt(3))/(28) = - (3 * sqrt(3))/(7) thousands of units per dollar
(c) Compute the elasticity of demand when x = 10.
Elasticity of Demand = (x'/x) * (p) = (- (3 * sqrt(3))/(7))/(10) * (2*sqrt(3)) = -6/7

3)
The weekly demand equation is given by p + x + 3xp = 53,
where x is the number of thousands of units demanded weekly and p is in dollars. If the price p is decreasing at a rate of 70 cents per week when the level of demand is 5000 units, then demand is increasing at a rate of (53 - 5000 - 70)/(3*70 + 1) = -4917/211 = -23.30 thousands of units per week.

4)
A company is decreasing production of math-brain protein bars at a rate of 100 cases per day. All cases produced can be sold. The daily demand function is given by p(x) = 20 − x/200,
where x is the number of cases produced and sold, and p is in dollars.
If the daily production is 800 cases, then revenue is decreasing at a rate of (20 - 800/200) * (-100) + (800) * (-1/200) * (-100) = (20 - 4) * (-100) + (800) * (1/2) = -1600 + 400 = -1200 dollars per day.

5)
The wholesale price p of e-tablet writing styluses in dollars is related to the supply x in thousands of units by 400p^2 − x^2 = 14375,
If 5,000 styluses are available at the beginning of a week, and the price is falling at 30 cents per week, then supply is rising at a rate of (2 * 400 * p * (-0.30) - 2 * x * x')/(2 * -1 * x) = (800 * p * (-0.30) - 0)/(2 * -5000) = (800 * sqrt(14375 + 5000^2)/400 * (-0.30) - 0)/(2 * -5000) = (800 * sqrt(14375 + 25000000)/400 * (-0.30))/(2 * -5000) = (2 * sqrt(14375 + 25000000) * (-0.30))/(-5000) = 0.012 * sqrt(14375 + 25000000) = 150.90 thousands of styluses per week.

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The result is the 95% confidence interval for µ.

The Metropolis-Hasting (M-H) algorithm is a Markov Chain Monte Carlo (MCMC) method used to sample from a probability distribution. In this case, we want to sample from the posterior distribution of µ, given the recorded data and the prior distribution. The M-H algorithm works by proposing a new value for µ, calculating the acceptance probability, and then deciding whether to accept or reject the proposed value. Here are the steps to construct the M-H algorithm:

Start with an initial value for µ, denoted as µ0.
Propose a new value for µ, denoted as µp, from the proposal distribution q(µ| µo) : N(θ: mean = µp, std = 2).
Calculate the acceptance probability, denoted as α, using the likelihood function L(µ) and the prior distribution p(µ):
α = min{1, [L(µp)/L(µ0)]*[p(µp)/p(µ0)]*[q(µ0| µp)/q(µp| µ0)]}
Generate a random number u from the uniform distribution U(0,1).
If u ≤ α, accept the proposed value and set µ0 = µp. Otherwise, reject the proposed value and keep µ0 unchanged.
Repeat steps 2 to 5 for a specified number of MCMC draws (15000 in this case), and discard the first 1500 draws as the burn-in phase.
Calculate the 95% confidence interval for µ using the remaining 13500 draws.

The 95% confidence interval for µ can be calculated by finding the 2.5th and 97.5th percentiles of the posterior distribution of µ. This can be done by sorting the 13500 draws of µ in ascending order and finding the values that correspond to these percentiles. The result is the 95% confidence interval for µ.

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