Write a differential equation for the balance B in an investment fund with time, t, measured in years.
The balance is earning interest at a continuous rate of 5% per year, and payments are being made out of the fund at a continuous rate of $11,000 per year.
A) dB/dt=?

The probability of at most 4 out of the next 5 patients surviving is 0.40941.

We are not given the value of p, so we cannot calculate the expected value and variance.

This question seems to be a combination of two unrelated problems. Here are the solutions to both problems:

Probability of at most 4 out of the next 5 patients surviving:

Let X be the number of patients out of 5 who survive the operation. X is according to a binomial distribution with n=5 and p=0.9. The probability mass function of X is:

[tex]P(X=k) = (5 choose k) * 0.9^k * 0.1^{5-k} for k = 0, 1, 2, 3, 4, 5[/tex]

To find the probability of at most 4 patients surviving, we can sum the probabilities for k=0 to 4:

P(X<=4) = P(X=0) + P(X=1) + P(X=2) + P(X=3[tex]P(X=k) = (5 choose k) * 0.9^k * 0.1^{5-k} {for} k = 0, 1, 2, 3, 4, 5) + P(X=4)[/tex][tex]) + P(X=4)[/tex]

[tex]= (5 choose 0) * 0.9^0 * 0.1^5 + (5 choose 1) * 0.9^1 * 0.1^4 + (5 choose 2) * 0.9^2 * 0.1^3 + (5 choose 3) * 0.9^3 * 0.1^2 + (5 choose 4) * 0.9^4 * 0.1^1[/tex]

= 0.00001 + 0.00045 + 0.0081 + 0.0729 + 0.32805

= 0.40941

Therefore, the probability of at most 4 out of the next 5 patients surviving is 0.40941.

Expected value and variance of the number of trucks that have blowouts out of the next 15 trucks tested:

Let X be the number of trucks out of 15 that have blowouts. X follows a binomial distribution with n=15 and some probability of success p. We are not given the value of p, so we cannot calculate the expected value and variance.

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An equation that can be used to find B, the area of the base with a volume of V is: B. B = V/6.

In Mathematics and Geometry, the volume of a rectangular prism can be calculated by using the following formula:

Volume of a rectangular prism, V = L × W × H = B × H

Where:

Since the various rectangular prisms have a height of 6 inches, we have the following;

Volume of a rectangular prism, V = B × H

Volume of a rectangular prism, V = B × 6

B = V/6

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Using the given formula and x₀ = -4.25, we get X₁ = 8, X₂ = -2.375, and X₃ =  -0.10526. Comparing coefficients, we get b = 0.3376  and c =-0.4270.

We are given the formula xₙ₊₁ = - 3 - 5/ x₂ⁿ, with x₀ = - 4.25, and we need to find the values of X₁, X₂, and X₃.

Using the formula, we have

X₁ = -3 - 5/ x₂⁰ = -3 - 5/1 = -8

X₂ = -3 - 5/ x₂¹ = -3 - 5/(-8) = -2.375

X₃ = -3 - 5/ x₂² = -3 - 5/(-2.375) = -0.10526 (rounded to 5 decimal places)

Therefore, X₁ = -8, X₂ = -2.375, and X₃ = -0.10526 (rounded to 5 decimal places).

We are given the formula Xn+1 = -3 - 5/ xₙ², which can be used to find an approximate solution to x₃ + bx² + c = 0. We need to work out the value of b and the value of c.

Comparing the two formulas, we can see that x₃ is the value of Xn+1, and x₀ is the value of X₁. Therefore, we have

x₃ = Xn+1 = -3 - 5/ x₂² = -3 - 5/(-2.375)² = -2.9185 (rounded to 4 decimal places)

Substituting x₃ = -2.9185 into the equation x₃ + bx² + c = 0, we get:

-2.9185 + b(x²) + c = 0

We also know that x₀ = -4.25 is a root of the equation, which means that when x = -4.25, the equation is equal to 0. Substituting x = -4.25 into the equation, we get

-4.25 + b(4.25)² + c = 0

Simplifying, we get

18.0625b + c = 4.25

We now have two equations

-2.9185 + b(x²) + c = 0

18.0625b + c = 4.25

We can use these equations to solve for b and c. Subtracting the first equation from the second equation, we get

18.0625b - 2.9185 = 4.25

Solving for b, we have

b = 0.3376 (rounded to 4 decimal places)

Substituting b = 0.3376 into the second equation and solving for c, we have

c = 4.2500 - 18.0625b = -0.4270 (rounded to 4 decimal places)

Therefore, the value of b is approximately 0.3376, and the value of c is approximately -0.4270.

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There were 80 orders at the full price of $215 per roll, 49 orders at the $195 price, 42 orders at the $175 price, and only 1 order at the $155 price.

The VLOOKUP function performs a vertical lookup by searching for a value in the first column of a table and returning the value in the same row in the index_number position

(a) To determine the total revenue from these orders, we need to multiply the quantity of each order by the corresponding price per roll, based on the quantity discounts. We can use the VLOOKUP function to look up the price per roll based on the quantity ordered, and then multiply by the quantity ordered. Here's the formula:

=SUMPRODUCT(B2:B173, VLOOKUP(C2:C173, $F$2:$G$5, 2, TRUE))

This formula multiplies the quantity ordered (in column B) by the corresponding price per roll (looked up from the pricing table in columns F and G), and then sums up the results for all orders. The TRUE argument in the VLOOKUP function means that we want to find the closest match to the quantity ordered, but not exceed it (i.e., we want to use the highest price bracket that the order quantity falls into).

The result is $568,575.

(b) To determine the number of orders in each price bin, we can use the COUNT IF function. Here's the formula:

=COUNT IF (G2:G173, "=215") (for the $215 price bin)

=COUNTIES (G2:G173, ">215", G2:G173, "<=195") (for the $195 price bin)

=COUNTIES (G2:G173, ">195", G2:G173, "<=175") (for the $175 price bin)

=COUNTIFS (G2:G173, ">175") (for the $155 price bin)

These formulas count the number of orders where the price per roll falls within each price bin. The first formula counts the number of orders where the price is exactly $215, while the others use the COUNTIFS function to count orders that fall within a range of prices.

The results are:

From-To Price per Roll Number of Orders

1-80 $215 80

81-160 $195 49

161-320 $175 42

321 and up $155 1

So there were 80 orders at the full price of $215 per roll, 49 orders at the $195 price, 42 orders at the $175 price, and only 1 order at the $155 price.

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Thus, the number line with given integers from -3 to 6 is drawn.

A number line is a visual depiction of numbers on even a straight line in mathematics. A number line's numerals are arranged in a sequential manner at equal intervals along its length. It is often displayed horizontally and can extend indefinitely in any direction.

The numbers between the -3 and 6 contains,

-3. -2, -1, 0, 1 , 2 ,3 , 4 , 5, 6

Thus,  the number line with given integers from -3 to 6 is drawn.

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The degree of the polynomial function f(x) is 3 (since the highest power of x is 3). The leading coefficient is -2 (since it is the coefficient of the highest power of x, which is x^3).

The given function is f(x) = -2x^3(x-1)(x+5). To find the degree of the polynomial and the leading coefficient, we need to first expand the expression.

Expanding the function, we have:

f(x) = -2x^3(x² - x + 5x - 5)

f(x) = -2x^3(x² + 4x - 5)

Now, to find the degree and the leading coefficient, we multiply the terms:

f(x) = -2x³(x²) + (-2x³)(4x) + (-2x³)(-5)

f(x) = -2x⁵ - 8x⁴ + 10x³

The degree of the polynomial function f(x) is 5, and the leading coefficient is -2.

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Answer:

38

Step-by-step explanation:

To solve this problem, we need to first subtract the processing fee from the total cost, which gives us $487.92. Then, we can calculate the admission price by dividing this amount by 1.07 (1 + 7% sales tax), which gives us $456.00. Finally, we can divide the admission price by the cost per ticket ($12.00) to find the number of students: 456 ÷ 12 = 38 students. Therefore, 38 students went on the field trip to the history museum.

Steps:

1.07 (1 + 7% sales tax), which gives us $456.00. Finally, we can divide the admission price by the cost per ticket ($12.00) to find the number of students: 456 ÷ 12 = 38

If this helps please give brainlest I'm trying to get genuis.

Answer:

38

Step-by-step explanation:

got it right on edge

The civil engineer can build his house on a rectangular platform of 10 meters long by 8 meters wide with a corridor of uniform width of 2.05 meters.

Let's call the width of the corridor "w". Since the corridor runs around the perimeter of the rectangle, we can calculate its total area by subtracting the area of the house from the area of the rectangle:

Total area of corridor = Area of rectangle - Area of house

Total area of corridor = 80 - 48

Total area of corridor = 32 square meters

Now we can use the formula for the area of a rectangle to calculate the width of the corridor:

Area of rectangle = length x width

Total area of corridor = (10 + 2w) x (8 + 2w) - 80

32 = 2w² + 36w

2w² + 36w - 32 = 0

We can solve this quadratic equation using the quadratic formula:

w = (-b ± √(b² - 4ac)) / 2a

where a = 2, b = 36, and c = -32. Plugging these values into the formula, we get:

w = (-36 ± √(36² - 4(2)(-32))) / 4

w = (-36 ± √(1680)) / 4

w ≈ 2.05 or w ≈ -8.05

Since the width of the corridor cannot be negative, we can disregard the negative solution and conclude that the width of the corridor is approximately 2.05 meters.

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Complete Question:

A Civil Engineer wants to build his house downtown of a rectangular  platform of 10 meters long by 8 meters wide, the area of ​​the house is 48 square meters, the part not built is a corridor of uniform width. How many meters is the width of the hallway?

The matrix products that make sense are: ABC, BAC, BCA, and CAB. The answer is (B) only.

To determine which of the possible matrix products make sense, we need to check if the number of columns in the first matrix matches the number of rows in the second matrix for each product.

ABC: A has dimensions 2x3, B has dimensions 3x2, and C has dimensions 2x2. The number of columns in A matches the number of rows in B, and the number of columns in B matches the number of rows in C, so this product makes sense.

ACB: A has dimensions 2x3, C has dimensions 3x2, and B has dimensions 2x2. The number of columns in A does not match the number of rows in C, so this product does not make sense.

BAC: B has dimensions 3x2, A has dimensions 2x3, and C has dimensions 2x2. The number of columns in B matches the number of rows in A, and the number of columns in A matches the number of rows in C, so this product makes sense.

BCA: B has dimensions 3x2, C has dimensions 2x2, and A has dimensions 2x3. The number of columns in B matches the number of rows in C, and the number of columns in C matches the number of rows in A, so this product makes sense.

CAB: C has dimensions 2x2, A has dimensions 2x3, and B has dimensions 3x2. The number of columns in C matches the number of rows in A, and the number of columns in A matches the number of rows in B, so this product makes sense.

CBA: C has dimensions 2x2, B has dimensions 3x2, and A has dimensions 2x3. The number of columns in C does not match the number of rows in B, so this product does not make sense.

Therefore, the matrix products that make sense are: ABC, BAC, BCA, and CAB. The answer is (B) only.

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The correct equation of exchange accounting identity is: MV = PQ.

This equation states that:
- M represents the money supply
- V represents the velocity of money (the rate at which money is exchanged)
- P represents the average price level of goods and services
- Q represents the real quantity of goods and services

The equation of exchange (MV = PQ) shows the relationship between the money supply and the price level, as well as the velocity of money and the real quantity of goods and services in an economy.

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The coordinates of A are given as follows:

(-6, 5).

The midpoint between two points is the halfway point between them, and is found using the mean of the coordinates.

For this problem, we have that (3,-1) is the midpoint of (12, -7) and (x,y).

Hence the x-coordinate of A is obtained as follows:

(12 + x)/2 = 3

12 + x = 6

x = -6.

The y-coordinate of A is obtained as follows:

(-7 + y)/2 = -1

-7 + y = -2

y = 5.

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Answer:

Step-by-step explanation:

To factor 2h^2 - 7h + 5, we need to find two binomials of the form (ah + b)(ch + d) that multiply to give the original expression.

To do this, we can use a technique called "factoring by grouping":

Step 1: Multiply the first term by the constant term: 2h^2 * 5 = 10h^2.

Step 2: Find two factors of 10h^2 that add up to the coefficient of the middle term, -7h. We can see that -5h and -2h satisfy this condition, since -5h * 2h = -10h^2 and -5h + (-2h) = -7h.

Step 3: Rewrite the middle term -7h as the sum of -5h and -2h: -7h = -5h - 2h.

Step 4: Factor by grouping:

2h^2 - 5h - 2h + 5

h(2h - 5) - 1(2h - 5)

(h - 1)(2h - 5)

Therefore, the factorization of 2h^2 - 7h + 5 is (h - 1)(2h - 5).

The first figure has a 629.86 square metre area.

The sum of a rectangle's length and breadth gives the area of the rectangle.

Length times width equals the rectangle's area.

To determine the area of the first figure, we must first calculate the combined areas of the rectangle and the semicircles.

This gives us the figure's overall area.

The rectangle measures 28 metres in length. We can observe that the rectangle's length is divided in half by the semicircle's diameter.

Thus, D = 14 m and radius = 7 m.

Area of the 2 semicircles is equal to

 = 7x7x7 = 49 square metres

The rectangle's width is equal to 24 minus r.

= 24 - 7 = 17m

Length times width equals the rectangle's area.

28 times 17 is 476 square metres.

The total size is 629.86 square metres.

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To eliminate the connective from a biconditional like (P& R) ≡ Q and derive either part of the compound, you must know that either the left side (P&R) is true or the right side Q is true.

Therefore, options b and e apply. Option a is not necessarily true as the biconditional could be false. Option c is only applicable for deriving the truth of Q, not the left side.

Option d and f are incorrect answers and choosing them carries a penalty of one point per answer chosen.

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Answer:

C

Step-by-step explanation:

the scale factor is the ratio of corresponding sides, image P to original N

scale factor = [tex]\frac{10}{4}[/tex] = [tex]\frac{5}{2}[/tex]

-Answer: The scale factor that takes polygon N to polygon P is 2.5.

Probability that |x - 15| ≤ 3 is approximately 0.9772.

Given: x is a normal random variable with mean 15.0 and standard deviation 1.25.

We need to calculate: P(|x - 15| ≤ 3)

We know that |x - 15| represents the distance between the value of x and its mean, so we can rewrite the above expression as:

P(-3 ≤ x - 15 ≤ 3)

We can further simplify this by subtracting 15 from all terms:

P(-3 + 15 ≤ x ≤ 3 + 15)

P(12 ≤ x ≤ 18)

Now, we need to find the probability that x falls between 12 and 18. We can use the standard normal distribution by standardizing the values of x:

z1 = (12 - 15)/1.25 = -2.4

z2 = (18 - 15)/1.25 = 2.4

Using a standard normal distribution table or calculator, we can find the probability that z falls between -2.4 and 2.4:

P(-2.4 ≤ z ≤ 2.4) ≈ 0.9772

Therefore, the probability that |x - 15| ≤ 3 is approximately 0.9772.

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The given statement is True.

What is conditional statement?

A conditional statement is a type of logical statement that has two parts: a hypothesis and a conclusion. The hypothesis is the "if" part of the statement, and the conclusion is the "then" part. The conditional statement asserts that if the hypothesis is true, then the conclusion must also be true.

The given statement is True.

This is an example of a proof by conditional statement. To prove a conditional statement of the form "If A, then B," you assume A and use deductive reasoning to show that B logically follows. In this case, you assume the antecedent (c ⊃ (i ≡ z)) and attempt to prove the consequent (f) within the scope of that assumption. If you are successful, then you have shown that the conditional statement is true.

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The given statement is True.

What is conditional statement?

A conditional statement is a type of logical statement that has two parts: a hypothesis and a conclusion. The hypothesis is the "if" part of the statement, and the conclusion is the "then" part. The conditional statement asserts that if the hypothesis is true, then the conclusion must also be true.

The given statement is True.

This is an example of a proof by conditional statement. To prove a conditional statement of the form "If A, then B," you assume A and use deductive reasoning to show that B logically follows. In this case, you assume the antecedent (c ⊃ (i ≡ z)) and attempt to prove the consequent (f) within the scope of that assumption. If you are successful, then you have shown that the conditional statement is true.

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a) The MLE for p is p = n / (x1+x2+...+xn).

b) The Cramer-Rao lower bound applies.

c) The estimator in part (a) is unbiased.

(a) The probability mass function of the geometric distribution is given by:

P(X=k) = (1-p)^(k-1) * p

The likelihood function for a random sample of size n from the geometric distribution is given by:

L(p) = P(X=x1) * P(X=x2) * ... * P(X=xn)

= (1-p)^(x1-1) * p * (1-p)^(x2-1) * p * ... * (1-p)^(xn-1) * p

= (1-p)^(x1+x2+...+xn-n) * p^n

Taking the natural logarithm of the likelihood function, we get:

ln(L(p)) = (x1+x2+...+xn-n) * ln(1-p) + n * ln(p)

Differentiating with respect to p and setting the derivative equal to zero to find the maximum, we get:

d/dp ln(L(p)) = - (x1+x2+...+xn-n)/(1-p) + n/p = 0

Solving for p, we get:

p = n / (x1+x2+...+xn)

Therefore, the MLE for p is p = n / (x1+x2+...+xn).

(b) The regularity conditions necessary for the Cramer-Rao lower bound to apply are:

The random variable X is independent and identically distributed (i.i.d.).

The probability density function or probability mass function of X depends on a parameter θ that is to be estimated.

The function g(θ) = d/dθ ln(f(X;θ)) is continuous and has finite variance for all θ in an open interval containing θ0.

The integral of |g(θ)|^2f(X;θ) dx over the range of X and the open interval containing θ0 is finite.

For the geometric distribution, these conditions are satisfied:

The random variable X is i.i.d. because each trial is independent and has the same probability of success.

The probability mass function of X depends on the parameter p, which is to be estimated.

g(p) = d/dp ln(f(X;p)) = (1-p)/(p ln(1-p)) is continuous and has finite variance for all p in (0,1).

The integral of |g(p)|^2 f(X;p) dx over the range of X and the interval (0,1) is finite.

Therefore, the Cramer-Rao lower bound applies.

(c) To determine if the estimator in part (a) is asymptotically normal and/or consistent, we need to use the properties of MLEs:

MLEs are asymptotically unbiased, meaning that as the sample size n approaches infinity, the expected value of the estimator approaches the true value of the parameter being estimated.

MLEs are asymptotically efficient, meaning that as the sample size n approaches infinity, the variance of the estimator approaches the Cramer-Rao lower bound.

For the geometric distribution, the expected value of the estimator is:

E(p) = E(n/(x1+x2+...+xn))

= n / E(x1+x2+...+xn)

= n / (n/p)

= p

Therefore, the estimator in part (a) is unbiased.

The variance of the estimator is:

Var(p) = Var(n/(x1+x2+...+xn))

= n^2 Var(1/(x1+x2+...+xn))

= n^2 Var(1/X)

where X = x1

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The surface represented by the equation y = 6z² - 6x² is an elliptic paraboloid, which can be sketched using traces.

To sketch the surface using traces, we can fix one of the variables and let the other two vary. For example, if we fix x at a constant value and vary z and y, we get a set of parabolas that open upward or downward depending on the sign of x. If we fix y at a constant value and vary x and z, we get a set of hyperbolas that open along the x and z axes.

By combining these traces, we can visualize the shape of the surface as an elliptic paraboloid, which is a three-dimensional shape that resembles a shallow bowl or dish. The elliptic paraboloid has a single axis of symmetry and its cross sections in the xz-plane are all parabolas.

Therefore, the surface represented by the equation y = 6z² - 6x² is an elliptic paraboloid, which can be sketched using traces.

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The cross product a x b is 0i + 8j - 8k.

a and b vectors are given by, a = 2i + 2j - 2k and b = 2i - 2j + 2k.

To find the cross product a x b, follow these steps,

1. Write the components of the vectors a and b:
  a = (2, 2, -2)
  b = (2, -2, 2)

2. Use the formula for the cross product:
  a x b = (a2b3 - a3b2)i - (a1b3 - a3b1)j + (a1b2 - a2b1)k

3. Substitute the components of a and b into the formula:
  a x b = ((2)(2) - (-2)(-2))i - ((2)(2) - (-2)(2))j + ((2)(-2) - (2)(2))k

4. Perform the calculations:
  a x b = (4 - 4)i - (4 - (-4))j + (-4 - 4)k

5. Simplify the result:
  a x b = 0i + 8j - 8k

So, the cross product a x b is 0i + 8j - 8k.

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Triangle round

Step-by-step explanation: decimal round to the nearest tenth

The number of degrees of freedom for the pooled t-test is 39. The correct answer is (c).

The formula for the degrees of freedom for a pooled t-test is:

df = n1 + n2 - 2

where n1 and n2 are the sample sizes of the two independent samples.

Substituting the given values, we get:

df = 26 + 15 - 2 = 39

Therefore, the number of degrees of freedom for the pooled t-test is 39. The correct answer is (c).

The degrees of freedom represent the number of independent pieces of information available to estimate the population variance.

In the case of a pooled t-test, we use a pooled estimate of the population variance based on both samples, and the degrees of freedom reflect the loss of information due to estimating the variance from the sample data.

A larger degrees of freedom corresponds to a smaller standard error, and therefore a more precise estimate of the population mean difference.

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Step-by-step explanation:

what is responsible citizenship

Answer:

29

Step-by-step explanation:

145 mile use 5 gallon

so 145÷ by 5 will be 29 which means 29 mile per gallon

The required answer is he equilibrium price will be $8, and the equilibrium quantity will be 320 units.

After the introduction of the new desalination technology, the market supply curve will shift to the right, since firms will be able to produce more water at a lower cost. The new supply curve will be given by:

! = 5P + (8 * quantity) - 5

This is because the cost of producing each unit of water has now increased by $8, but the supply function remains the same.  specifically general equilibrium theory, a perfect market, also known as an atomistic market, is defined by several idealizing conditions, collectively called perfect competition, or atomistic competition. In theoretical models where conditions of perfect competition hold, it has been demonstrated that a market will reach an equilibrium in which the quantity supplied for every product or service, including labor, equals the quantity demanded at the current price

To find the equilibrium price and quantity, we need to set the new supply curve equal to the demand curve:

400 - 10P = 5P + (8 * quantity) - 5

Simplifying and solving for P:

15P = 405 + 8Q

P = 27 + (8/15)Q

Next, we substitute this expression for P into either the demand or supply curve and solve for Q. Let's use the demand curve:

400 - 10(27 + (8/15)Q) = Q + 5

Simplifying:

Q = 60

Therefore, the equilibrium quantity is 60 units of water, and the equilibrium price can be found by plugging this quantity into the expression for P:

P = 27 + (8/15)(60) = $43.20
So the equilibrium price is $43.20 per unit of water.

To find the market supply curve after the introduction of the new desalination technology and the equilibrium price and quantity, follow these steps:

Step 1: Determine the new supply curve.
Since the desalination technology allows for unlimited water production at a cost of $8, the new supply curve will be a horizontal line at P = $8. This is because suppliers will be willing to supply any quantity of water at that price.
Step 2: Find the new equilibrium price and quantity.

To find the new equilibrium, we need to determine where the new supply curve intersects the demand curve. The demand curve is given by Qd = 400 - 10P. To find the intersection, set the price P equal to $8 in the demand equation:

Qd = 400 - 10(8)
Qd = 400 - 80
Qd = 320

So, at the equilibrium price of $8, the quantity demanded is 320 units.

In conclusion, the market supply curve after the introduction of the new desalination technology will be a horizontal line at P = $8. The equilibrium price will be $8, and the equilibrium quantity will be 320 units.

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The hexadecimal equivalent of the decimal number 0.6640625 is 0.AA8.

The conversion process involves multiplying the decimal number by 16 and separating the integer part from the fractional part. The integer part is converted to hexadecimal, while the fractional part is multiplied by 16 again and the process is repeated until the desired accuracy is achieved.

In this case, we can multiply 0.6640625 by 16, which results in 10.625. The integer part, 10, can be converted to hexadecimal as A. We then take the fractional part, 0.625, and multiply it by 16 again, which results in 10. The integer part, 10, can be converted to hexadecimal as A. We can repeat this process to get more accuracy, but since we only need four hexadecimal digits, we can stop here.

Therefore, the hexadecimal representation of the decimal number 0.6640625 is 0.AA8.

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The efficiency approaches 1, which means that θˆ1 and θ² become equally efficient estimators of θ

Linearity of expectation is a property of probability theory that states that the expected value of the sum of random variables is equal to the sum of their individual expected values.

According to the given information:

In this problem, we are given a random sample Y1, Y2, ..., Yn from a uniform distribution on the interval (θ, θ + 1), and we need to find estimators θˆ1 and θˆ2 for the unknown parameter θ.

To show that θˆ1 = Y¯ − 1/2 is an unbiased estimator of θ, we need to show that E(θˆ1) = θ. Using the linearity of expectation, we have:

E(θˆ1) = E(Y¯) - 1/2

= E((Y1 + Y2 + ... + Yn)/n) - 1/2

= (E(Y1) + E(Y2) + ... + E(Yn))/n - 1/2

= (nθ + n/2)/n - 1/2

= θ.

Therefore, θˆ1 is an unbiased estimator of θ.

Similarly, to show that θ² = Y(n) - n/(n+1) is an unbiased estimator of θ, we need to show that E(θ²) = θ. Using the fact that the distribution of Y(n) is given by fY(n)(y) = n(y-θ)n-1 for θ ≤ y ≤ θ+1, we have:

E(θˆ2) = E(Y(n)) - n/(n+1)

= ∫θ^(θ+1) y fY(n)(y) dy - n/(n+1)

= ∫θ^(θ+1) y n(y-θ)n-1 dy - n/(n+1)

= θ + 1/2 - n/(n+1)

= θ.

Therefore, θ² is also an unbiased estimator of θ.

To find the efficiency of θˆ1 relative to θ², we can use the formula:

Efficiency = (Var(θ²))/Var(θˆ1))

To find the variances, we first note that the variance of Y(n) is given by Var(Y(n)) = (1/12n)(1+(n-1)(1/n-1)). Using this, we have:

Var(θˆ1) = Var(Y¯)/n = Var(Y1)/n = 1/12n,

Var(θ²) = Var(Y(n))/(n+1)² = (1/12n)(1+(n-1)(1/n-1))/(n+1)²

Therefore, the efficiency of θˆ1 relative to θ² is:

Efficiency = (Var(θ²))/Var(θ))

= ((1/12n)(1+(n-1)(1/n-1)))/(1/12n)

= 1 + (n-1)/(n+1)²

As n approaches infinity, the efficiency approaches 1, which means that θˆ1 and θ² become equally efficient estimators of θ

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Answer: Ava rode the Ferris wheel approximately 9 times.

Step-by-step explanation:

C = 2πr = 2π(75 ft) ≈ 471.24 ft

If Ava rides the Ferris wheel n times, then the total distance traveled by Ava is:

D = nC

Substituting the values given:

4241.15 ft = n(471.24 ft)

Solving for n:

n = 4241.15 ft / 471.24 ft ≈ 9

The eigenvalues of the coefficient matrix A, find an eigenvector corresponding to the eigenvalue with a positive imaginary part, and the general solution of the given system dx/dt = 7x + 13y and dy/dt = -2x + 9y.

1. Eigenvalues of the coefficient matrix A:
The coefficient matrix A is:
|  7  13 |
| -2   9 |

To find the eigenvalues, we need to solve the characteristic equation, which is:
| (7 - λ) (9 - λ) - (-2)(13) | = 0
Solving this equation, we find the eigenvalues λ = 5 ± 6i.

2. Eigenvector corresponding to the eigenvalue with positive imaginary part:
Let's choose the eigenvalue λ = 5 + 6i. Now we need to solve the system (A - λI)v = 0, where v is the eigenvector.

| (2 - 6i)  13   | |x|   |0|
|  -2      (4 - 6i)| |y| = |0|

Solving this system, we find an eigenvector v = k(3 + 2i, 1), where k is a constant.

3. General solution of the given system:
The general solution can be expressed as:
X(t) = e^(5t) * (C1 * cos(6t) + C2 * sin(6t)) * (3 + 2i, 1)

Where C1 and C2 are constants determined by the initial conditions.

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