During the spring of 2020, the state of Indiana was on lock down orders due to COVID-19. The state's business sales dropped exponentially and are modeled after the following equation:
Sales = 500 (1 - 0.10)^t
where t = number of days and sales = number of millions of dollars.
When sales have reached $23.5 million, it will be declared a statewide economic crisis. How many days until sales reach the economic crisis?

The solutions to the equation to the nearest tenth are x = 10.1 and x = 2.9.

We have,

-2x² - 13x + 20 = -3x²

Combining like terms

-2x² - 13x + 20 = -3x²

x² - 13x + 20 = 0 (adding 3x² to both sides)

Now we can use the quadratic formula to solve for x:

x = (-b ± √(b² - 4ac)) / 2a

In this case,

a = 1, b = -13, and c = 20.

Substituting these values into the quadratic formula:

x = (-(-13) ± √((-13)² - 4(1)(20))) / 2(1)

x = (13 ± √(169 - 80)) / 2

x = (13 ± √(89)) / 2

So the solutions are:

x = (13 + √(89)) / 2

x ≈ 10.1

and

x = (13 - √(89)) / 2

x ≈ 2.9

Therefore,

The solutions to the equation to the nearest tenth are x ≈ 10.1 and x ≈ 2.9.

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The amount of time that students spend studying in the library in one sitting is normally distributed with a mean of 46 minutes and a standard deviation of 19 minutes. Hence the distribution of X is X ~ N (46, 19).

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1. The modal number of pairs of shoes owned by the children is 3.

2. The median number of pairs of shoes owned by the children is 3.

3. The Mean is 3.

1. The modal number of pairs of shoes owned by the children is 3.

2. The median number of pairs of shoes owned by the children

= 14/2 th term

= 7 th term

= 3

3. The Mean

= (1 x 2+ 2 x 3+ 3 x 5+ 4 x 2 + 5 x 1+ 6x 1)/ (2 +3 +5 +2 + 1 +1)

= 42/14

= 3

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Billy's health score go down  by 75% if his sugar resources are quadrupled

Let the amount of sugar in Billy's kitchen be denoted by S and the number of cookies he can bake be denoted by C. Let his health score be denoted by H. Then we have the following relationships:

C ∝ S (directly proportional)

C ∝ 1/H (inversely proportional)

Combining these two relationships, we get:

C ∝ S/H

If S is quadrupled, then C will also quadruple according to the first relationship. However, H will decrease by some percentage x according to the second relationship. To find x, we can use the fact that C is proportional to S/H:

C = k*S/H

where k is a constant of proportionality. If S is quadrupled, then C will also quadruple, so we have:

4C = k4S/H

C = kS/(H/4)

This tells us that if S is quadrupled, then C will be divided by H/4. In other words, C/H will be divided by 4. So, the percentage decrease in H can be found as follows:

C/H → (C/H)/4 = (S/H)/(4/k) → x = 100%*(1 - 1/4) = 75%

Therefore, if Billy's sugar resources are quadrupled, his health score will go down by 75%.

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we can extend {1+x,1} to a basis of P3 by adding x^2.

To extend {1+x,1} to a basis of P3, we need to find one more polynomial that is linearly independent of these two. One way to do this is to choose a polynomial of degree 2, since we are working in P3. Let's try x^2.

We need to check if x^2 is linearly independent of {1+x,1}. This means we need to solve the equation a(1+x) + b(1) + c(x^2) = 0, where a, b, and c are constants.

Expanding this equation gives us a + ax + b + cx^2 = 0. Since x and x^2 are linearly independent, this means that a = 0 and c = 0. Therefore, we are left with just b(1) = 0, which means that b = 0 as well.

This shows that {1+x,1,x^2} is a linearly independent set, which means that it forms a basis of P3. Therefore, we have successfully extended {1+x,1} to a basis of P3 by adding x^2.

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The lengths of the diagonals in the isosceles trapezoid are 11.045 units and 7.2 units.

From the given figure, the vertices of the quadrilateral are (1, 6), (3, 0), (-5, 0) and (-1, 6).

From lower left to upper right: (-5, 0) and (1, 6)

Here, length = √(6+5)²+(1-0)²

= √122

= 11.045 units

From lower right to upper left: (3, 0) and (-1, 6)

Here, length = √(-1-3)²+(6-0)²

= √52

= 7.2 units

Therefore, the lengths of the diagonals in the isosceles trapezoid are 11.045 units and 7.2 units.

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To prove that df ^ dg = |df/dx df/dy| dxdy.|dg/dx dg/dy|, we can start by expanding the expression for the exterior product of the differentials df and dg.

df ^ dg = (df/dx dx + df/dy dy) ^ (dg/dx dx + dg/dy dy)

Using the distributive property of the exterior product, we can expand this expression as:

df ^ dg = (df/dx dx) ^ (dg/dx dx) + (df/dx dx) ^ (dg/dy dy) + (df/dy dy) ^ (dg/dx dx) + (df/dy dy) ^ (dg/dy dy)

Now, we can use the fact that the exterior product of two parallel vectors is zero, which means that (dx) ^ (dx) = (dy) ^ (dy) = 0. This allows us to simplify the expression as:

df ^ dg = (df/dx df/dy dy ^ dx) ^ (dg/dx dg/dy dy ^ dx)

Since dy ^ dx = -dx ^ dy, we can further simplify the expression as:

df ^ dg = -|df/dx df/dy| dx ^ dy ^ (dg/dx dg/dy) dx ^ dy

Now, we can use the fact that dx ^ dy = -dy ^ dx, which means that (dx ^ dy) ^ (dx ^ dy) = 0. This allows us to simplify the expression as:

df ^ dg = -|df/dx df/dy dg/dx dg/dy| (dx ^ dy) ^ (dx ^ dy)

Since (dx ^ dy) ^ (dx ^ dy) = 0, we can conclude that:

df ^ dg = 0

Therefore, we have proven that df ^ dg = |df/dx df/dy| dxdy.|dg/dx dg/dy|.

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The volume of the piñata that Mrs. Powell is making for her son's birthday, would be 2, 016 in ³

The piñata that Mrs. Powell is making, has a composite shape which means that you can find the volume by first finding the volume of the two composite shapes.

The volume of the cube is:

= Length x Width x Height

= 12 x 12 x 12

= 1, 728 in ³

Then the volume of the triangular prism :

= 1 / 2 x base x height x width

= 1 / 2 x 8 x 12 x 6

= 288 in ³

The volume of the pinata is:

= 1, 728 + 288

= 2, 016 in ³

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a. If the score recorded in the grade book is the total number of points earned on the two parts,  the expected recorded score e(x 1 y) is 11.6.

b.  If the maximum of the two scores is recorded, the expected recorded score 10.08.

a) The expected recorded score is given by:

e(x + y) = ΣΣ(x + y) * p(x, y)

So, we have:

e(x + y) = (0+0)*0.02 + (5+0)*0.04 + (10+0)*0.06 + (15+0)*0.02 + (5+10)*0.15 + (10+10)*0.20 + (15+10)*0.15 + (5+15)*0.01 + (10+15)*0.14 + (15+15)*0.01

Simplifying:

e(x + y) = 0.02(0 + 0) + 0.04(5 + 0) + 0.06(10 + 0) + 0.02(15 + 0) + 0.15(5 + 10) + 0.20(10 + 10) + 0.15(15 + 10) + 0.01(5 + 15) + 0.14(10 + 15) + 0.01(15 + 15)

e(x + y) = 11.6

So, the expected recorded score is 11.6.

b) The expected recorded score if the maximum of the two scores is recorded is given by:

e(max(x, y)) = ΣΣ(max(x, y)) * p(x, y)

So, we have:

e(max(x, y)) = max(0, 5)*0.06 + max(5, 0)*0.04 + max(10, 0)*0.06 + max(15, 0)*0.02 + max(5, 10)*0.15 + max(10, 10)*0.20 + max(15, 10)*0.15 + max(5, 15)*0.01 + max(10, 15)*0.14 + max(15, 15)*0.01

Simplifying:

e(max(x, y)) = 0.065 + 0.045 + 0.0610 + 0.0215 + 0.1510 + 0.2010 + 0.1515 + 0.0115 + 0.1415 + 0.0115

e(max(x, y)) = 10.08

So, the expected recorded score is 10.08.

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Hi! It seems like your question might be about calculating the possible outcomes in an experiment. Based on the terms provided, I'll try my best to help you.

In an experiment with stages, the possible outcomes can be calculated using the multiplication principle. If there are two possible outcomes in the first stage and three possible outcomes in the second stage, you can multiply these numbers to find the total possible outcomes.

Total outcomes = (Outcomes in stage 1) x (Outcomes in stage 2)

Total outcomes = 2 x 3 = 6

Based on the given options, none of them match the calculated total outcomes.

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Answer:

$71.75

Rounded : $72

Step-by-step explanation:

To budget for gasoline this month, you can calculate the average amount spent on gasoline over the past four months:

Average = (67 + 78 + 53 + 89) / 4 = amount you should budget (x)

Average = 287 / 4 = x

71.75 = x

(Answer Rounded if that’s what you need but you didn’t ask: $72)

Therefore, you should budget around $71.75 or $ 72 for gasoline this month, assuming your driving habits and gas prices remain relatively constant. However, keep in mind that unexpected changes in gas prices or driving habits may affect your actual spending.

The height of the second cylinder should be 56.25% greater than the height of the first cylinder. (Option 1)

Let's assume the radius of the first cylinder to be 'r' and its height to be 'h'. So, its volume can be represented as V1 = πr^2h.

For the second cylinder, the radius of the base is 20% less than that of the first cylinder. So, the radius of the second cylinder can be represented as 0.8r. Let the height of the second cylinder be represented as 'H'. So, its volume can be represented as V2 = π(0.8r)²H.

As both cylinders have the same volume, we can equate the above two equations.

πr²h = π(0.8r)²H

h = (0.8)²H

H = (1/(0.8)²)h

H = (1.5625)h

Therefore, the height of the second cylinder should be 56.25% greater than the height of the first cylinder.

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Complete Question:

Two cylinders have the same volume, but the radius of the base of the second cylinder is 20% less than the radius of the base of the first. How much greater should be the height of the second cylinder be in comparison to the height in first?

Options:

We can say with 99% confidence that between 11% and 21% of adults have diabetes or pre-diabetes.

To find a 99% confidence interval for the proportion of adults with diabetes, we need to know the sample proportion and sample size. Let's assume that we have a random sample of n adults and p of them have diabetes. Then, the sample proportion is:

P = p/n

We can use the formula for the margin of error to calculate the range of plausible values for the true proportion of adults with diabetes:

margin of error = z*√(P(1-P)/n)

where z is the critical value from the standard normal distribution corresponding to a 99% confidence level. From a standard normal distribution table, we find that z = 2.576.

Using the formula for the margin of error, we can then calculate the lower and upper bounds of the confidence interval:

lower bound = P - margin of error

upper bound = P + margin of error

Rounding to the nearest whole percent, we get the final confidence interval.

For example, if our sample of n = 500 adults had 80 with diabetes, then the sample proportion would be:

P = 80/500 = 0.16

The margin of error would be:

margin of error = 2.576√(0.16(1-0.16)/500) = 0.045

The lower and upper bounds of the confidence interval would be:

lower bound = 0.16 - 0.045 = 0.115 (rounded to 11%)

upper bound = 0.16 + 0.045 = 0.205 (rounded to 21%)

Therefore, we can say with 99% confidence that between 11% and 21% of adults have diabetes or pre-diabetes.

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The experimental probability of landing on heads is 12% greater than the theoretical probability of landing on heads. The correct option is C

Flipping a fair coin, the theoretical chance of landing on heads is 0.5, or 50%. The experimental probability is the ratio of the total number of coin flips to the number of times the coin landed on heads.

The experiment's experimental probability is 62/100 = 0.62 or 62% since the student flipped the coin 100 times and it came up heads 62 times.

We can see that by comparing the experimental and theoretical probabilities, 62% - 50% = 12%

So the experimental probability is 12% greater than the theoretical probability.

Therefore, The experimental probability of landing on heads is 12% greater than the theoretical probability of landing on heads.

Therefore, The correct option is C

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From the trigonometric ratios, the value of sine trigonometric ratio for angle C, i.e., sin(C) in above right angled triangle CDE, is equals to the 0.55.

Trigonometry is a branch of mathematics. The trigonometric ratios are special measurements of a right triangle the right angle trigonometric ratios, these ratios describe the relationship between the sides and angles in a right triangle. The six trigonometric ratios in a right angled triangle are defined as sine, cosine, tangent, cosecant, secant, and cotangent. The symbols used for them are sin, cos, sec, tan, csc, cot. The three main ratios are defined as below

We have a right angled triangle CDE with 90° measure of angle D present in above figure. We have to determine the value of sine angle of C. Consider angle C priority,

Height or opposite of triangle = 11

Length of hypotenuse of triangle = 20

Using the above formula for sine trigonometric ratio, [tex]sin \: C = \frac{opposite}{hypotenuse}[/tex]

Substitute all known values in above formula, [tex]= \frac{11}{20}[/tex]

= 0.55

Hence, required value is 0.55.

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Complete question :

The above figure complete the question.

Find the value of sin C rounded to the nearest hundredth, if necessary.

20

E

11

D

Help PLEASE

Answer:

they jumped

Step-by-step explanation:

They jump because they want to escape Mario and Javier. Paula is very nervous because there are many people, it is not possible to escape quickly

I hope I’m right if not I’m sorry

Answer:

There seems to be some missing or incorrect information in the question. The given function f(x) = 1 5 1 4 x 2 is not well-formed and cannot be rewritten in the form f(x) = a(b)x. Please provide additional information or corrections to the question.

Step-by-step explanation:

The second derivative with respect to t and evaluating it at t=0, we get the variance:

Var(X+Y) = Mx+y''(0) - [Mx+y'(0)]^2 = [-0.18(4e^-2t) + 0

Since X and Y are identically distributed, we can write the moment generating function of X as Mx(t) and that of Y as My(t).

Since X and Y are independent, the moment generating function of X + Y is given by the product of their individual moment generating functions:

Mx+y(t) = Mx(t)My(t)

We are given the moment generating function of X + Y as:

Mx+y(t) = 0.09e^-2t + 0.24e^t + 0.34 + 0.24e^t + 0.09e^2t

We can rewrite this as:

Mx+y(t) = 0.09(e^-2t + e^2t) + 0.48e^t + 0.34

Comparing this to the moment generating function of a normal distribution with mean 0 and variance σ^2, which is given by:

M(t) = e^(μt + σ^2t^2/2)

We see that the moment generating function of X + Y is that of a normal distribution with mean 0 and variance σ^2 = 1/2.

Thus, X + Y ~ N(0, 1/2).

Since X and Y are identically distributed, X ~ N(0, 1/4) and Y ~ N(0, 1/4).

Therefore,

P(X ≤ 0) = P(X - Y ≤ -Y) = P(Z ≤ -Y/√(1/2)),

where Z ~ N(0,1).

Since X and Y are identically distributed, we have

P(X - Y ≤ -Y) = P(Y - X ≤ X) = P(-Y + X ≤ X) = P(X ≤ Y)

So,

P(X ≤ 0) = P(X ≤ Y) = P(X - Y ≤ 0)

= P[(X+Y) - 2Y ≤ 0]

= P[Z ≤ 2(Y - X)/√2]

where Z ~ N(0,1).

Now, let's find the mean and variance of X + Y:

E[X + Y] = E[X] + E[Y] = 2E[X]

Since X and Y are identically distributed, we have E[X] = E[Y].

Thus, E[X + Y] = 2E[X] = 2E[Y]

And,

Var(X + Y) = Var(X) + Var(Y) = 2Var(X)

Since X and Y are identically distributed, we have Var(X) = Var(Y).

Thus, Var(X + Y) = 2Var(X)

Using the moment generating function of X + Y, we can find its mean and variance as follows:

Mx+y(t) = E[e^(t(X+Y))]

Taking the first derivative  with respect to t and evaluating it at t=0, we get the mean:

E[X+Y] = Mx+y'(0) = [0.09(-2e^-2t) + 0.48e^t + 0.24e^t + 0.18(2e^2t)]|t=0

= -0.18 + 0.24 + 0.18 = 0.24

Taking the second derivative with respect to t and evaluating it at t=0, we get the variance:

Var(X+Y) = Mx+y''(0) - [Mx+y'(0)]^2 = [-0.18(4e^-2t) + 0.

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(a) Letting X1, ..., Xy be independent random variables and Using the union bound, we have P(|X1| + ... + |XN| ≥ t) ≤ P(|X1| ≥ t/N) + ... + P(|XN| ≥ t/N) ≤ 2N[tex]e^{(-tε/N)}[/tex] for all t > 0.

(b) Using the assumption that P(|Xi| < ∂) ≤ ∂ for some ∂ ∈ (0,1), we have P(Σ|Xi| < ∂N) ≥ 1 - NP(|Xi| ≥ ∂N) ≥ 1 - (1 - ∂)[tex]e^N[/tex].

Setting t = 2N[tex]e^ε[/tex], we obtain

which is equivalent to

By setting ∂ = 2Ne**ε/N, we get

P(Σ|Xi| < ∂) ≥ 1 - e**(-ε), and therefore,

Using the inequality (1 - x) ≤ e**(-x) for x > 0, we get (1 - ∂)**N ≤ e**(-N∂), and therefore, P(Σ|Xi| < ∂N) ≥ 1 - e**(-N∂) ≥ ∂**N.

Thus, we have shown that NP(Σ|Xi| < ∂N) ≥ ∂**N for some ∂ ∈ (0,1) and P(|X1| + ... + |XN| ≥ t) ≤ P(|X1| ≥ t/N) + ... + P(|XN| ≥ t/N) ≤ 2N[tex]e^{(-tε/N)}[/tex] for all t > 0

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Answer:

Step-by-step explanation:

If the squares of the two shorter sides add up to the square of the hypotenuse, the triangle contains a right angle.

10.-8.=2h

12:00-10:30=1.5h

2h-1.5h=0.5h

13km÷0.5h×2h=52km

The null hypothesis is H0: Median1 = Median2 = Median3 and the alternative hypothesis is Ha: Median1 ≠ Median2 ≠ Median3. The test statistic is H = 9.73. The p-value is 0.007. Reject H0. There is sufficient evidence to conclude that there is a significant difference in the time required by the three methods.

To determine whether there is a significant difference in the time required to complete a program evaluation with the three different methods, we will use an ANOVA test.

1. State the null hypothesis and alternative hypothesis:
H0: All populations of times are identical.
Ha: Not all populations of times are identical.

2. Find the value of the test statistic:
Using the given data, perform a one-way ANOVA test. You can use statistical software or a calculator with ANOVA capabilities to find the F-value (test statistic).

3. Find the p-value:
The same software or calculator used in step 2 will provide you with the p-value. Remember to round your answer to three decimal places.

4. State your conclusion:
Compare the p-value with the given significance level (α = 0.05).
- If the p-value is less than α, reject H0. There is sufficient evidence to conclude that there is a significant difference in the time required by the three methods.
- If the p-value is greater than or equal to α, do not reject H0. There is not sufficient evidence to conclude that there is a significant difference in the time required by the three methods.

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[tex]\textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ h=96 \end{cases}\implies V=\cfrac{\pi r^2 (96)}{3} \\\\[-0.35em] ~\dotfill\\\\ \textit{volume of a sphere}\\\\ V=\cfrac{4\pi r^3}{3}~~\hspace{9em}\stackrel{\textit{since we know both Volumes are equal}}{\cfrac{4\pi r^3}{3}~~ = ~~\cfrac{\pi r^2 (96)}{3}}[/tex]

[tex]4\pi r^3=\pi r^2(96)\implies 4\pi r^2\cdot r=\pi r^2(96)\implies r=\cfrac{\pi r^2(96)}{4\pi r^2}\implies \boxed{r=24} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \stackrel{ \textit{\LARGE cone} }{\cfrac{\pi (24)^2(96)}{3}}\implies \stackrel{ \textit{\LARGE sphere} }{\cfrac{4\pi (24)^3}{3}}\implies18432\pi ~~ \approx ~~ \text{\LARGE 57905.84}~units^3[/tex]

Answer:

Step-by-step explanation:

The height of the cylinder is 7/2 inches.

Remember that for a cylinder of radius R and height H, the volume is:

V = pi*R²*H

Where pi = 22/7

We know that:

R = (1/3) in

V = (1 + 2/9) in³ = 11/9 in³

Replacing these values we will get:

11/9  = (22/7)*(1/3)²*H

11/9 = (22/7)*(1/9)*H

11 =(22/7)*H

11*(7/22) = H

7/2 = H

The answer is 7 halves inches.

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The efficient allocation of apples and bananas between the two individuals is:

A1 = B1 = 50 (allocated to individual 1)

A2 = 50 and B2 = 100 (allocated to individual 2)

What is utility?

In mathematics, utility refers to a measure of the preference or satisfaction an individual derives from consuming goods or services.

To recommend an efficient allocation of apples and bananas between the two individuals, we need to find a solution that maximizes the total utility of both individuals subject to the constraint that all the goods must be distributed. In other words, we need to solve the following optimization problem:

Maximize U1(A1, B1) + U2(A2, B2) subject to A1 + A2 = 100 and B1 + B2 = 150

Let's begin by solving for individual 1's optimal allocation. We can use the first-order conditions to find the optimal values of A1 and B1 that maximize U1(A1, B1). Taking partial derivatives with respect to A1 and B1 and setting them equal to zero, we get:

∂U1/∂A1 = 0 => 0 = 0

∂U1/∂B1 = 0 => 2 = 2B1/B2

Solving for B1/B2, we get B1/B2 = 1. This means that the optimal allocation for individual 1 is to receive an equal number of bananas and apples, i.e., A1 = B1 = 50.

Next, we solve for individual 2's optimal allocation. Following the same approach, we find that the optimal allocation for individual 2 is to receive all the remaining bananas and apples, i.e., A2 = 50 and B2 = 100.

Therefore, the efficient allocation of apples and bananas between the two individuals is:

A1 = B1 = 50 (allocated to individual 1)

A2 = 50 and B2 = 100 (allocated to individual 2)

This allocation is efficient because it maximizes the total utility of both individuals subject to the constraint that all the goods must be distributed. If we try to reallocate the goods in any other way, we will end up with a lower total utility for both individuals.

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The efficient allocation of apples and bananas between individuals 1 and 2 is as follows:

Individual 1 gets 60 apples and 75 bananas

Individual 2 gets 40 apples and 75 bananas

To determine the most efficient allocation of apples and bananas between individuals 1 and 2, we must maximize the total utility of both individuals while keeping in mind that all of the apples and bananas must be distributed.

From the constraint equation:

A1 + A2 = 100

B1 + B2 = 150

Now, let's write out the total utility function:

U = U1 + U2

U = 2A1 + B1 + 2A2 + B2 + 2A2B2

Using the Lagrangian method:

L = 2A1 + B1 + 2A2 + B2 + 2A2B2 - λ1(A1 + A2 - 100) - λ2(B1 + B2 - 150)

Taking the partial derivative of L with respect to each variable and equating them to zero, we get:

∂L/∂A1 = 2 - λ1 = 0

∂L/∂A2 = 2 + 4B2 - λ1 = 0

∂L/∂B1 = 1 - λ2 = 0

∂L/∂B2 = 1 + 2A2 - λ2 + 4A2B2 = 0

∂L/∂λ1 = A1 + A2 - 100 = 0

∂L/∂λ2 = B1 + B2 - 150 = 0

Solving these equations, we get:

λ1 = 2, λ2 = 1, A1 = 60, A2 = 40, B1 = 75, B2 = 75

Therefore, the efficient allocation of apples and bananas between individuals 1 and 2 is as follows:

Individual 1 gets 60 apples and 75 bananas

Individual 2 gets 40 apples and 75 bananas

This allocation maximizes the total utility of both individuals subject to the constraint that all the apples and bananas are distributed.

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Answer:

Alot

Step-by-step explanation:

Think abt it

In exercise 9.38(b), we are given a random sample Y1, Y2, ..., Yn from a normal distribution with mean μ and unknown variance σ^2. The likelihood function for this sample is:

L(μ, σ^2) = (2πσ^2)^(-n/2) exp[-∑(Yi-μ)^2/(2σ^2)]

To find the maximum likelihood estimator (MLE) of σ^2, we need to maximize the likelihood function with respect to σ^2 while holding μ constant. Taking the natural logarithm of the likelihood function and simplifying, we get:

ln L(μ, σ^2) = -n/2 ln(2π) - n/2 ln(σ^2) - ∑(Yi-μ)^2/(2σ^2)

Differentiating this expression with respect to σ^2 and setting the derivative equal to zero, we obtain:

d/dσ^2 ln L(μ, σ^2) = -n/(2σ^2) + ∑(Yi-μ)^2/(2σ^4) = 0

Solving for σ^2, we get:

σ^2 = ∑(Yi-μ)^2/n

Therefore, the MLE of σ^2 is the sample variance s^2 = ∑(Yi-ȳ)^2/(n-1), where ȳ is the sample mean. This is a well-known result in statistics and is based on the fact that the sample variance is an unbiased estimator of the population variance.

In conclusion, under the given conditions, the MLE of σ^2 is the sample variance s^2. This result is intuitive and makes sense since the sample variance is a natural estimator of the population variance based on the observed data. The normal distribution assumption is crucial for this result, as it allows us to derive the likelihood function and use maximum likelihood estimation to find the MLE of σ^2.

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The population proportion of members who favor holding the charity event in the civil club is approximately 94.12%. To calculate the population proportion of members in the civil club who favor holding the charity event, follow these steps:

The population proportion of members who favor holding the charity event can be calculated by dividing the number of members who said yes by the total number of members in the club.Proportion = Number of members who said yes / Total number of members in the club
Step:1. Identify the total number of members in the civil club: 85 members.
Step:2. Identify the number of members who said yes to holding the charity event: 80 members.
Step:3. Divide the number of members who said yes by the total number of members: 80 / 85.
Step:4. Convert the result to a percentage by multiplying by 100: (80 / 85) x 100.
So, the population proportion of members who favor holding the charity event in the civil club is approximately (80 / 85) x 100 = 94.12%.

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