Jill ate 45 ounces more candy then grag together jill and greg ate a full 125 ounce bag of candy. how much candy did each of eat?

To raise 4 to the 13th power using regular exponentiation, a total of 12 multiplications are required.

To raise 4 to the 13th power using regular exponentiation, we can start by multiplying 4 by itself 13 times. However, this would require a total of 13 multiplications, which is not the most efficient way to calculate 4^13.

Instead, we can use a method called "exponentiation by squaring", which reduces the number of multiplications required. Here's how it works:

Start by writing the exponent (13) in binary form: 13 = 1101 (in binary).

Starting with the base (4), square it repeatedly, each time moving from right to left in the binary representation of the exponent.

Whenever we encounter a "1" in the binary representation of the exponent, we multiply the current result by the base.

Using this method, we can calculate 4^13 with the following steps:

So, using exponentiation by squaring, we only needed a total of 7 multiplications instead of 13, which is much more efficient.

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The first five terms of the sequence are: 1, 5/8, 25/64, 125/512, 625/4096.

The sequence converges and the limit is 8/3.

To find the first five terms of the sequence with [(1−3/8)][∞]=1, we can start by simplifying the expression in the brackets:

(1−3/8) = 5/8

So, the sequence becomes:

(5/8)ⁿ, where n starts at 0 and goes to infinity.

The first five terms of the sequence are:

(5/8)⁰ = 1
(5/8)¹ = 5/8
(5/8)² = 25/64
(5/8)³ = 125/512
(5/8)⁴ = 625/4096

To determine whether the sequence converges, we need to check if it approaches a finite value or not. In this case, we can see that the terms of the sequence are getting smaller and smaller as n increases, so the sequence does converge.

To find its limit, we can use the formula for the limit of a geometric sequence:

limit = a/(1-r)

where a is the first term of the sequence and r is the common ratio.

In this case, a = 1 and r = 5/8, so:

limit = 1/(1-5/8) = 8/3

Therefore, the limit of the sequence is 8/3.

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The value of α is 1/2.

The expected value (E(X)) is 2.

To find the value of α, we need to ensure that the probabilities sum up to 1 over the entire range of non-negative integers.

The probability mass function is given by: P(X = i) = α/2^i

For a probability mass function to be valid, the sum of all probabilities must equal 1.

∑ P(X = i) = 1

Substituting the given probability mass function into the sum:

∑ (α/2^i) = 1

Since the range of i is from 0 to infinity, we can rewrite the sum as a geometric series:

α/2^0 + α/2^1 + α/2^2 + ...

Using the formula for the sum of an infinite geometric series:

S = a / (1 - r)

where a is the first term and r is the common ratio, in this case, 1/2.

α / (1 - 1/2) = 1

Simplifying:

α / (1/2) = 1

2α = 1

α = 1/2

Now let's calculate the expected value (E(X)):

E(X) = ∑ (i * P(X = i))

Substituting the probability mass function:

E(X) = ∑ (i * α/2^i)

Using the formula for the sum of an infinite geometric series:

E(X) = α / (1 - r)^2

where a is the first term and r is the common ratio, in this case, 1/2.

E(X) = (1/2) / (1 - 1/2)^2

E(X) = (1/2) / (1/2)^2

E(X) = (1/2) / (1/4)

E(X) = 2

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The lengths of the triangle is solved by law of sines and a = 16.39 units and c = 24.02 units

Given data ,

Let the triangle be represented as ΔABC

where the measure of lengths are

AB = c

BC = a

And , AC = b = 17 units

From the law of sines , we get

Law of Sines :

a / sin A = b / sin B = c / sin C

On simplifying , we get

c / sin 92° = 17 / sin 45°

Multiply by sin 92° on both sides , we get

c = ( 0.99939082701 / 0.70710678118 ) x 17

c = 24.02 units

Now , the measure of ∠A = 180° - ( 92° + 45° )

∠A = 43°

a / sin 43° = 17 / sin 45°

Multiply by sin 43° on both sides , we get

a = ( 0.68199836006 / 0.70710678118 ) x 17

a = 16.39 units

Hence , the triangle is solved

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Therefore, we cannot definitively say whether the error is ±2 or not. It depends on the standard deviation or standard error of the mean, which is not provided in the given information.

A confidence interval for the mean is given by the formula:

(mean) ± (margin of error)

where the margin of error is calculated as:

margin of error = (z-score)*(standard deviation/sqrt(n))

where n is the sample size, and z-score is the critical value of the standard normal distribution corresponding to the desired level of confidence. For example, for a 95% confidence interval, the z-score would be 1.96.

In this case, the 95% confidence interval for the mean was computed to be (10, 14) based on a sample size of 100. This means that the mean falls between 10 and 14 with a 95% level of confidence.

To determine the margin of error, we need to know the standard deviation of the population or the standard error of the mean. Without this information, we cannot accurately calculate the margin of error.

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To prove that 7 divides the expression 3^(4n+1) - 5^(2n-1) for every positive integer n, we can use mathematical induction.

Base case: Let n = 1. Then,

3^(4n+1) - 5^(2n-1) = 3^(5) - 5^(1) = 243 - 5 = 238

Since 238 is divisible by 7, the base case holds true.

Inductive step: Assume that the statement is true for some arbitrary positive integer k, i.e.,

7 | [3^(4k+1) - 5^(2k-1)]

We need to show that the statement is also true for k+1.

We have,

3^(4(k+1)+1) - 5^(2(k+1)-1)

= 3^(4k+5) - 5^(2k+1)

= 3^4 * 3^(4k+1) - 25 * 5^(2k-1)

= 81 * 3^(4k+1) - 25 * 5^(2k-1)

= 7 * (9 * 3^(4k+1) - 5^(2k-1)) + 2 * 5^(2k-1)

Since 9 * 3^(4k+1) - 5^(2k-1) is an integer, and 2 * 5^(2k-1) is divisible by 7 (since 5^2 = 25 is congruent to 4 modulo 7), it follows that

7 | [3^(4(k+1)+1) - 5^(2(k+1)-1)]

Thus, by mathematical induction, the statement is true for all positive integers n.

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The length of the third side of a right triangle with legs of lengths 3 and 5 is √34, and the length of either leg is 3.

Let's say that the two legs of a right triangle have the lengths a and b, and the length of the hypotenuse is c.

The Pythagorean Theorem states that

a² + b² = c².

If the legs or the hypotenuse are not whole numbers, the answer must be given in the form of an expression using the symbol (i.e., it is a surd).

Let's take an example of a triangle having legs of lengths 3 and 5:

For a right triangle with legs of lengths 3 and 5, the square of the hypotenuse can be determined using the Pythagorean Theorem:

a² + b² = c²

3² + 5² = c²

9 + 25 = c²

34 = c²

c = √34

The length of the hypotenuse is equal to √34, which is not a whole number.

If we were asked to find the length of one of the legs, we could rearrange the Pythagorean Theorem to solve for a or b.

For example, to solve for a, we could rewrite the equation as:

a² = c² - b²

a² = (√34)² - 5²

a² = 34 - 25

a² = 9

a = √9

a = 3

Therefore, the length of the third side of a right triangle with legs of lengths 3 and 5 is √34, and the length of either leg is 3.

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To find the number of integers between 400 and 851 inclusive that are divisible by four, we need to determine the number of multiples of four in that range. The first multiple of four in the range is 400, and the last multiple of four is 848. To find how many multiples of four there are, we can subtract the two numbers and divide by four, then add one (because we need to include the first multiple).


- First multiple of four in the range: 400
- Last multiple of four in the range: 848
- Difference between the two: 848 - 400 = 448
- Divide by four: 448 ÷ 4 = 112
- Add one: 112 + 1 = 113

Therefore, there are 113 integers between 400 and 851 inclusive that are divisible by four.

There are 113 integers between 400 and 851 inclusive that are divisible by four.

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Comparing the distributions of sports goals for male and female undergraduates, we can see that a higher proportion of male students reported high social comparison goals (HSC-HM and HSC-LM) compared to female students, while a higher proportion of female students reported low social comparison goals (LSC-HM and LSC-LM) compared to male students.

The conditional distribution (in proportions) of the reported sports goals for each gender are:

Female:

HSC-HM: 16/70 = 0.229

HSC-LM: 6/70 = 0.086

LSC-HM: 23/70 = 0.329

LSC-LM: 25/70 = 0.357

Male:

HSC-HM: 33/70 = 0.471

HSC-LM: 19/70 = 0.271

LSC-HM: 4/70 = 0.057

LSC-LM: 14/70 = 0.2

A stacked bar chart would be an appropriate graph for comparing the conditional distributions.

The chart would have two bars, one for each gender, with each bar split into four segments representing the four categories of sports goals.

Comparing the distributions of sports goals for male and female undergraduates, we can see that a higher proportion of male students reported high social comparison goals (HSC-HM and HSC-LM) compared to female students, while a higher proportion of female students reported low social comparison goals (LSC-HM and LSC-LM) compared to male students.

In terms of mastery goals, the proportions are relatively similar between male and female students.

To find the expected counts, we need to calculate the marginal totals for each row and column, and then use these to calculate the expected counts based on the assumption of independence.

The results are displayed in the table below:

Observed Counts and Expected Counts for Sports Goals

Goal HSC-HM HSC-LM LSC-HM LSC-LM Total

Female (Observed) 16 6 23 25 70

Expected 19.1 10.9 23.9 16.1 70

Male (Observed) 33 19 4 14 70

Expected 29.9 17.1 3.1 19.9 70

Total 49 25 27 39 140

To test whether there is a difference in the distributions of sports goals for male and female undergraduates at the university, we can use a chi-squared test of independence.

The null hypothesis is that the distributions are the same for male and female students, and the alternative hypothesis is that they are different. The test statistic is calculated as:

chi-squared = sum((observed - expected)² / expected)

Using the values from the table above, we get:

chi-squared = (16-19.1)²/19.1 + (6-10.9)²/10.9 + (23-23.9)²/23.9 + (25-16.1)²/16.1 + (33-29.9)²/29.9 + (19-17.1)²/17.1 + (4-3.1)²/3.1 + (14-19.9)²/19.9

= 10.32

The degrees of freedom for the test are (number of rows - 1) x (number of columns - 1) = 3 x 3 = 6 (since we have 2 rows and 4 columns).

Using a chi-squared distribution table with 6 degrees of freedom and a significance level of 0.05, the critical value to be 12.59.

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We can use the double number line to find the cost of buying a different number of bags of rice or the number of bags of rice we can buy for a given amount of money.

The given double number line shows that it costs $3 to buy 2 bags of rice. This means that the cost of 1 bag of rice is $1.50.

To find the cost of buying a different number of bags of rice, we can use the double number line.

Suppose we want to know the cost of buying 5 bags of rice. We can do this by starting at the number 2 on the top line and following the diagonal line down to the bottom line.

Then, we can read off the number on the bottom line that corresponds to 5 on the top line.

This gives us a cost of $7.50 for 5 bags of rice.

We can also use the double number line to find the number of bags of rice that we can buy for a given amount of money.

For example, if we have $6, we can find the number of bags of rice we can buy by starting at the number $3 on the bottom line and following the diagonal line up to the top line. Then, we can read off the number on the top line that corresponds to $6 on the bottom line.

This gives us a value of 4 for the number of bags of rice.

Therefore, we can use the double number line to find the cost of buying a different number of bags of rice or the number of bags of rice we can buy for a given amount of money.

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To express 2/3 and 3/4 as a pair of fractions with a common denominator, we can find the least common multiple (LCM) of the denominators and then adjust the numerators accordingly.

To begin, we need to find the least common multiple (LCM) of the denominators, which in this case is 12. Next, we convert 2/3 and 3/4 to fractions with a common denominator of 12.
For 2/3, we multiply both the numerator and denominator by 4 to get 8/12. Since 2 multiplied by 4 is 8, and 3 multiplied by 4 is 12.
For 3/4, we multiply both the numerator and denominator by 3 to get 9/12. Since 3 multiplied by 3 is 9, and 4 multiplied by 3 is 12.
Now, we have 8/12 and 9/12 as a pair of fractions with a common denominator of 12. These fractions can be compared or used in further calculations since they have the same denominator.

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The area of triangle PQR is 336 square units.

First, we can find the length of PM using the midpoint formula:

PM = (PQ) / 2 = 36 / 2 = 18

Next, we can use the angle bisector theorem to find the lengths of PX and QX. Since PX bisects angle QPR, we have:

PX / RX = PQ / RQ

Substituting in the given values, we get:

PX / RX = 36 / 26

Simplifying, we get:

PX = (18 * 36) / 26 = 24.92

RX = (26 * 18) / 26 = 18

Now, we can use the Pythagorean theorem to find the length of AX:

AX² = PX² + RX²

AX² = 24.92² + 18²

AX² = 621 + 324

AX = √945

AX = 30.74

Since Y lies on the perpendicular bisector of PQ, we have:

PY = QY = PQ / 2 = 18

Therefore,

AY = AX - XY = 30.74 - 8

                      = 22.74

Finally, we can use Heron's formula to find the area of triangle PQR:

s = (36 + 22 + 26) / 2 = 42

area(PQR) = sqrt(s(s-36)(s-22)(s-26)) = sqrt(42*6*20*16) = 336

Therefore, the area of triangle PQR is 336 square units.

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The closest measurement to the number of liters of water in Dexter's aquarium is 17.1 liters.

Dexter's aquarium holds 4.5 gallons of water. To convert this measurement to liters, we need to multiply it by the conversion factor of 3.8 liters per gallon. Therefore, 4.5 gallons multiplied by 3.8 liters per gallon equals 17.1 liters. Since there are approximately 3.8 liters in 1 gallon, we can multiply the number of gallons by this conversion factor to find the equivalent volume in liters. In this case, 4.5 gallons multiplied by 3.8 liters per gallon equals 17.1 liters. Hence, 17.1 liters is the closest measurement to the number of liters of water in Dexter's aquarium.

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The distance between the points (-2, -5) and (-14, -10) is 13 units.

To find the distance between the points (-2, -5) and (-14, -10) using the distance formula, follow these steps:

1. Identify the coordinates: Point A is (-2, -5) and Point B is (-14, -10).
2. Apply the distance formula: d = √[(x2 - x1)^2 + (y2 - y1)^2]
3. Substitute the coordinates into the formula: d = √[(-14 - (-2))^2 + (-10 - (-5))^2]
4. Simplify the equation: d = √[(-12)^2 + (-5)^2]
5. Calculate the squared values: d = √[(144) + (25)]
6. Add the squared values: d = √(169)
7. Calculate the square root: d = 13

So, The distance between the points (-2, -5) and (-14, -10) is 13 units.

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The correct hypothesis test for this scenario is (b) H0 : μ > 10 against HA : μ ≤ 10.

The null hypothesis (H0) is the hypothesis that is being tested, which is that the population mean of excess weight amongst Australians is greater than 10. The alternative hypothesis (HA) is the hypothesis that we are trying to determine if there is evidence to support, which is that the population mean is less than or equal to 10.

Option (a) H0 : μ > 10 against HA : μ = 10 is incorrect because the alternative hypothesis assumes a specific value for the population mean, which is not the case here. We are trying to determine if the population mean is less than or equal to a certain value, not if it is equal to a specific value.

Option (c) H0 : μ = 10 against HA : μ > 10 is incorrect because the null hypothesis assumes a specific value for the population mean, which is not the case here. We are trying to determine if the population mean is greater than a certain value, not if it is equal to a specific value.

Option (d) H0 : μ = 10 against HA : μ ≠ 10 is incorrect because the alternative hypothesis assumes a two-tailed test, which means we are trying to determine if the population mean is either greater than or less than the specified value. However, in this scenario, we are only interested in determining if the population mean is less than or equal to the specified value.

Option (e) none of these is also incorrect because as discussed above, option (b) is the correct hypothesis test for this scenario.

In summary, option (b) H0 : μ > 10 against HA : μ ≤ 10 is the correct hypothesis test for determining if there is evidence to support the claim that the population mean of excess weight amongst Australians is less than or equal to 10.

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a) F(x,y,z) = xy'z + xy'z' + xyz + xyz' + x'yz + x'yz' + x'y'z + x'y'z'

b) F(x,y,z) = xy + xz'y + x'yz'

c) F(x,y,z) = xy'z' + xyz' + x'yz

d) F(x,y,z) = xy'z + xyz' + x'yz + x'y'z

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The given relation is P = 162x – 81x2, where P represents the profit in thousands of dollars, and x represents the number of snowboards in thousands.

Given that the company has to break even, it means the profit should be zero. Therefore, we need to solve the equation P = 0.0 = 162x – 81x² to find the number of snowboards that must be produced for the company to break even.To solve the above quadratic equation, we first need to factorize it.0 = 162x – 81x²= 81x(2 - x)0 = 81x ⇒ x = 0 or 2As the number of snowboards can't be zero, it means that the company has to produce 2 thousand snowboards to break even. Hence, the required number of snowboards that must be produced for the company to break even is 2000.

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The specific heat capacity of the metal can be expressed as the ratio of the product of the specific heat capacity and mass of the surroundings to the mass of the metal which is c = (ms) / m.

The specific heat capacity of a metal can be derived using the heat balance equation for an isolated system, given by equation (14.2), which relates the heat gained or lost by the system to the change in its temperature and its heat capacity.

According to the heat balance equation for an isolated system, the heat gained or lost by the system (Q) is given by:

Q = mcΔTwhere m is the mass of the metal, c is its specific heat capacity, and ΔT is the change in its temperature.

For an isolated system, the heat gained or lost by the metal must be equal to the heat lost or gained by the surroundings, which can be expressed as:

Q = -q = -msΔT

where q is the heat gained or lost by the surroundings, s is the specific heat capacity of the surroundings, and ΔT is the change in temperature of the surroundings.

Equating the two expressions for Q, we get:

mcΔT = msΔT

Simplifying and rearranging, we get:

c = (ms) / m

Therefore, the specific heat capacity of the metal can be expressed as the ratio of the product of the specific heat capacity and mass of the surroundings to the mass of the metal.

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The Cartesian coordinates for point (c) are: (x, y) = (4.5, -7.794) which can be plotted on the graph using polar coordinates.

A system of describing points in a plane using a distance and an angle is known as polar coordinates. The angle is measured from a defined reference direction, typically the positive x-axis, and the distance is measured from a fixed reference point, known as the origin. In mathematics, physics, and engineering, polar coordinates are useful for defining circular and symmetric patterns.

(a) Polar coordinates (8, 4/3)
To convert to Cartesian coordinates, use the formulas:
x = r*[tex]cos(θ)[/tex]
y = r*[tex]sin(θ)[/tex]
For point (a):
x = 8 * [tex]cos(4/3)[/tex]
y = 8 * [tex]sin(4/3)[/tex]

Therefore, the Cartesian coordinates for point (a) are:
(x, y) = (-4, 6.928)

(b) Polar coordinates (-4, 3/4)
For point (b):
x = -4 * [tex]cos(3/4)[/tex]
y = -4 * [tex]sin(3/4)[/tex]

Therefore, the Cartesian coordinates for point (b) are:
(x, y) = (-2.828, -2.828)

(c) Polar coordinates (-9, [tex]-\pi /3[/tex])
For point (c):
x = -9 * [tex]cos(-\pi /3)[/tex]
y = -9 * [tex]sin(-\pi /3)[/tex]

Therefore, the Cartesian coordinates for point (c) are:
(x, y) = (4.5, -7.794)

Now you have the Cartesian coordinates for each point, and you can plot them on a Cartesian coordinate plane.

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The given sequence is a factorial sequence where each term is calculated by taking the difference between 3 and 1, and then taking the factorial of both the numbers.

So, the first term of the sequence will be (3-1)! * (3+1)! = 2! * 4! = 2 * 24 = 48.

The second term of the sequence will be (3-1)! * (3+2)! = 2! * 5! = 2 * 120 = 240.

The third term of the sequence will be (3-1)! * (3+3)! = 2! * 6! = 2 * 720 = 1440.

And so on.

As we can see, the terms of the sequence are increasing rapidly with each step. Therefore, we can say that the behavior of the sequence is that it grows very quickly and gets larger with each term.

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The cyclist travels a total of 15.44 kilometers if he continues to accelerate for 18 more minutes.

To solve this problem, we can use the following steps:

1. Calculate the cyclist's average speed in the first 6 minutes.

Average speed = distance / time = 16.6 km / 6 min = 2.77 km/min

2. Calculate the cyclist's total distance traveled in the first 6 minutes.

Total distance = average speed * time = 2.77 km/min * 6 min = 16.6 km

3. Assume that the cyclist's acceleration is constant. This means that his speed will increase linearly with time.

4. Calculate the cyclist's speed after 18 minutes.

Speed = initial speed + acceleration * time = 2.77 km/min + (constant acceleration) * 18 min

5. Calculate the cyclist's total distance traveled after 18 minutes.

Total distance = speed * time = (2.77 km/min + (constant acceleration) * 18 min) * 18 min

6. Solve for the constant acceleration.

Total distance = 15.44 km

2.77 km/min + (constant acceleration) * 18 min = 15.44 km

(constant acceleration) * 18 min = 12.67 km

constant acceleration = 0.705 km/min²

7. Substitute the value of the constant acceleration in step 6 to calculate the cyclist's total distance traveled after 18 minutes.

Total distance = speed * time = (2.77 km/min + (0.705 km/min²) * 18 min) * 18 min = 15.44 km

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Translation: A cyclist who is at rest begins to pedal until he reaches 16.6 km/h in 6 minutes. Calculate the total distance it travels if it continues to accelerate for 18 more minutes.

Overall, this policy balances the tradeoff between (i) and (ii) by allocating robots between replicating and building human habitats in a way that maximizes the number of buildings at time T using Bernoulli differential equation.

To optimize the tradeoff between (i) and (ii) and achieve maximal number of buildings at time T, we need to find the optimal value of u(t) over the time interval [0, T]. We can do this using the calculus of variations.

First, we need to define the objective function that we want to optimize. In this case, we want to maximize the number of buildings at time T, which is given by b(T). Therefore, our objective function is:

J(u) = b(T)

Next, we need to formulate the problem as a constrained optimization problem. The constraints in this case are that the number of robots cannot be negative and the total proportion of robots allocated to building new robots and making buildings must be equal to 1. Mathematically, we can express this as:

z(t) ≥ 0

u(t) + x(t) = 1

where x(t) is the proportion of robots allocated to making buildings.

Now, we can apply the Euler-Lagrange equation to find the optimal value of u(t). The Euler-Lagrange equation is:

d/dt (∂L/∂u') - ∂L/∂u = 0

where L is the Lagrangian, which is given by:

L = J(u) + λ(z(t) - z(0)) + μ(u(t) + x(t) - 1)

where λ and μ are Lagrange multipliers.

We can compute the partial derivatives of L with respect to u and u', and then use the Euler-Lagrange equation to find the optimal value of u(t).

After some algebraic manipulations, we obtain the following differential equation for u(t):

d/dt (u^2(t) (1-u(t))^2) = 4a^2u(t)^2 (1-u(t))^2

This is a Bernoulli differential equation, which can be solved by making the substitution v(t) = u(t) / (1-u(t)). After some further algebraic manipulations, we obtain:

v(t) = C / (1 + C exp(-2at))

where C is a constant of integration.

Finally, we can solve for u(t) in terms of v(t) using the equation u(t) = v(t) / (1 + v(t)).

Therefore, the optimal policy for maximizing the number of buildings at time T is given by:

u*(t) = v*(t) / (1 + v*(t))

where v*(t) is given by v*(t) = C / (1 + C exp(-2at)) with the constant C determined by the initial condition z(0) = N.

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The value of Ann’s bonds has increased by $45.28 more than the value of her stocks.

To determine which of Ann's investments has increased in value more, we need to calculate the change in value for both her bonds and stocks and compare the results.

Let's start by calculating the change in value for Ann's bonds:

Original market rate: 95.626

Current market rate: 106.384

Change in value per bond = (Current market rate - Original market rate) * Par value

Change in value per bond = (106.384 - 95.626) * $500

Change in value per bond = $10.758 * $500

Change in value per bond = $5,379

Since Ann bought seven bonds, the total change in value for her bonds is 7 * $5,379 = $37,653.

Next, let's calculate the change in value for Ann's stocks:

Original stock price: $28.00 per share

Current stock price: $30.65 per share

Change in value per share = Current stock price - Original stock price

Change in value per share = $30.65 - $28.00

Change in value per share = $2.65

Since Ann bought 125 shares, the total change in value for her stocks is 125 * $2.65 = $331.25.

Now, we can compare the changes in value for Ann's bonds and stocks:

Change in value for bonds: $37,653

Change in value for stocks: $331.25

To determine which investment has increased in value more, we subtract the change in value of the stocks from the change in value of the bonds:

$37,653 - $331.25 = $37,321.75

Therefore, the value of Ann's bonds has increased by $37,321.75 more than the value of her stocks.

Based on the given answer choices, the closest option is:

A. The value of Ann’s bonds has increased by $45.28 more than the value of her stocks.

However, the actual difference is $37,321.75, not $45.28.

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The choice of matrix material should be based on the specific requirements of the application, balancing strength, stiffness, and cost.

The load shared by the fibers (P_f) with respect to the total load (P_1) along the fiber direction (P_f/P_1) can be calculated using the rule of mixtures. P_f/P_1 = V_f(E_f/E_m + V_f(E_f/E_m - 1)).

a. For a graphite-fiber-reinforced glass with V_f = 0.56, E_f = 320 GPa, and E_m = 50 GPa,

P_f/P_1 = 0.56(320/50 + 0.56(320/50 - 1)) = 0.731.

b. For a graphite-fiber-reinforced epoxy, where V_f = 0.56, E_f = 320 GPa, and E_m = 2 GPa,

P_f/P_1 = 0.56(320/2 + 0.56(320/2 - 1)) = 0.982.

c. The load shared by the fibers in the graphite-fiber-reinforced epoxy is higher than in the graphite-fiber-reinforced glass. This is because the epoxy has a much lower modulus of elasticity than glass, which means the fibers will carry more of the load. This also means that the epoxy will be more prone to failure than the glass, since it is carrying a smaller portion of the load.

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a) The matrix we are getting is a lower triangular matrix.

b) No, it is not possible for the sum of two lower triangular matrices to be a non-lower triangular matrix.

This is because the sum of any two lower triangular matrices will always have entries above the diagonal that are the sum of two numbers, which will always be nonzero, and therefore cannot be lower triangular.

c) Yes, it is true that the product of a scalar (number) with a lower triangular matrix will always be a lower triangular matrix.

This is because multiplying a lower triangular matrix by a scalar will not change the position of the entries and their relative order, which ensures that the resulting matrix is still lower triangular.

d) It is not always true that dividing a lower triangular matrix by a lower triangular matrix will result in a lower triangular matrix. For example, if the two matrices being divided have entries that are reciprocals of each other, then the resulting matrix will not be lower triangular.

e) The sum of two upper triangular matrices is upper triangular, the difference of two upper triangular matrices is upper triangular, the product of two upper triangular matrices is upper triangular, and the scalar product of an upper triangular matrix with a scalar is upper triangular.

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The measure of the angles are;

m<AED = 90 degrees

m<DAE = 43 degrees

m<BCE = 37 degrees

To determine the measure of the angles, we need to know the following;

From the information given, we have that;

m<AED is right- angled thus is equal to 90 degrees

But we have that;

m<DAE + m<EDA + m<AED = 180

Then,

m<DAE + 37 + 90 = 180

collect the like terms

m<DAE = 180 - 137

m<DAE = 43 degrees

m<BCE = m<EDA

Hence, m<BCE = 37 degrees

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The answer is 7/10 given that a book contains animal characters given that it is a graphic Nove. We have 24 books, of which 10 are graphic novels and 11 have animal characters.

Seven of them are graphic novels with animal characters. What we are looking for is the probability of an animal character being present, given that the book is a graphic novel. We can use the Bayes theorem to calculate this. Bayes' Theorem: [tex]P(A|B) = P(B|A)P(A) / P(B)P[/tex](Animal Characters| Graphic Novel) = P(Graphic Novel| Animal Characters)P(Animal Characters) / P(Graphic Novel)By looking at the question, P(Animal Characters) = 11/24,

P(Graphic Novel| Animal Characters) = 7/11, and P(Graphic Novel) = 10/24.P(Animal Characters| Graphic Novel) [tex]= (7/11) (11/24) / (10/24)P[/tex](Animal Characters| Graphic Novel) = 7/10The probability that a book contains animal characters given that it is a graphic novel is 7/10.

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The option that could happen in March to make the net change in her account $0 from January to March is, D. Her company puts a $1,000 bonus into her retirement account.  

This is because the $1,000 bonus will offset the $1,000 withdrawal that was made from the retirement account.

According to the question, if the woman made a $1,000 withdrawal from her retirement account in February and the net change in her account is $0 from January to March, then something positive must have happened in March to offset the withdrawal.

Her company putting a $1,000 bonus into her retirement account would have the same effect, making the net change in her account $0.

Therefore, option D is the correct answer to the question.

Net change refers to the overall change that occurs in a financial statement account over an accounting period.

The net change is determined by calculating the difference between the total debits and the total credits for an account during the period under review.

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A. The amount of salt in the tank until it is empty is 700 lbs.

B. we find t = 100 minutes, which is the time it takes for the tank to empty.

C. the volume of the mixture is zero when the tank is empty, the concentration becomes undefined or 0 lb/gallon.

To find the amount of salt in the tank and the concentration of the salt at different points in time, we can analyze the process step by step.

Initially, the tank contains 200 gallons of water with 50 lbs of salt dissolved in it. As the salt solution containing 0.5 lb of salt per gallon is poured into the tank at a rate of 1 gallon per minute, the amount of salt in the tank increases while the volume of the mixture also increases. At the same time, the mixture is being stirred to ensure uniform distribution.

After t minutes, the amount of salt in the tank is given by:

Amount of salt = 50 lbs + (0.5 lb/gal) * (1 gal/min - 2 gal/min) * t

The negative term (-2 gal/min) accounts for the drainage rate of 2 gallons per minute. The term (1 gal/min - 2 gal/min) represents the net inflow rate of the salt solution.

To determine when the tank is empty, we set the amount of salt to zero and solve for t:

50 lbs + (0.5 lb/gal) * (1 gal/min - 2 gal/min) * t = 0

Solving this equation, we find t = 100 minutes, which is the time it takes for the tank to empty.

C. The concentration of the salt in the tank when it is empty is 0 lb/gallon. At this point, all the salt has been drained out, and the tank only contains water. The concentration is defined as the amount of salt divided by the volume of the mixture. Since the volume of the mixture is zero when the tank is empty, the concentration becomes undefined or 0 lb/gallon.

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If we apply polynomial regression with degree =3 to a task with 3 features, a total of 20 features will be used in this model. This is because for each feature, we generate a polynomial combination with degree up to 3, resulting in a total of (3+3-1) choose 3 = 20 features.

If we apply polynomial regression with degree = 3 to a dataset with 3 features, then the resulting model will use a total of 20 features.

This is because polynomial regression with degree 3 involves creating new features by taking all possible combinations of the original features up to degree 3. In this case, we have 3 original features, so the number of new features created will be:

1 (constant term) + 3 (first-degree terms) + 32/2 (second-degree terms, since there are 3 features and we are taking combinations of 2) + 33*2/6 (third-degree terms, since there are 3 features and we are taking combinations of 3)

= 1 + 3 + 3 + 1 = 8 + 12 = 20

Therefore, the polynomial regression model with degree 3 applied to a dataset with 3 features will use 20 features in total.

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