A certain small country has $10 billion in paper currency in circulation, and each day $50 million comes into the country's banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Since both old bills and new bills will come into the banks while the new currency is gradually introduced, we will need to solve a differential equation to track the amount of new currency in circulation at a given time. Let x (t) denote the amount of new currency, in billions of $, in circulation after t days. We've shown that new currency is introduced at the rate 10 - x (t) / 10 0.05, which simplifies to 0.005 (10 - x (t)). This justifies that x (t) satisfies the differential equation dx / dt = 0.005 (10 - x). (a) Solve the differential equation to find x (t). (b) At what time t will new bills make up 90% of the currency in circulation?

Answer: 18480

i used a calculator :)

Therefore, the measure of angle G in the right triangle is approximately 31.0 degrees.

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used to study the properties and behavior of triangles, and is also widely used in fields such as engineering, physics, surveying, and navigation. The three main trigonometric functions are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively. These functions relate the ratios of the sides of a right triangle to the angles of the triangle.  Trigonometry also involves other concepts such as the Pythagorean theorem, which relates the sides of a right triangle, and the unit circle, which is a circle of radius 1 used to define trigonometric functions of angles beyond 90 degrees. Trigonometry is widely used in a variety of applications, such as calculating the heights of buildings and mountains, designing bridges and other structures, and predicting the motion of objects in physics and engineering.

Here,

In a right triangle, the tangent of an acute angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Using this formula, we can find the tangent of angle G:

tan(G) = opposite / adjacent

tan(G) = 6 / 10

tan(G) = 0.6

To find the measure of angle G, we need to take the inverse tangent (or arctangent) of 0.6:

G = tan⁻¹(0.6)

G ≈ 31.0 degrees

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The fraction of sweets that does she eat is 7/20

In this case, we will be using fractions to describe the portion of sweets that Hanna eats from her bag of 20.

Hanna has a bag of 20 sweets, and she eats 7 of them. To find out what fraction of the sweets she eats, we need to determine the ratio of the number of sweets she eats to the total number of sweets in the bag.

So in this case, Hanna eats 7 sweets, and there are 20 sweets in the bag. Therefore, the fraction of sweets she eats can be written as:

=> 7/20

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The 99% confidence interval for the true difference between testing averages for students using method 1 and students using method 2 is [15.8, 20.8].

The 99% confidence interval for the true difference between testing averages for students using method 1 and students using method 2 can be calculated using a two-sample t-test.

First, calculate the sample difference in means. Subtract the mean for method 1 (50.3) from the mean for method 2 (68.6) to get 18.3.

Next, calculate the pooled standard deviation, which is the weighted average of the standard deviations for each sample:

Pooled standard deviation = [tex]√(((n1-1)×s12 + (n2-1)×s22)/ (n1+n2-2))[/tex]

n1=116, s12=15.62, n2=151, s22=17.21

Pooled standard deviation = [tex]√(((116-1)×15.62 + (151-1)×17.21)/ (116+151-2))[/tex]
Pooled standard deviation = [tex]√(1422.3788/265)[/tex]

Pooled standard deviation = 4.89

Finally, calculate the 99% confidence interval for the true difference between testing averages for students using method 1 and students using method 2:

99% Confidence Interval = Sample Difference in Means ± (Critical Value) × (Pooled Standard Deviation/√(n1+n2))

99% Confidence Interval =[tex]18.3 ± (2.58) × (4.89/√(116+151))[/tex]

99% Confidence Interval = 18.3 ± 2.50

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Answer:

they bought 62 item from noodles and 51 items from hot chips

Answer: [tex]c-34=21[/tex]; 55 cups of lemonade.

Step-by-step explanation:

We are given the amount sold and the amount leftover so we need to figure out how many cups were there at the start. Since you are subtracting the amount of cups you sold from the amount you start with and it equals an end amount, the cups can be modeled by the equation [tex]c-34=21[/tex] rather than [tex]21+c=34[/tex].

Now to solve for c

[tex]c-34=21\\[/tex]

add 34 to both sides

[tex]c=55[/tex]

Therefore, the probability of selecting a defective battery followed by another defective battery is about 12/145 or 8.28%.

The probability of selecting a defective battery on the first draw is 9/30, or 3/10, since there are 3 defective batteries out of 30 total.

Since one battery has already been drawn, there are now 29 batteries left in the box, and 8 defective batteries remaining. The probability of selecting another defective battery on the second draw, without replacement, is 8/29.

To find the probability of both events happening (selecting a defective battery followed by another defective battery), we multiply the probabilities of each event:

P(defective battery and then another defective battery) = P(defective battery on first draw) × P(defective battery on second draw)

P(defective battery and then another defective battery) = (3/10) × (8/29)

P(defective battery and then another defective battery) = 24/290 = 0.0828 or 8.28%

Therefore, the probability of selecting a defective battery followed by another defective battery out of 30 AAA batteries in a box where 9 are defective and two batteries are selected without replacement is 0.0828 or 8.28%.

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These theories are all true and can be used to demonstrate the congruence of two triangles.

Three straight sides and three angles define the two-dimensional geometric form known as a triangle. It is created by joining three non-collinear plane lines. The three places where a triangle's sides intersect are known as the triangle's vertices, and the line segments that make up the triangle's sides are known as its edges or sides. A triangle's inner angles are always added up to 180 degrees. Triangles can be categorised according to their edges and side lengths.

given

Many triangular congruence theorems exist, and they are all true.

The two triangles are congruent if their two sides and included angles match those of another triangle, according to the Side-Angle-Side (SAS) congruence theory.

The Angle-Side-Angle (ASA) congruence theory states that two triangles are congruent if their included sides and two angles are congruent with one another.

The two triangles are congruent if all three sides of one triangle are congruent with all three sides of the other triangle, according to the Side-Side-Side (SSS) congruence theory.

The right triangles are congruent if their hypotenuses and legs are congruent with those of another right triangle, according to the Hypotenuse-Leg (HL) congruence theory.

These theories are all true and can be used to demonstrate the congruence of two triangles.

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If A = [tex]PDP^{-1}[/tex], Then [tex]A^{k}[/tex] = [tex](PDP^{-1}) ^{k}[/tex] = [tex]PDP^{-1}[/tex][tex]PDP^{-1}[/tex] .....[tex]PDP^{-1}[/tex] = [tex]PD^{k} P^{-1}[/tex]since [tex]P^{-1}[/tex] P = I, the identity matrix.

Further, the representation can be made as in the following attachment.

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We have that, The probability that a randomly selected young man is at least 2 inches taller than a randomly selected young woman is 0.709

1. The shape, center, and spread of the M-W distribution are:Shape: NormalCenter: 4.8 inches (69.3 - 64.5)Spread: The standard deviation of the difference between two normally distributed variables is given by the formula:

[tex]\sigma (M-W) = \sqrt{(\sigma_1^2 + \sigma_2^2)} = \sqrt{(2.8^2 + 2.5^2)} \approx 3.64[/tex]

Let us define [tex]Z = (M - W - 2)/3.64[/tex]. So the probability that a randomly selected young man is at least 2 inches taller than a randomly selected young woman is:

[tex]P(M > W + 2) = P((M - W - 2)/\sigma(M-W) > (- 2)/\sigma (M-W)) = P(Z > -0.55) = 1 - P(Z \leq  - 0.55)[/tex]

Using a standard normal table or a calculator, we find that [tex]P(Z \leq  -0.55) \approx 0.291[/tex]. Therefore, the probability that a randomly selected young man is at least 2 inches taller than a randomly selected young woman is: [tex]P(M > W + 2) \approx  1 - 0.291 = 0.709[/tex].

The shape, center, and spread of the M-W distribution are:

Shape: Regular Center: 4.8 inches (69.3 - 64.5)

Dispersion: The standard deviation of the difference between two normally distributed variables is given by the formula:

[tex]\sigma (M-W) = \sqrt{(\sigma_1^2 + \sigma_2^2)} = \sqrt{(2.8^2 + 2.5^2)} \approx 3.64[/tex]

The probability that a randomly selected young man is at least 2 inches taller than a randomly selected young woman is: [tex]P(M > W + 2) \approx 0.709[/tex].

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Answer:

  138°

Step-by-step explanation:

You want the measure of the angle at two tangents when they intercept an arc of 42°.

The short answer is that the exterior angle x is the supplement of the measure of the arc:

  x = 180° -42°

  x = 138°

An exterior angle where secants meet is half the difference of the arcs of the circle they intercept. Here, the secants have been located so the corresponding chord length between the near and far circle intercept points have degenerated to zero. That is, they are tangents.

The angle relation still holds:

  x = (long arc - short arc)/2 = ((360° -42°) -42°)/2 = (360° -2·42°)/2

  x = 180° -42° = 138°

The tangents, together with their associated radii form a quadrilateral. The angles at the tangents are 90°, and the total of all angles is 360°. This gives us the relation ...

  x + 90° +42° +90° = 360°

  x +42° = 180° . . . . . . . . . . . . . subtract 180°

  x = 180° -42° = 138°

(We solved this with an extra step, so you could see the same "supplementary angles" relationship between x and 42°.)

$50 is one estimate for the cost of Carlotta's fees per individual. This is based on information from the article, which claims that some consumers pay $50 a session for Carlotta's services,

which are less expensive than conventional therapy. The precise price, however, is not stated and may change based on the client's financial status and the services rendered. Carlotta's services are priced in the article, although the details are not totally apparent. It states that Carlotta charges less than conventional therapy, indicating that her costs are reasonable and competitive. The article also mentions that some clients pay $50 for each session, which gives a particular pricing range. However, it is crucial to remember that the precise cost may change .

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Answer: Infinitely many solutions

Step-by-step explanation:

On the graph I have attached, you can see the two lines overlap each other. This means the linear equations have an infinite amount of solutions.

Answer:

Another order pair could be (30,10).

Every y value is "one-third of" every x value.

The number of pitchers that she should produce and the price that she should charge in order to maximize profit is 0 and $5, respectively, and the maximum profit is $0.

Given that a retired potter can produce china pitchers at a cost of $5 each. She estimates her price function to be where p is the price at which exactly x pitchers will be sold per week.

The formula for the profit is given by,

Profit = Revenue - Cost

The revenue for selling x pitchers at a price p is xp.

The total cost of producing x pitchers is 5x.

Therefore, Profit [tex]= xp - 5x[/tex]

Profit can be expressed as a function of x using the formulae of the function,

[tex]y = xp - 5x= (p - 5)x[/tex] .............(1)

Now, the number of pitchers that she should produce and the price that she should charge in order to maximize profit can be found by finding the maximum value of the profit function (1).

Let the maximum profit be P. Maximize P with respect to x. To find the maximum of a function of x, we can find its derivative and equate it to zero.

[tex]dP/dx = 0P = (p - 5)x[/tex]

Differentiate P with respect to x again to determine whether it is a maximum or minimum.

[tex]d^2P/dx^2 = -5 < 0[/tex]

Since [tex]d^2P/dx^2[/tex] is negative, the value of P is a maximum.

Substitute the value of x from equation (1) in terms of P to find

[tex]p.P = (p - 5)*P/(p - 5) \\\\p = 5[/tex]

Therefore, the price to maximize profit is $5 and the number of pitchers to produce is,

[tex]x = P/(p - 5) = P/0 =[/tex] undefined

Maximum profit is [tex]P = (p - 5)x = (5 - 5)x = $0[/tex]

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Answer:

b all real numbers are greater then or equal-2

Answer:

4

Step-by-step explanation:

count 3 what comes next

If a large bag of the same candy has 75 pieces, then the expected average for each color is 15 pieces.

Now, let's apply this concept to our candy problem. If the bag of candy has 75 pieces, we can assume that each color makes up approximately 1/5 or 20% of the total. That means each color should have an equal chance of appearing, with an expected average of 15 pieces per color (75 pieces divided by 5 colors).

However, this is only an expected average. In reality, there may be some variation in the number of pieces of each color that we actually find in the bag. This variation can be caused by a variety of factors, such as the manufacturing process, the mixing of different batches of candy, or even random chance.

To determine which color is likely to appear the most, we can look at the range of variation for each color. If the range of variation for a particular color is higher than the expected average of 15 pieces, then that color is more likely to appear more often than the others. On the other hand, if the range of variation is lower than the expected average, then that color is less likely to appear as often.

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It is correct to state that if there is a sample, the standard error is used in the denominator of the z stat or t stat formula.

The t-distribution is used when we have the standard deviation for the sample instead of the population, hence the equation is given as follows:

[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]

In which:

The standard error is modeled as follows:

[tex]s_e = \frac{s}{\sqrt{n}}[/tex]

Which is the denominator of the formula for the test statistic, hence the statement is true.

When we have the standard deviation for the population, we use the z-distribution, however the formula for the standard error is similar.

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If Yumi bought 13 fish for total then yumi bought 8 zebra fish and 5 neon tetras.

Substitution is a mathematical operation that involves replacing a variable or expression in an equation, function, or formula with another variable, expression, or value.

Let's assume Yumi bought x zebra fish and y neon tetras.

x + y = 13 (equation 1)

2x + 2.3y = 27.5 (equation 2)

Using equation 1, we can solve for one variable in terms of the other:

x = 13 - y

Then, change x in equation 2 to the following expression:

2(13 - y) + 2.3y = 27.5

Simplifying and solving for y:

26 - 2y + 2.3y = 27.5

0.3y = 1.5

y = 5

So Yumi bought 5 neon tetras. To find how many zebra fish she bought, we can substitute y = 5 into equation 1:

x + 5 = 13

x = 8

Therefore, Yumi bought 8 zebra fish and 5 neon tetras.

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The answer is 250 kilometers. Peter and his mother traveled a total distance of 250 kilometers on their mail delivery.

Distance is a numerical measurement of how far apart two objects or points are. It is a scalar quantity, meaning that it is only expressed as a magnitude, or numerical value, without any direction.

1 centimeter is equal to 0.01 kilometers, so 50 kilometers would be equal to 5,000 centimeters. The total distance between the four cities is 5,000 centimeters, which is equal to 50 kilometers.

Charltc to Thornton is 1,000 centimeters, Thornton to Avon is 1,500 centimeters, Avon to Ashton is 1,000 centimeters, and Ashton to Charltc is 1,500 centimeters. This gives us a total of 5,000 centimeters, or 50 kilometers.

This can be calculated by multiplying the total distance between the four cities (50 kilometers) by the number of times they traveled the route (5 times, since they flew from Charltc to Thornton, then to Avon, then to Ashton, and then back to Charltc). This gives us 250 kilometers, which is the answer to the question.

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Peter and his mother traveled a total distance of 250 kilometers on their mail delivery. The answer is 250 kilometers.

Distance is a numerical measurement of how far apart two objects or points are. It is a scalar quantity, meaning that it is only expressed as a magnitude, or numerical value, without any direction.

1 centimeter =  0.01 kilometers,

so 50 kilometers = 5,000 centimeters.

Charlton to Thornton= 1,000 centimeters,

Thornton to Avon= 1,500 centimeters,

Avon to Ashton= 1,000 centimeters,

and Ashton to Charlton= 1,500 centimeters.

Total distance= 1,000+1,500+1,000+1,500

= 5,000 centimeters, or 50 kilometers.

The total distance between the four cities is 5,000 centimeters, which is equal to 50 kilometers.

This can be calculated by multiplying the total distance between the four cities (50 kilometers) by the number of times they traveled the route (5 times.

=50*5

=250

Since they flew from Charlton to Thornton, then to Avon, then to Ashton, and then back to Charlton.

250 kilometers is the answer to the question.

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Question:

Peter's mother is a pilot. She often makes deliveries near their community. Peter and his mother flew from Charlton to Thornton to make a mail delivery. Then they continued on to Avon and Ashton and returned to Charlton. How many kilometers did Peter and his mother travel in all?

A crime reporter's null hypothesis is:H0: The actual number of burglaries per month is 3000.

A null hypothesis is a hypothesis that claims that there is no significant relationship between two variables. The term "null" implies that no effect exists or that the effect is negligible. The null hypothesis is typically utilized in scientific research and experiments. It is also utilized in statistical studies, such as those investigating cause-and-effect relationships or the effects of a treatment or intervention.

The null hypothesis of a crime reporter regarding 3000 burglaries per month is that there is no significant difference between the observed results and what is expected or hypothesized. This is because 3000 burglaries per month is the average, and the null hypothesis aims to verify whether this is the actual result.

Therefore, a crime reporter's null hypothesis is:H0: The actual number of burglaries per month is 3000.

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Answer:

A

Step-by-step explanation:

a) ✔️

This is the principle of mathematical induction. If P holds for the base case k=1 and we can show that if it holds for any arbitrary k (e.g. k=n) then it must also hold for the next value (e.g. k=n+1), then we have shown it holds for all natural numbers.

b) ❌

There is no guarantee that P holds for all natural numbers from the statement alone. It only guarantees that for any k where P does not hold, there exists a smaller number i where P does not hold.

c) ❌

This is the principle of weak mathematical induction. It only shows that if P holds for a given k and for all smaller values i then it must hold for k+1. It does not guarantee that P holds for all natural numbers.

d) ❌

This statement is the negation of the principle of mathematical induction. It is known as the "strong induction" principle, which assumes that if P does not hold for k, then there exists a smaller i where P does not hold. However, this principle is not sufficient to prove that P holds for all natural numbers k.

Question 7. C (rectangular pyramid)

Question 8. 17 cm

Answer:

Question 7:C rectangular pyramid

Question 8: C 120 in

A=2(wl+hl+hw)=2·(10·2+5·2+5·10)=160

Step-by-step explanation:

Answer: The volume of the cylinder is approximately 21.2 cubic meters

Step-by-step explanation:

To find the volume of a cylinder, we use the formula:

V = πr^2h

where:

V = volume

r = radius

h = height

In this case, we are given that the cylinder has a height of 3m, but we need to determine the radius.

Since we are not given the radius directly, we can estimate it from the diameter. If the cylinder has a diameter of 3m, then the radius would be half of that, or 1.5m.

So, substituting the values into the formula, we get:

V = π(1.5)^2(3)

V ≈ 21.2 m^3

Rounding to the nearest tenth, the volume of the cylinder is approximately 21.2 cubic meters.

If x is a positive integer , 4x^1/2 is equivalent to product of 2 and square root of x, wherein it would surely be a positive value greater than 2.

Positive integers are the numbers on the number line which are greater then zero and extend on the right hand side of the number line till infinity. These numbers are also whole numbers in itself such as 1, 2, 3...,∞. When 4x^1/2 is calculated, it is assumed that 4x is raised to power half, which will provide the answer as 2√x.

It is because square root of 4 will be 2 and that of x will be √x. Square roots are the numbers obtained by multiplying a specific number by the number itself. For example: 3×3 = 9 or square root of 9 is 3.

If some positive integer is fixed in the equation, the desired outcome would be obtained as follows:

If x=4, (4×4)^1/2 = 4

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The shortest distance from the origin to the curve x2 + xy + y2 = 3 is √(6-2√7) and the longest distance is √(6+2√7).

We have to find the shortest and longest distances from the origin to the curve x^2 + xy + y^2 = 3. This can be done using the Lagrange multiplier technique.

Given, x^2 + xy + y^2 = 3.

We have to minimize and maximize the distance of the origin from the given curve. The distance of the origin from the point (x, y) is given by √(x²+y²).

Therefore, we have to minimize and maximize the function f(x, y) = √(x²+y²) subject to the constraint x^2 + xy + y^2 = 3.

Now, we have to form the Lagrange function.

L(x, y, λ) = f(x, y) + λ(g(x, y))

where, g(x, y) = x2 + xy + y2 - 3L(x, y, λ) = √(x²+y²) + λ(x2 + xy + y2 - 3)

Now, we have to find the partial derivatives of L with respect to x, y, and λ.

∂L/∂x = x/√(x²+y²) + 2λx+y = 0 ............. (1)

∂L/∂y = y/√(x²+y²) + λx+2λy = 0 ............. (2)

∂L/∂λ = x² + xy + y² - 3 = 0 ............. (3)

Solving equations (1) and (2), we get x/√(x²+y²) = 2y/x.

Since x and y cannot be equal to 0 simultaneously, we can say that x/y = ±2.

Substituting x = ±2y in equation (3), we get y²(5±2√7) = 9.

Now, we can solve for x and y to get the values of (x, y) at which the minimum and maximum value of the distance of the origin occurs.

Using x = 2y, we get y²(5+2√7) = 9 ⇒ y = ±3/√(5+2√7)

Using x = -2y, we get y²(5-2√7) = 9 ⇒ y = ±3/√(5-2√7)

Therefore, the four points at which the distance is minimum and maximum are {(2/√(5+2√7), 1/√(5+2√7)), (-2/√(5+2√7), -1/√(5+2√7)), (2/√(5-2√7), -1/√(5-2√7)), (-2/√(5-2√7), 1/√(5-2√7))}.

To find the minimum and maximum distances, we can substitute these points in f(x, y) = √(x²+y²).

After substituting, we get the minimum distance as √(6-2√7) and the maximum distance as √(6+2√7).

Therefore, the shortest distance from the origin to the curve x^2 + xy + y^2 = 3 is √(6-2√7) and the longest distance is √(6+2√7).

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The two sides of a right triangle opposite the non-right angles are called as adjacent and opposite angle or Legs.

A triangle is a three-sided regular polygon in which the total of any two sides is always larger than the sum of the third side.

A right-angled triangle is one with one of its internal angles equal to 90 degrees, or any angle is a right angle. As a result, this triangle is also known as the right triangle or the 90-degree triangle. In trigonometry, the correct triangle is very significant.

A right-angled triangle is a triangle in which one of the angles is 90 degrees. The total of the other two angles is 90 degrees. The sides that include the right angle are perpendicular and form the triangle's base. The third side is known as the hypotenuse, and it is the longest of the three sides.

The three sides of the right triangle are connected. Pythagoras' theorem explains this relationship. This theorem states that in a right triangle,

Perpendicular² + Base² = Hypotenuse²

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Answer:

3 packages

Step-by-step explanation:

61 - 20 * 2 (small grain sandpaper) = $21 remain. The largest grain costs $7, so 21/7 = 3 packages of the largest grain sandpaper was bought.

Hope this helped!