The radius of a circle is 7 kilometers. What is the length of a 90° arc?

a) The rate at which the top of the ladder is sliding down the wall when the bottom of the ladder is 5 m from the wall is -0.1 m/s.

b) The rate of change of the angle between the ground and the ladder when the bottom of the ladder is 5 m from the wall is -25/676 rad/s.

c) The rate of change of the area of the triangle bounded by the ladder, the building, and the ground, when the bottom of the ladder is 5 m from the wall is 3.5 m^2/s.

a) To find the rate at which the top of the ladder is sliding down the wall, we start by expressing the length of the ladder, z, in terms of the distances x and y using the equation z^2 = x^2 + y^2.

By differentiating this equation with respect to time, we obtain 2z(dz/dt) = 2x(dx/dt) + 2y(dy/dt), where dz/dt represents the rate of change of z, dx/dt is the rate at which x is changing, and dy/dt is the rate at which y is changing.

Given that dx/dt = 0.6 m/s, x = 5 m, and z = 13 m, we can substitute these values into the equation and simplify to find 13(dz/dt) = 3 + y(dy/dt).

To isolate dy/dt, we differentiate equation (1) with respect to t, resulting in dy/dt = [2z(dz/dt) - 2x(dx/dt)] / (2y).

Substituting the given values and dz/dt = 0.6, we find dy/dt = (13/12)(dz/dt) - (1/2). Plugging in dz/dt = 0.6, we obtain dy/dt = (13/12) * 0.6 - 0.5 = -0.1 m/s. The negative sign indicates that the top of the ladder is sliding down the wall.

b)

This can be determined by differentiating the equation involving the tangent of the angle and applying the chain rule.

To find the rate of change of the angle, θ, between the ground and the ladder, we start with the equation tan θ = y/x. By differentiating both sides with respect to t,

we get sec^2θ(dθ/dt) = (1/x)dy/dt,

where dθ/dt represents the rate of change of θ.

Substituting x = 5, y = 12, and dy/dt = -0.1, we find sec^2θ = 25/169.

Taking the square root of both sides, we get secθ = 13/5.

To find dθ/dt, we have (dθ/dt) = [(1/x)dy/dt] / sec^2θ = (5/169)(-0.1) / (169/25) = -25/676 rad/s.

c)

This can be determined by differentiating the equation for the area of the triangle.

The area of the triangle, A, can be expressed as A = (1/2)xy. By differentiating with respect to t, we find dA/dt = (1/2)[x(dy/dt) + y(dx/dt)], where dA/dt represents the rate of change of the area.

Substituting the given values and calculating, we find

dA/dt = (1/2)[5*(-0.1) + 12*0.6] = 3.5 m^2/s.

Thus, the rate of change of the triangle's area is 3.5 m^2/s.

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A result is called "statistically significant" when the p-value is less than the significant level, typically 0.05.

This means that there is less than a 5% chance that the results are due to chance alone, and suggests that there is a real effect present in the data.

The Null Hypothesis is the assumption that there is no significant difference between the measured phenomenon and a certain value or set of values. A statistically significant result means that the observed data are inconsistent with the assumption of no difference, or no effect.

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The area of the region between y=x^(1/2) and y=x^(1/5) for 0<=x<=1 is 1/6 square units.

To find the area between the two curves y=x^(1/2) and y=x^(1/5) for 0<=x<=1, we will use the definite integral.

Step 1: Identify the bounds of integration. Since the question specifies 0<=x<=1, these are our bounds.

Step 2: Determine the difference between the two functions. Subtract the smaller function from the larger one: (x^(1/2)) - (x^(1/5)).

Step 3: Set up the integral. We will integrate the difference of the two functions with respect to x, from 0 to 1.

area = ∫(x^(1/2) - x^(1/5)) dx from 0 to 1

Step 4: Evaluate the integral. Using the power rule for integration, we get:

area = (2/3 * x^(3/2) - 5/6 * x^(6/5)) evaluated from 0 to 1

Step 5: Substitute the bounds and find the area.

area = [(2/3 * (1)^(3/2) - 5/6 * (1)^(6/5)) - (2/3 * (0)^(3/2) - 5/6 * (0)^(6/5))]

area = (2/3 - 5/6)

area = 1/6

The area of the region between y=x^(1/2) and y=x^(1/5) for 0<=x<=1 is 1/6 square units.

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Check the picture below.

\(\textit{area of a triangle}\\\\ A=\cfrac{1}{2}bh~~ \begin{cases} b=base\\ h=height\\[-0.5em] \hrulefill\\ b=8\\ h=10 \end{cases}\implies A=\cfrac{1}{2}(8)(10)\implies A=40\)

The area of a vertical cross-section through the centre of the base of the party hat is 40 square inches.

It is defined as the three-dimensional shape in which the base is a circular shape and if we go from circular base to top the diameter of the circle reduces and at the vertex, it becomes almost zero.

Clyde has a cone-shaped party hat.

The height of the hat = 10 inches

The radius of the base of the hat = 4 inches.

The diameter of the base of the hat = 2×4 ⇒8 inches

The vertical cross-section of the cone exactly looks like a triangle.

In this scenario, the vertical cross-section of the hat is a triangle with a height of 10 inches, and base length of 8 inches.

We know the formula for the area of a triangle is given by:

\(\rm A = \frac{1}{2} (b\times h)\)

Here b = 8 inches, and h = 10 inches

\(\rm A = \frac{1}{2} (8\times 10)\)

A = 40 square inches

Thus, the area of a vertical cross-section through the centre of the base of the party hat is 40 square inches.

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Answer:

d x=+-3

Step-by-step explanation:

divide both sides by 2

Answer:

The last one.

Step-by-step explanation:

You can begin by dividing by - 2

-2*abs(x) =-6

-2*abs(x)/-2 = - 6/-2

abs(x) = 3

So x can be -3 or 3. Both are solutions

That's the last one.

The five large cities in India are:

The population of large cities in India are:

The distance between the large cities in India are:

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The sum of the perimeters of all the squares in this infinite process is 128 cm.

To find the sum of the perimeters of all the squares in this infinite process, we can consider the relationship between the side lengths of each square.

In the initial square, the side length is 16 cm. When we inscribe the second square by joining the midpoints of the sides, the side length of the second square becomes half of the side length of the first square, which is 8 cm.

Similarly, for each subsequent square, the side length will be half of the previous square's side length.

Therefore, the side lengths of the squares in the process can be represented as 16 cm, 8 cm, 4 cm, 2 cm, 1 cm, and so on.

To find the sum of the perimeters, we need to sum the perimeters of all the squares.

The perimeter of each square is calculated by multiplying the side length by 4.

For the initial square: 16 cm * 4 = 64 cm

For the second square: 8 cm * 4 = 32 cm

For the third square: 4 cm * 4 = 16 cm

For the fourth square: 2 cm * 4 = 8 cm

For the fifth square: 1 cm * 4 = 4 cm

And so on.

To find the sum, we can add all these values together:

64 cm + 32 cm + 16 cm + 8 cm + 4 cm + ...

This is an infinite geometric series with a common ratio of 1/2.

The sum of an infinite geometric series can be calculated using the formula: sum = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.

In this case, the first term (a) is 64 cm and the common ratio (r) is 1/2.

Using the formula, we have:

sum = 64 cm / (1 - 1/2) = 64 cm / (1/2) = 128 cm

Therefore, the sum of the perimeters of all the squares in this infinite process is 128 cm.

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Answer:

y is 1

Step-by-step explanation:

The above question was not written properly

Complete Question

Everyone loved the Chocolate eclairs and they all got eaten during the party. Mr Watson friend, James, ate 6/8 of them if there were 16 of the chocolates, how many did James eat?

Answer:

12 chocolates

Step-by-step explanation:

From the above question, we know that

Total number of Chocolates = 16

The fraction of Chocolate that James ate = 6/8

Therefore, the amount of chocolates that James ate is calculated as:

= 6/8 of 16 chocolates

= 6/8 × 16 chocolates

= 12 chocolates

Therefore, James ate 12 chocolates.

Answer: 12

Step-by-step explanation:

Number of chocolates = 16.

Fraction eaten by James = 6/8.

The number of chocolates eaten by James will be gotten by multiplying 6/8 by 16. This will be:

= 6/8 × 16

= 12

James ate 12 chocolate

The probability that Sue will sit next to Bob is 96/24 = 4. So, the odds are 4:1.

We need to find out the odds that Sue will sit next to Bob. There is a round table with five different people. Therefore, the total number of ways that five different people can be seated at a round table is (5 - 1)! = 4!.

Thus, there are 24 ways that these five different people can be seated around the table. Now, let's say Sue and Bob are sitting next to each other. Thus, there are 4! = 24 ways in which they can be seated around the table. Now, Sue and Bob can be treated as a single unit since they are sitting together. So, there are a total of 48 + 48 = 96 possible ways that Sue will sit next to Bob in a 5 person round table. There are 24 possible ways that 5 people can be seated around the table.

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The value of x is 2.045 by simple algebra.

Use the time formula, t = d/s, which states that time is equal to distance divided by speed, to solve for time. Time is a function of both speed and distance.

Until it reaches its destination 200 miles distant, a car is driving on a little highway at either 55 or 35 miles per hour, depending on the posted speed restrictions.

x is the duration, in hours, that the vehicle is travelling at 55 mph.

Y is the duration, in hours, that the vehicle is travelling at 35 mph.

55x + 35y = 200 is the equation defining the relationship.

We must locate:

If the trip takes the car 2.5 hours at 35 miles per hour, how long does it take to travel at 55 miles per hour?

∵ 55x + 35y = 200

The drive takes the car 2.5 hours at a speed of 35 mph.

Y shows the duration of time for an automobile travelling at 35 mph.

∴ y = 2.5

- To obtain the value of x, substitute the value of y into the equation.

∴ 55x + 35(2.5) = 200

∴ 55x + 87.5 = 200

- Take 87.5 off of both sides.

∴ 55x = 112.5

- Subtract 55 from both sides.

x equals 2.045 hours.

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The length of the cable is 13.4 ft

Here, we want to find the height of the cable

By the given information in the question, we start with a diagrammatic representation of the information

We have this as follows;

As seen from above, what we have is a right-triangle, with the cable length representing the hypotenuse (the longest side)

Using Pythagoras' theorem, we can get the cable length

The square of the length of the hypotenuse is equal to the sum of the squares of the two other sides

Mathematically, we have this as follows (we are representing the length of thecable by c0

Thus, we have it that;

A purchaser of a single ticket can anticipate losing, on average, $28.

The likelihood of winning each prize multiplied by the prize's worth, then adding up all the prizes, can be used to determine the estimated earnings for a single-ticket purchaser.

The big prize has a 1/9000 chance of being won, and it is worth $45000. Hence, the following are the anticipated profits from the main prize:

45000/9000 = 5

The odds of winning one of the three second-place prizes, each worth $9000, are 2/9000. The following is the anticipated profits from the second prize:

2/9000 * 9000 = 2

Finally, the price of the ticket itself is the projected cost of the ticket:

$35

Consequently, the difference between the expected value of the prizes and the ticket's price can be used to compute the expected wins for a single ticket purchaser:

$5 + $2- $35 = -$28

This indicates that a purchaser of a single ticket can anticipate losing, on average, $28.

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if the roots of the polynomial function are -9 and 7 - i, then the other factor of the polynomial function is x - (7  + i)

The roots of the polynomial function are given as:

-9 and 7 - i

The root is 7 - i is a complex number.

This means that its conjugate must also be a root of the factor.

The conjugate of 7 - i is 7 + i

So, we have:

x = 7 + i

Equate to 0

x - (7  + i) = 0

Hence, the other factor of the function is x - (7  + i)

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Answer:

A on edge 23

Step-by-step explanation:

trust me bro

Based on the inscribed angle theorem, the measure of angle BED in the diagram given is: 57°.

The inscribed angle theorem states the measure of the inscribed angles subtended by the same arc are the same.

∠BCD and ∠BED are subtended by the same arc BD.

Therefore, based on the inscribed angle theorem, we have:

m∠BCD = m∠BED = 57°.

m∠BED = 57°

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Answer:

30%

Step-by-step explanation:

Purchases:

Coat £20

Hat £15

Gloves for £5

Total amount of purchases: £20 + £15 + £5 = £40

Sales:

Coat at 50% profit: 150% × £20 = 1.5 × £20 = £30

Hat: £18

Gloves at a 20% loss: 80% of £5 = £4

Total amount of sales: £30 + £18 + £4 = £52

Amount of profit: £52 - £40 = £12

Percent of profit: £12 / £40 × 100% = 30%

Answer:

60

Step-by-step explanation:

The bride and groom can form 252 different groups of five people from the total of 10 people at the rehearsal dinner, as they need to assign them to two tables of five people each.

To determine the number of different groups of five people that can be formed, we can use the concept of combinations. In this scenario, we have 10 people who need to be divided into two groups of five people each.

The number of ways to choose five people out of 10 can be calculated using the combination formula. The formula for combinations is given by:

C(n, r) = n! / (r! * (n - r)!)

Where n represents the total number of people and r represents the number of people to be chosen. In this case, n = 10 and r = 5.

Plugging these values into the formula, we get:

C(10, 5) = 10! / (5! * (10 - 5)!)

= (10 * 9 * 8 * 7 * 6 * 5!) / (5! * 5 * 4 * 3 * 2 * 1)

= (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1)

= 252

Therefore, the bride and groom can form 252 different groups of five people from the total of 10 people at the rehearsal dinner. This means that there are 252 distinct ways to divide the 10 people into two tables of five people each, providing flexibility in creating diverse groups for the dinner.

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The evolution of the function call (things '(11 -2 31)) is as follows:

(things '(11 -2 31)) -> (things '(-2 31)) -> (things '(31)) -> (things '()) -> '() the final result of the given call is '().

The given function is a recursive function called "things" that takes a list as input. It checks if the list is empty (null), and if so, it returns an empty list. Otherwise, it checks if the first element of the list (car x) is greater than 10. If it is, it adds 1 to the first element and recursively calls the "things" function on the rest of the list (cdr x). If the first element is not greater than 10, it subtracts 1 from the first element and recursively calls the "things" function on the rest of the list. The function then returns the result.

Now, let's see the evolution resulting from the call (things '(11 -2 31)):

1. (things '(11 -2 31))

  Since the list is not empty, we move to the next if statement.

  The first element (car x) is 11, which is greater than 10, so we add 1 to it and recursively call the "things" function on the rest of the list.

  The recursive call is (things '(-2 31)).

2. (things '(-2 31))

  Again, the list is not empty.

  The first element (car x) is -2, which is not greater than 10, so we subtract 1 from it and recursively call the "things" function on the rest of the list.

  The recursive call is (things '(31)).

3. (things '(31))

  The list is still not empty.

  The first element (car x) is 31, which is greater than 10, so we add 1 to it and recursively call the "things" function on the rest of the list.

  The recursive call is (things '()).

4. (things '())

  The list is now empty, so the function returns an empty list.

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The direct and Inverse variation relationship,x = 2 and z = 1, the value of y is 12.

the direct and inverse variation relationship:

y = k * (x^2) / z

where k is the constant of variation.

Given that y = 4 when x = 4 and z = 12, we can substitute these values into the equation:

4 = k * (4^2) / 12

Simplifying, we have:

4 = k * 16 / 12

To solve for k, we can cross-multiply and divide:

4 * 12 = k * 16

48 = 16k

Dividing both sides by 16:

48 / 16 = k

k = 3

Now that we have determined the value of k, we can use it to find y when x = 2 and z = 1.

Substituting these values into the equation, we have:

y = 3 * (2^2) / 1

Simplifying further

y = 3 * 4 / 1

y = 12

Therefore, when x = 2 and z = 1, the value of y is 12.

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The correct box-and-whisker plot is the one labeled:

Average Monthly Temperatures

64   74   84  94  

The box ranges from 68 to 90, with the median at 84. The whiskers extend from 64 to 94. So, third option is the right answer.

To create a box-and-whisker plot, we need to organize the data in ascending order:

64, 68, 68, 74, 76, 82, 84, 86, 90, 90, 92, 94

Now, we can identify the key components of the box-and-whisker plot:

Minimum: 64 (the lowest value in the data set)

Lower Quartile (Q1): 68 (the median of the lower half of the data set)

Median (Q2): 84 (the middle value of the data set)

Upper Quartile (Q3): 90 (the median of the upper half of the data set)

Maximum: 94 (the highest value in the data set)

Using these values, we draw a box from Q1 to Q3, with a line marking the median (Q2). We also draw "whiskers" extending from the box to the minimum and maximum values. In this case, the box ranges from 68 to 90, with the median at 84. The whiskers extend from 64 to 94.

Therefore, the correct box-and-whisker plot is the one labeled:

Average Monthly Temperatures

64   74   84  94

The right answer is  third option.

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Answer:

3

Step-by-step explanation:

Answer:

3

Step-by-step explanation:

The number is x, then:

The number is 3

The solution to the given system of linear equations is x = 3, y = -3. That is, (3, -3)

The graph of the system of equations is shown below

From the question, we are to determine the apparent solution to the given system of linear equations

The given system of equations is

y = -2x + 3

y = 1/3x - 4

Set the two equations equal

That is,

-2x + 3 = 1/3x - 4

Multiply through by 3

-6x + 9 = x - 12

Add 6x to both sides of the equation

-6x + 6x + 9 = x + 6x - 12

9 = 7x - 12

Add 12 to both sides of the equation

9 + 12 = 7x - 12 + 12

21 = 7x

x = 21/7

x = 3

Substitute the value of x into the first equation

y = -2x + 3

y = -2(3) + 3

y = - 6 + 3

y = - 3

The solution is (3, -3)

The graph is shown below

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The no-arbitrage price today of a 5-year $1,000 US Treasury note with a 2% coupon rate and a 4.25% yield to maturity is approximately $908.44, closest to option B: $900.53.

To determine the no-arbitrage price of a 5-year $1,000 US Treasury note with a coupon rate of 2% and a yield to maturity (YTM) of 4.25%, we can use the present value of the future cash flows.First, let's calculate the annual coupon payment. The coupon rate is 2% of the face value, so the coupon payment is ($1,000 * 2%) = $20 per year.The yield to maturity of 4.25% is the discount rate we'll use to calculate the present value of the cash flows. Since the coupon payments occur annually, we need to discount them at this rate for five years.

Using the present value formula for an annuity, we can calculate the present value of the coupon payments:PV = C * (1 - (1 + r)^-n) / r,

where PV is the present value, C is the coupon payment, r is the discount rate, and n is the number of periods.

Plugging in the values:PV = $20 * (1 - (1 + 0.0425)^-5) / 0.0425 = $85.6427.

Next, we need to calculate the present value of the face value ($1,000) at the end of 5 years:PV = $1,000 / (1 + 0.0425)^5 = $822.7967.

Finally, we sum up the present values of the coupon payments and the face value:No-arbitrage price = $85.6427 + $822.7967 = $908.4394.

Rounding to the penny, the no-arbitrage price is $908.44, which is closest to option B: $900.53.

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Answer:

y = -3x + 8

Step-by-step explanation:

When the height is 38 cm and the base is 32 cm, the rate at which the area of the triangle changes is 10 cm²/min.

The rate at which the area of a triangle changes can be found by multiplying the rate at which the base is shrinking by the rate at which the height is increasing.

Given:

Rate of shrinking of the base = -2 cm/min

Rate of increasing of the height = 3 cm/min

Height of the triangle = 38 cm

Base of the triangle = 32 cm

To find the rate at which the area of the triangle changes, we use the formula for the area of a triangle:

Area = (1/2) * base * height

Differentiating the area formula with respect to time gives us:

dA/dt = (1/2) * (db/dt) * height + (1/2) * base * (dh/dt)

Substituting the given values, we have:

dA/dt = (1/2) * (-2) * 38 + (1/2) * 32 * 3

Simplifying, we get:

dA/dt = -38 + 48

dA/dt = 10 cm²/min

Therefore, when the height is 38 cm and the base is 32 cm, the rate at which the area of the triangle changes is 10 cm²/min.

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The general case when G₁⊕G₂⊕G₃...⊕Gₙ is cyclic generated by (g₁, g₂, g₃,...,gₙ) is that the subgroups G₁, G₂, G₃,...., Gₙ are cyclic generated by  g₁, g₂, g₃, ..., gₙ respectively.

G⊕H is cyclic

That is G⊕H = <(g, h)>

Let us choose an arbitrary element g₁ ∈ G, then as G⊕H is cyclic

(g₁, h₁) ∈ G⊕H = <( g, h)>

This means that there exists an integer x such that

(g₁, h₁) = (g, h)ˣ

(g₁, h₁) = (gˣ, hˣ)

that is g₁ = gˣ

implies g₁ ∈ <g>

that is G ⊆ <g> as g₁ is an arbitrary element belonging to G

But we know <g> ⊆ G

Hence G = <g>

Thus G is a cyclic group generated by g.

Similarly we can prove that H group is cyclic generated by h, that is H = <h>

So the general case is that G₁⊕G₂⊕G₃...⊕Gₙ is cyclic generated by (g₁, g₂, g₃,...,gₙ), then the subgroups G₁, G₂, G₃,...., Gₙ are cyclic generated by  g₁, g₂, g₃, ..., gₙ respectively.

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Answer:

140°

Step-by-step explanation:

a° = 180°-(75°-35°) = 180°-40° = 140°