If a right triangle is formed when three squares are joined at their vertices, the area of the largest square is the same as the combined area of the two smaller squares.

The flight-path angle is equal to 45' when v =90' on all parabolic trajectories is proved using the trajectories.

Using trajectories equation :

r = p/(1+cosv)     at v= 90 degrees

r{90} = p/ (1+cos90°)

        = p       since, cos 90 = 0

also, v= √2mue/r

        v{90} = √2mue/p

where r {90}  and v{90} are radius and velocity at v = 90 degrees.

the angular momentum is given as , h = r.v cosФ

and at periapsis on a parabolic trajectory , rp = p/2

  v{90} = √2mue/p

applying the concentration of angular momentum,

rp.vp = r90.v90.cosФ

p/2.√2mue/p = p.√2mue/p.cosФ

cos Ф = 1/√2

therefore,Ф = 45

hence the flight-path angle is equal to 45' when v =90' on all parabolic trajectories is proved.

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The square of the canonical correlation is a measure of the strength of association between two sets of variables. It ranges from 0 to 1, with values closer to 1 indicating a stronger association between the two sets of variables. In this case, with a value of .64, it suggests that there is a moderate to strong association between the two sets of variables being analyzed.

The discriminant function and the square of the canonical correlation are both measures used in statistical analysis to understand the relationship between two sets of variables.

The discriminant function is used in discriminant analysis, a type of statistical analysis that helps to predict which group an individual observation belongs to based on a set of predictor variables. The estimated discriminant function tells us the relationship between the predictor variables and the group membership.

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Answer:  15 (cupcakes)

Step-by-step explanation:

So, as you understand, 3 cupcakes represent 20% of the order.

20% = 20/100 = 1/5

3=1/5

3x5=15

1/5x5=1

Therefore, your one whole order is 15 cupcakes

Answer:

K^2 - 100 = (K + 10)(K - 10)

16p^2 - 36 = 4(4p^2 - 9) = 4(2p + 3)(2p - 3)

9d^2 - 32 = (3d - 4)(3d + 4)

24a^2 - 54b^2 = 6(4a^2 - 9b^2) = 6(2a + 3b)(2a - 3b)

100b^3 - 36b = 4b(25b^2 - 9) = 4b(5b - 3)(5b + 3)

The missing side is 29, because in a triangle, the sum of the angles is 180°.  

A triangle is a three-sided polygon, a two-dimensional shape with three straight sides and three angles. It is one of the most basic shapes in geometry, and can be used to construct more complex shapes and figures such as rectangles and circles. Triangles come in many different varieties, including equilateral, isosceles, and scalene. The angles in a triangle add up to 180°, and the sides of a triangle are always connected.

In this case, 39° + X° = 180°, and X° = 180° - 39° = 141°. Since the angle is given in degrees, the side length can be found using the law of cosines, which states c^2 = a^2 + b^2 - 2ab cos C. In this case, c = 29, a = b = 11, and C = 141°. Plugging this into the equation, we get 29^2 = 11^2 + 11^2 - 2(11)(11)cos(141°). Simplifying and solving for c, we get c = 29. Therefore, the missing side is 29.

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The probability of obtaining 5 impulses in one minute is 0.067.

The calculation of the probability of obtaining 5 impulses in one minute is based on the Poisson distribution. The Poisson distribution is used to calculate the probability of a given number of events occurring in a fixed interval of time or space, when the average rate of occurrences is known. In this case, the given rate of occurrences is 10 impulses per minute.

The probability of obtaining 5 impulses in one minute can be calculated by using the following formula:

P(x) = (e-λ) (λ^x) / x!

Where,

P(x) is the probability of x events occurring in a given interval

e is the base of the natural logarithm, approximately 2.718

λ is the average rate of occurrences, which is 10 in this case

x is the number of events, which is 5 in this case

Therefore, the probability of obtaining 5 impulses in one minute can be calculated as:

P(5) = (2.718-10) (10^5) / 5!

P(5) = 0.067

Therefore, the probability of obtaining 5 impulses in one minute is 0.067.

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Both fractions in the expression 1/3+ 1/3 are examples of proper fractions

A fraction can be defined as the part of a whole variable, element or number.

In mathematics, there are different types of fractions. They include;

Some examples of simple fractions; 1/3, 5/6

Some examples of improper fractions; 6/4, 7/5, 4/3

Some examples of proper fractions; 3/4, 1/2, 2/4

Some examples of mixed fractions; 2 1/2, 3 1/4, 5 2/5

From the information given, we have that;

1/3 + 1/3

Hence, they are proper fractions.

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The probability that x will take on a value between 42 and 44 is 0.25.

The continuous uniform distribution's probability density function is:

f(x) = 1/(b - a), when a < x < b

     = 0, when x < a or x > b

The internal form of the continuous random variable x in this case ranges from 40 to 48. Next, P.D.F is

f(x) = 1/(48 - 40)

f(x) = 1/8

The probability that x will take on a value between 42 and 44 is:

P(42 ≤ x ≤ 44) = [tex]\int\limits^{44}_{42} \frac{1}{8}\, dx[/tex]

P(42 ≤ x ≤ 44) = [tex]\frac{1}{8}[x]^{44}_{42}[/tex]

P(42 ≤ x ≤ 44) = 1/8(44 - 42)

P(42 ≤ x ≤ 44) = 1/8(2)

P(42 ≤ x ≤ 44) = 1/4

P(42 ≤ x ≤ 44) = 0.25

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Answer:

  C. Exponential growth

Step-by-step explanation:

You want to know how to describe the relation that represents a 10% increase per year.

The problem statement tells you the number of guests "increased by approximately 10% each year". The nature of the function depends on what "10%" refers to.

If the amount of increase is constant year after year (10% of the original number of guests), then the relationship is linear.

If the amount of increase is 10% of the previous year's number, so the amount increases each year, then the relationship is exponential.

When the number increases year on year, the number is said to be growing. If the relationship is exponential, it is described as exponential growth.

If the number were decreasing by 10% per year, it would be an exponential decay relationship.

__

Additional comment

Whenever you're dealing with percentages, you always need to understand what is being used as the base amount. (What does 100% represent?)

Usually, a percentage increase or decrease is referring to the current value. Not always.

Sometimes a change from 10% to 6% is referred to as a 4% decrease, and sometimes it is referred to as a 40% decrease. It depends on who is selling what to whom.

Given the area code of 07893 followed by 5 more digits, the total phone number would be 07893xxxxx. Tammy requires the last digit to contain only 2 different digits.

To find the total number of different phone numbers possible, we can start by using a mathematical approach.

There are a total of 10 digits that can be used to form the phone numbers, 0 to 9. As Tammy wants the last digit to contain only 2 different digits, we have to choose 2 digits from the 10 available digits. This can be calculated using the formula for combinations, nCr, which gives us the number of ways to choose r elements from a set of n elements. In this case, n = 10 and r = 2, so the number of combinations would be 10C2 = 45.

Therefore, there are 45 different combinations of 2 digits possible from the 10 digits. As each combination can be used as the last digit of the phone number, there are a total of 45 different phone numbers possible for Tammy's office. To summarize, the number of different phone numbers possible for Tammy's office is 45.

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Therefore , the solution of the given problem of linear equation comes out to be it has infinitely many solutions.

The foundation for a linear regression curve is the equation y=mx+b. B is the slope, and m is the y-intercept. The previous sentence is sometimes referred as a "mathematical equation involving two variables," even though that y or y are separate components. Two variables make up bivariate linear equations. There are no known solutions to application issues involving linear equations. Y=mx+b is the equation for a single set of equations.

Here,

Given :

=> y = -1/2x - 3

=> x + 2y = -6

So,

=> x = -6 -2y

=> y = -1/2(-6-2y) -3

=> y = 3 + y -3

=> y=y= 0

Thus , infinitely many solutions

Therefore , the solution of the given problem of linear equation comes out to be it has infinitely many solutions .

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Answer:

See proof below

Step-by-step explanation:

We will denote the sum of n elements of the sequence as S(n)

What is proof by induction?

We first assume that for n=1 , S(1) is true
This step is called the basis

Then for n= k ≥ 1, prove S(n) is true and then prove S(k+1) also true. This is called the induction step

For this specific problem, we assume S(1) is true

[tex]\mbox{\large S(1) = 2}[/tex]

[tex]\mbox{\large S(k) = 2 + 4 + 6 + ..... + 2k = k(k+1) }[/tex]

We are using 2k because the values are twice the index values 1, 2, 3...
2.1, 2.1, 2.3 ...2k

[tex]\mbox{\large We now show }[/tex] [tex]\mbox{\large S(k + 1) }[/tex]  [tex]\mbox{\large is true }[/tex]

[tex]\mbox {\large S(k + 1) = S(k) + 2(k + 1)}[/tex]

[tex]\mbox{\large 2+4+6+...+2k+2(k+1)}\\\\\mbox{\large=k(k+1)+2(k+1)}\\\\\mbox{\large=(k+1)(k+2)}\\\\= \mbox{\large (k+1)(k+1+1)}}\\\\[/tex]

[tex]\mbox{\large Substituting n for k + 1 we get }[/tex]
[tex]\mbox{\large S(n) = n(n+1)}[/tex]

I hope the proof is comprehensible. If not please ask for clarifications

Answer:

  the proof is below

Step-by-step explanation:

You want a proof by induction of the formula for the sum of consecutive even numbers.

The given formula is ...

  Sn = n(n+1)

For n=1, the one-term sequence is the first term: 2. Its sum is 2.

  2 = (n)(n+1) = (1)(1+1) = 2 . . . . . true

For this step, we assume the formula is correct. Then ...

  Sk = k(k+1)

The sequence sum for k+1 terms is ...

  2 + 4 + 6 + ... + 2k = k(k+1) . . . . . assumption from step 2

  (2 + 4 + 6 + ... + 2k) +2(k+1) . . . . . one more term added; sum of k+1 terms

  k(k+1) +2(k+1) . . . . . . . . substitute assumed value for (2 + 4 + ...)

  (k +2)(k +1) . . . . . . . . . . factor out k+1

And the formula for k+1 terms is ...

  = (k+1)((k+1) +1) . . . . . substitute (k+1) for n

  = (k+1)(k+2) . . . . . . . perform the addition

Note that the sum of terms is (k+2)(k+1), and is identical to the formula for the same sum: (k+1)(k+2). (Remember, multiplication is commutative.)

  (k+2)(k+1) = (k+1)(k+2) . . . . . QED

In a class of 50 students, there are 365! / 315! ways that no two students will have the same birth date.

The total number of outcomes of two or more independent events equals the product of the number of outcomes of each individual event, according to this concept. A youngster, for example, who chooses six flavors of ice cream and three types of cones will have 6 x 3 = 18 distinct icecream options. According to the fundamental counting principle, if there are p ways to do one thing and q ways to do another, then there are pq ways to accomplish both.

Here,

There are 365 ways to choose the first student's birth date, 364 ways to choose the second student's birth date (since it can't be the same as the first student's), and so on, down to 316 ways to choose the 50th student's birth date (since it can't be the same as any of the first 49 students' birth dates).

So, the number of ways to choose the birth dates of 50 students so that no two are the same is,

= 365 × 364 × ... × 316

= 365! / (365 - 50)!

= 365! / 315!

There are 365! / 315! ways that no two students will have the same birth date in a class of size 50.

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Answer:

40 feet

Step-by-step explanation:

We need to find the scale factor.  Two corresponding sides are AB and ED.

Let s = scale factor

20x = 12  Divide both sides by 20

x = .6

To find CD take the corresponding side 25 and multiply it by the scale factor of .6

25 x .6 = 15

Side CD is 15

CD + AC = AD

25 + 15 = 40

The equation of direct variation given that y varies directly as x is 125= 25k.

The relationship between two variables in which one is a constant multiple of the other is referred to as direct variation. For instance, two variables are said to be in proportion when one affects the other. If b and an are directly proportional, then b = ka is the equation.

The equation of direct variation is given as:

y = kx

where, k is the proportionality constant.

Substituting the value of y = 125 and x = 25:

125 = k (25)

Hence, the equation of direct variation given that y varies directly as x is 125= 25k.

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The equation to model the depth (d) of the snow n days after the storm

d = 15-3n

What is the percentage?

It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol %.

An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used. In the equation 3x + 5 = 14, for instance, the two expressions 3x + 5 and 14 are separated.

20% of 15 = (20/100)*15 = 3

The equation to model the depth (d) of the snow n days after the storm

d = 15-3n

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To find the distance traveled by a car given the average speed and time traveled, we can use the formula:

distance = speed x time

In this case, the average speed of the car is 48 miles per hour (mph) and the time traveled is 5 hours and 45 minutes.

Before we use this formula, we need to convert the time of 5 hours and 45 minutes into hours. Since there are 60 minutes in 1 hour, we can convert 45 minutes into hours by dividing it by 60:

45 minutes / 60 minutes/hour = 0.75 hours

Now, we can add the time in hours and minutes to get the total time traveled in hours:

5 hours + 0.75 hours = 5.75 hours

Now we can use the formula to find the distance traveled:

distance = speed x time

distance = 48 mph x 5.75 hours

distance = 277 miles

So, the car travels 277 miles in 5 hours and 45 minutes.

Answer: 275 Miles

Step-by-step explanation:

1. Convert the hours into minutes:
We'll take the 5 hours and 45 minutes and divide the minutes by 60, giving you 5.75 hours.

2. Multiply the hours by the rate
The 48 mph is given to us, so we should multiply that by the converted hours we got in the first step. 5.75*48 = 275

The 272 demonstrates the number of miles that are driven in that total time span.

Answer:

She pays 63 quid (are you mad bruv?!!! 63 quid for a jacket?!!! you must be joking cuz!!!!)

Step-by-step explanation:

10% of 70 = (10/100) * 70

0.1 * 70 = 7

70 - 7 = 63

The formula for the interior angles is (3-2)180 which equals 180 degrees

It is important to note the properties of a polygon. They include;

A triangle is a two-dimensional shape and has three sides, thus, it is a polygon.

The formula for calculating the sum of the interior angles of any polygon is expressed as;

Sum of interior angles = (n-2)180

where, n is the number of sides

Substitute the number of sides as 3

Sum of interior angles = (3-2)180

expand the bracket

Sum of interior angles = 180 degrees

Hence, a triangle has a measure of 180 degrees

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The length of a rectangle= (10/3) cm or 3.334 cm

The area can be defined as the amount of space covered by a flat surface of a particular shape

The formula for the area of rectangle= length × breadth

It is given in the question that the area of a rectangle= 20 sq. cm

Also, Breadth is given as 6 cm.

We have to find the length.

Putting the given values in the above formula:

20 sq. cm = 6cm × length

length = 20 sq. cm/ 6cm

length =10/3 cm = 3.334 cm

Final length = (10/3) cm or 3.334 cm

Therefore, the length of a rectangle is 3.334cm.

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Answer:

6

Step-by-step explanation:

Δy =6-0 =6

Δx = 2-1 =1

slope = 6/1 = 6

The required points (x ,y ) for the curve x = 6t² + 5 , y = t³ -7  has slope 1/2  are ( 29,1).

Slope of the tangent line to the curve = dy/dx

dy/dx = ( dy / dt ) × ( dt / dx )  __(1)

y = t³ -7

⇒ dy /dt = 3t²

x = 6t² + 5

⇒ dx / dt = 12t

Now substitute the value of dy/dt and dx/dt in (1) we get,

dy / dx = 3t² / 12t

⇒ dy /dx = t /4

Slope (dy /dx ) = 1 /2

⇒ t /4 = 1 /2

⇒ t = 2

Point ( x , y ) = ( 6(2)² + 5 , 2³ -7 )

                    = ( 29, 1 )

Therefore, the points (x ,y ) for the curve with tangent line to the curve has slope 1/2 is equal to ( 29, 1) .

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By dividing the output power (work completed) by the input power ratio, the motor's efficiency may be determined. In this instance, the input power is 300 W, and the output power is 2400 joules (400 N * 6.0 m). As a result, the motor has an efficiency of 2400 J/300 W, or 8.0.

The input power is 300 W.

Time (t) = 20 s

Height is 6.0 metres.

(F) = 400 N Load

Output power (Pout) is calculated as follows: F x h = 400 N x 6.0 m = 2400 J

Efficiency (e) is calculated as Pout / P = 2400 J / 300 W = 8.0.

A motor's efficiency can be determined by dividing its output power by its input power. Assuming that the input power is 300 W and the output power equals the amount of work completed, the load (F) and height can be multiplied to determine the output power (h). The output power in this example is 2400 joules (400 N * 6.0 m) since the load is 400 N and the height is 6.0 m. The motor's efficiency is 2400 J/300 W, or 8.0, as a result. Knowing the input power, output power, and length of time it took the motor to lift the load will help you determine the efficiency. In this illustration, the time is 20 s, and the input power is 300 W.

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The function inverse of f(x) = 2x2, = -2 is also a function.

In the case of a square matrix, inverse of a matrix is a crucial operation. It only works with square matrices. Finding the determinant and adjoint of the given matrix is necessary to calculate the inverse. The transposition of the cofactor of the specific matrix results in adjoint.

Although every function has an inverse, not all inverses are also functions. The vertical line test might not pass for the inverse. A function will be its inverse if the original function is a one-to-one function. In addition to passing the vertical line, a one-to-one function.

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The equation of a line y=2x+8.

The equation of a straight line can be represented in the slope-intercept form, y = mx + c

Where c = intercept

Slope, m =change in the value of y on the vertical axis/change in the value of x on the horizontal axis

Looking at the given line,

y = 2x + 7

Compared with the slope-intercept equation,

Slope, m = 2

If a line is parallel to another line, it means that both lines have equal or the same slope. This means that the slope of the line passing through the point (-2, 4) is 2

Substituting m= 2, y = 4 and x = -2 into the equation, y = mx + c , it becomes

4 = 2 × - 2 + c

4 = - 4 + c

c = 4 + 4 = 8

The equation becomes y=2x+8.

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This statement is called an Income Statement.

December 3 I Income Statement

Revenue

Sales:          $1,027,800

Rental of Equipment to Customers:   $47,000

Total Revenue:        $1,074,800

Cost of Goods Sold:       $537,400

Gross Profit:        $537,400

Expenses

Salaries and Wages:       $15,000

Rent of Facilities:       $70,600

Depreciation of Equipment:     $3,000

Electricity:        $2,900

Supplies:         $1,850

Other Expenses:        $700

Total Expenses:        $93,150

Net Profit:         $444,250

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The surface area of the pyramid will be 128 square inches.

A three-dimensional object's surface area is the sum of all of its faces. Real-world applications of the concept of surface areas include wrapping, painting, and eventually building things to achieve the best possible design.

Given that each side of the base of a square pyramid measures 8 inches. The height of each lateral face of the pyramid is also 8 inches.

The surface area of the square pyramid will be calculated as:-

SA = 1 / 2 x  Perimeter x Height

SA = 1 / 2 x ( 8 x 4 ) x 8

SA = 16 x 8

SA = 128 square inches

Therefore, the surface area will be 128 square inches.

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Answer:

-25/8

Step-by-step explanation:

(-4 3/8) - (-1 1/4)

-35/8 - (-5/4)

-35/8 + 5/4

-35/8 + 10/8

-25/8