A(n) is a typical, highly representative example of a concept. It is often described as an "average" or an "ideal member."

Answer:

The common difference of given AP is Option 2) -10.

Step-by-step explanation:

We are given the following information in the question:

First term of AP, a  = 100

The sum of its first 6 terms = 5(the sum of its next six terms)

We have to find the common difference of AP.

The sum of n terms of AP is given by:

[tex]S_n = \dfrac{n}{2}\big(2a + (n-1)d\big)[/tex]

where a is the first term and d is the common difference.

Thus, we can write:

[tex]S_6 = 5 (S_{12}-S_6)\\\dfrac{6}{2}\big(200 + (6-1)d\big) = 5\bigg(\dfrac{12}{2}\big(200 + (12-1)d\big)-\dfrac{6}{2}\big(200 + (6-1)d\big)\bigg)\\\\600 + 15d =5(1200+66d-600-15d)\\600+15d=3000+255d\\2400 = -240d\\d = -10[/tex]

Thus, the common difference of given AP is -10.

It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.

Step-by-step explanation:

Given,

Climbing rate of James = 3 mph

Running back rate of James = 4 mph

Climbing rate of Lucas = 2 mph

Running back rate = 5 mph

Total distance = 2 miles

We know that;

Distance = Speed * Time

As we have to find, we will rearrange the formula in terms of time

[tex]Time=\frac{Distance}{Speed}[/tex]

Time took by James for climbing = [tex]\frac{2}{3}\ hours[/tex]

Time took for running back = [tex]\frac{2}{4}\ hours[/tex]

Total time = [tex]\frac{2}{3}+\frac{2}{4}=\frac{8+6}{12}=\frac{14}{12}[/tex]

Total time taken by James = [tex]\frac{7}{6}\ hours[/tex]

1 hour = 60

Total time taken by James = [tex]\frac{7}{6}*60=70\ minutes[/tex]

Time took by Lucas for climbing = [tex]\frac{2}{2}=\ 1\ hour[/tex]

Time took by Lucas for climbing = 60 minutes

Time took on return = [tex]\frac{2}{5} of\ an\ hour=\frac{2}{5}*60=24\ minutes[/tex]

Total time taken by Lucas = 60+24 = 84 minutes

Therefore,

It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.

Keywords: distance, speed

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Answer:

Step-by-step explanation:

It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.

Step-by-step explanation:

Given,

Climbing rate of James = 3 mph

Running back rate of James = 4 mph

Climbing rate of Lucas = 2 mph

Running back rate = 5 mph

Total distance = 2 miles

We know that;

Distance = Speed * Time

As we have to find, we will rearrange the formula in terms of time

Time took by James for climbing =

Time took for running back =

Total time =

Total time taken by James =

1 hour = 60

Total time taken by James =

Time took by Lucas for climbing =

Time took by Lucas for climbing = 60 minutes

Time took on return =

Total time taken by Lucas = 60+24 = 84 minutes

Therefore,

It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.®

Answer:

256

Step-by-step explanation:

a₁ = 2

aₙ = (aₙ₋₁)²

a₂ = (a₂₋₁)²

a₂ = (a₁)²

a₂ = (2)²

a₂ = 4

a₃ = (a₃₋₁)²

a₃ = (a₂)²

a₃ = (4)²

a₃ = 16

a₄ = (a₄₋₁)²

a₄ = (a₃)²

a₄ = (16)²

a₄ = 256

Answer:

It's D; degree- 5; leading coefficient-10 :)

Step-by-step explanation:

I just took the test and got it right

The given polynomial function f(x) = 10x⁵ + 7x⁴ + 5 has Degree: 5; leading coefficient: 10 which is the correct answer would be option (D).

A polynomial is defined as a mathematical expression that has a minimum of two terms containing variables or numbers. A polynomial can have more than one term.

The degree of a polynomial is the highest exponent of the variables in the polynomial. In this case, the highest exponent is 5, so the degree of the polynomial is 5.

The leading coefficient is the coefficient of the term with the highest degree.

In this case, the coefficient of the term with the highest degree (x⁵) is 10, so the leading coefficient is 10. Therefore, the correct answer is D).

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Answer:

76302 grams

Step-by-step explanation:

Diameter of the balloon = 9 m

Density of the gas (weight occupied per m³) = 200 g per m³

Volume of the spherical balloon = [tex]\frac{4}{3}\pi r^{3}[/tex]

Here r = 4.5 m

Then V= [tex]\frac{4}{3}\times\ 3.14 \times (4.5)^{3}[/tex]

V = 381.51 m³

Total weight = Total volume occupied * Density

                     = 381.51*200 = 76302 grams.

Final answer:

The total weight of the gas in a filled balloon with a 9-meter diameter is 76,340.25 grams, calculated by first finding the volume of the balloon and then multiplying it by the weight of the gas per cubic meter.

Explanation:

To find the total weight of the gas in a filled spherical balloon, first, we need to find the volume of the balloon using the formula for the volume of a sphere, which is V = (4/3)πr³. The diameter of the balloon is given as 9 meters, so the radius r is 9/2 meters or 4.5 meters.

Calculating the volume of the balloon:

Now, we'll multiply the volume of the balloon by the weight per cubic meter of the gas to get the total weight of the gas:

Answer:

Step-by-step explanation:

Start with the unknown, which is the number of hours J worked and the number of hours A worked.  If A worked twice as many hours as J, then J worked x hours and A worked 2x hours.  If J can iron 25 shirts per hour, x, then the number of shirts he can iron in his shift is 25x.  If A can iron 35 shirts per hour, x, then the number of shirts he can iron in his shift is 35(2x).  The number of shirts they iron together in x hours is

25x + 35(2x) = 380 and

25x + 70x = 380 and

95x = 380 so

x = 4

This means that J worked 4 hours and A worked 8 hours.

Answer: the measure of the indicated angle is 100 degrees

Step-by-step explanation:

The sum of angles in a triangle is 180 degrees. Let x represent the unknown angle in the bigger triangle. Therefore,

x + 80 + 25 = 180 degrees

x + 105 = 180

x = 180 - 105 = 75 degrees.

Let z represent the other unknown angle in the smaller triangle. Since the sum of the angles on a straight line is 180 degrees, therefore

75 + 55 + z = 180

130 +z = 180

z = 180 - 130 = 50 degrees

Let y represent the unknown angle that we are looking for. Therefore,

50 + y + 30 = 180

80 + y = 180

y = 180 - 80 = 100 degrees

Answer:

55

Step-by-step explanation:

Answer:

216 sq. units

Step-by-step explanation:

From the figure, we can see that only one pair of opposite angles are equal. So, the quadrilateral is a kite.

Formula to find the area of a kite:

Area, A = [tex]$ \frac{1}{2} \times d_1 \times d_2 $[/tex]

where, [tex]$ d_1 $[/tex] and [tex]$ d_2 $[/tex] are the lengths of the diagonals.

Here, [tex]$d_1 = 18 $[/tex] units.

And, [tex]$ d_2 = 24 $[/tex] units.

Therefore, the area A = [tex]$ \frac{1}{2} \times 18 \times 24 $[/tex]

= [tex]$ \frac{432}{2} $[/tex]

= 216 sq. units which is the required answer.

Answer:the length of the rectangle is 100m

the width of the rectangle is 80m

Step-by-step explanation:

Let L represent the length of the rectangle.

Let W represent the width of the rectangle.

The formula for determining the perimeter of a rectangle is expressed as

Perimeter = 2(L + B)

The perimeter of a rectangle is 360m. This means that

2(L + B) = 360

L + B = 180 - - - - - - - - - -1

If it's length is decreased by 20%, its new length would be

L - 0.2L = 0.8L

it's breadth is increased by 25%, it means that the new breadth

B + 0.25B = 1.25B

Since the perimeter remains the same,

2(0.8L + 1.25B) = 360

0.8L + 1.25B = 180 - - - - - - - - -2

Substituting L = 180 - B into equation 2, it becomes

0.8(180 - B) + 1.25B = 180

144 - 0.8B + 1.25B = 180

- 0.8B + 1.25B = 180 - 144

0.45B = 36

B = 36/0.45 = 80

L = 180 - B = 180 - 80

L = 100

Answer:

  b.  is $34 per ounce

Step-by-step explanation:

If the production cost were less, a competitor would drive the price down. If the production cost were more, the supplier would go out of business.

Since we're at equilibrium, the production cost must be equal to $34 per ounce.

Answer:

There were 22 men in the sight-seeing tour

Step-by-step explanation:

If the number f women on the sight-seeing tour was less than 30, the closest number that can be divided by 5 is 25, so we suppose that there were 25 women, so with the ratio women(5): children(2) we can apply a rule of three, so we multiply 25*2 divided by 2, now we know that there were 10 children, now with the ratio of men(11): children(5) we apply another rule of three, and multiply 10*11 and then divide it by 5, and now we know that there were 22 men in the sight-seeing tour.

Answer:

a) [tex]5xz + xy + 4yz[/tex]

b) 10

Step-by-step explanation:

a) Here [tex]F(x,y,z)=(5z+y)i+(4z+x)j+(4y+5x)k[/tex]

Since the case [tex]F[/tex]  =  ∇[tex]f[/tex] holds, then

∇[tex]f = f_xi+f_yj+f_zk[/tex] = [tex](5z+y)i+(4z+x)j+(4y+5x)k[/tex]

So, [tex]f_x = 5z + y[/tex]

If we integrate [tex]f_x[/tex] with respect to x, we will get an integration constant C which is also a function that depends to y and z.

Hence,

[tex]f = \int f_xdx = 5xz + xy + g(y,z)[/tex]

Now we need to find g(y,z).

So first let's take the derivative of g(y,z) with respect to y.

[tex]f_y = x + g_y(y,z) = 4z + x[/tex]

Hence, [tex]g_y(y,z) = 4z[/tex]

So now, if we integrate [tex]g_y[/tex] with respect to y to find g(y,z)

[tex]g = \int g_ydy = 4yz + C[/tex]

Thus,

[tex]f = 5xz + xy + g(y,z) = 5xz + xy + 4yz + C[/tex]

And since [tex]f(0,0,0)=0[/tex], then [tex]C=0[/tex]

Thus,

[tex]f = f(x,y,z) = 5xz + xy + 4yz[/tex]

b) By the Fundamental Theorem of Line Integrals, we know that

[tex]\int\limits^a_b F. dr = F[r(b)]-F[r(a)][/tex]

Hence,

[tex]\int\limits^a_b F. dr = F(1,1,1)-F(0,0,0) =[(5+1+4)-(0+0+0)]=10[/tex]

To find ????, solve the system of partial differential equations. Use the function ???? from part a to compute the line integral.

To find a function ???? such that ????=∇???? and ????(0,0,0)=0, we can solve the system of partial differential equations. Let ????=????????+????????+????????, then compute the partial derivatives of ???? with respect to each variable. Equating these partial derivatives to the given vector field components, we can solve for the unknown function ???? and find its value at the point (0,0,0).

To compute the line integral ∫????????⋅????????, we can use the fundamental theorem of calculus for line integrals. Since ????(x,y,z) is the gradient of ????, the line integral is equal to the change in ???? along the curve C from (0,0,0) to (1,1,1). We can use the function ???? found in part a to evaluate this change in ????.

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Answer:

Price of  hamburgers = $2

Price of white-dogs = $3

Step-by-step explanation:

Let

The number price of hamburgers be x

The number price of white-dogs  be  y

Then the first family buys 3 hamburgers and 2 white -dogs for $13

3x + 2y = 13-----------------------------(1)

Another first family buys 2 hamburgers and 5 white -dogs for $16

2x + 5y = 16----------------------------(2)

To solve let us multiply eq(1) by 2 and eq(2) by 3, we get

6x + 4y = 26-----------------------------(3)

6x + 15y = 48----------------------------(4)

subracting (3) from (4)

6x + 15y = 48

6x + 4y = 26

(-)

-------------------------

        11y  = 22

---------------------------

[tex]y =\frac{22}{11}[/tex]

y =2

Now substituting the value of y in eq(1),

3x + 2(2) = 13

3x + 4 = 13

3x = 13-4

3x = 9

x =[tex]\frac{9}{3}[/tex]

x =3

To calculate the rate at which the distance between the bicycle and the balloon is changing, take into account the speeds of the balloon and bicycle and use Pythagoras's theorem and the formula dd/dt = sqrt((dx/dt)^2 + (dy/dt)^2). This calculates out to approximately 17.03 ft/sec.

This problem can be solved using the concepts of relative velocity and Pythagoras' theorem. It's a classic problem of a balloon moving vertically upwards at a speed of 1 ft/sec and a bicycle moving horizontally at a speed of 17 ft/sec.

After 3 seconds, the balloon would have ascended an additional 3 feet (because it's rising at 1 foot per second), so it will be at a height of 68 feet. Meanwhile, in those same 3 seconds, the bicycle will have travelled 51 feet (because it's going at 17 feet per second).

At this point, you can use Pythagoras's theorem to find the distance (d) between the bicycle and the balloon. The distance d is the hypotenuse of a right triangle where one side (the vertical side) is 68 feet and the other side (the horizontal side) is 51 feet. Using Pythagoras' theorem, d² = 68² + 51². So, d = sqrt(68^2+51^2) = 85 feet.

Then, find the change in d, known as dd/dt, which tells us how fast the distance between the balloon and the bicycle is changing. Given the constant velocities of the balloon and the bicycle, dd/dt is also a constant. You can use the formula dd/dt = sqrt((dx/dt)^2 + (dy/dt)^2), where dx/dt = 17 ft/sec (the speed of the bicycle), and dy/dt = 1 ft/sec (the speed of the balloon). So, dd/dt = sqrt(1^2 + 17^2) = sqrt(290) = approximately 17.03 ft/sec. So, the distance between the balloon and the bicycle is increasing at a rate of 17.03 ft/sec after 3 seconds.

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The distance between the bicycle and the balloon is changing at a rate of approximately 11 ft/sec, 3 seconds after the bicycle has passed under the balloon.

We need to determine how fast the distance between the bicycle and the balloon is changing 3 seconds after the bicycle passes under the balloon.

Let's denote:

We are given:

After 3 seconds:

Apply the Pythagorean theorem to find z:

z = √(x² + y²)
z = √(51² + 68²)
z = √(2601 + 4624)
z = √7225
z = 85 ft

To find the rate of change of the distance z, we differentiate with respect to time (t):

2z (dz/dt) = 2x (dx/dt) + 2y (dy/dt)

Continue with substituting the known values:

85 (dz/dt) = 51 (17) + 68 (1)
85 (dz/dt) = 867 + 68
dz/dt = 935 / 85
dz/dt ≈ 11 ft/sec

Thus, the distance between the bicycle and the balloon is changing at a rate of approximately 11 ft/sec after 3 seconds.

Answer:

Step-by-step explanation:

Initial amount that was deposited into the savings account is $600 This means that the principal,

P = 600

The account earns 2.1 % interest compounded annually.. This means that it was compounded once in a year. So

n = 1

The rate at which the principal was compounded is 2.1%. So

r = 2.1/100 = 0.021

It was compounded for t years. So

t = t

a) The formula for compound interest is

A = P(1+r/n)^nt

A = total amount in the account at the end of t years. Therefore

A = 600 (1+0.021/1)^1×t

A = 600(1.021)^t

b)when A =$800, it becomes

800 = 600(1.021)^t

Dividing both sides by 600, it becomes

1.33 = (1.021)^t

Taking the tth root of both sides

t = 14 years

It will take 14 years

Answer:

1. B.

2. B.

Step-by-step explanation:

Trigonometry

1) Coterminal angles can be found by adding or subtracting 360° (or 2\pi radians) to a given angle. If we have 120°, adding 360° gives 480°, adding again 360° gives 840°. There is no way to get -180°, so this option is not a coterminal angle to 120°

2)

A. [tex]Cos (-90^o)=0[/tex], and not undefined

B. [tex]Sin (-90^o)=-1[/tex]. This is correct

C.  [tex]Tan (-90^o)[/tex] is undefined, not zero

Thus the only correct option is B.

Why is this the answer?

The x value x = 4 shows up in the point (4, 3), which is in the given function set. Adding (4, -3) to this set will have x = 4 show up twice. We cannot have one x value pair up with more than one y value. In other words, any input cannot map to more than one output. Visually, the two points (4,3) and (4,-3) will fail the vertical line test, which means we wouldnt have a function.

Answer: check explanation

Step-by-step explanation:

The question is not complete but nonetheless, gross profit is easy to calculate. Let us start by the definition of gross profit, to how to calculate gross profit and, calculation of gross profit margin.

GROSS PROFIT: gross profit can be defined as the profit gained after subtracting the costs of making and selling the products. Gross profit is also known as gross margin.

HOW TO CALCULATE GROSS PROFIT: Gross profit can be calculated by subtracting the total revenue from the cost of goods sold. That is, gross profit= total revenue - cost of goods sold.

So, for example; Company A makes women handbags. Assuming the company made $10million in total revenue for the year and cost of goods sold is $5 million. We can use the formula above to find the company's gross profit margin.

Hence, company A's gross profit= total revenue($10,000,000) - cost of goods sold ($5,000,000).

= $5,000,000.

That is, the gross profit for company A= $5,000,000.

GROSS PROFIT MARGIN: This can be calculated using the formula below;

Gross profit margin= (total revenue - cost of goods)/ revenue.

Hence, from the example above;

Gross profit margin= $10,000,000 - $5,000,000) / $10,000,000.

= 50%

Answer:

The function

{\ displaystyle f (z) = {\ frac {z} {1- | z | ^ {2}}}} {\ displaystyle f (z) = {\ frac {z} {1- | z | 2}

It is an example of real and bijective analytical function from the open drive disk to the Euclidean plane, its inverse is also an analytical function. Considered as a real two-dimensional analytical variety, the open drive disk is therefore isomorphic to the complete plane. In particular, the open drive disk is homeomorphic to the complete plan.

However, there is no bijective compliant application between the drive disk and the plane. Considered as the Riemann surface, the drive disk is therefore different from the complex plane.

There are bijective conforming applications between the open disk drive and the upper semiplane and therefore determined as Riemann surfaces, are isomorphic (in fact "biholomorphic" or "conformingly equivalent"). Much more in general, Riemann's theorem on applications states that the entire open set and simply connection of the complex plane that is different from the whole complex plane admits a bijective compliant application with the open drive disk. A bijective compliant application between the drive disk and the upper half plane is the Möbius transformation:

{\ displaystyle g (z) = i {\ frac {1 + z} {1-z}}} {\ displaystyle g (z) = i {\ frac {1 + z} {1-z}}}

which is the inverse of the transformation of Cayley.

if an analytic function on the unit disk has a pole on the unit circle, its power series representation diverges on the unit circle, as the singularity prevents the power series from converging outside the disk of convergence.

To understand why a power series diverges on the unit circle when the analytic function it represents has a pole on the unit circle, we can use the concept of analytic continuation and the properties of poles and singularities.

Here's a step-by-step explanation:

1.Analytic function on the unit disk: Let's consider an analytic function defined on the open unit disk, denoted by[tex]\(D = \{z \in \mathbb{C} : |z| < 1\}\)[/tex]. This means the function is holomorphic (complex differentiable) at every point within this disk.

2.Power series representation: Since the function is analytic on[tex]\(D\)[/tex], it can be represented by a power series expansion around any point [tex]\(z_0\) in \(D\)[/tex]. Let's denote this function by[tex]\(f(z)\)[/tex], and its power series representation centered at[tex]\(z_0\) by \(\sum_{n=0}^{\infty} a_n (z - z_0)^n\)[/tex].

3.Pole on the unit circle: Suppose[tex]\(f(z)\)[/tex] has a pole (a point where the function becomes unbounded) on the unit circle[tex]\(|z| = 1\)[/tex], i.e., there exists a point [tex]\(z_1\)[/tex] on the unit circle such that [tex]\(f(z_1)\)[/tex] is infinite. Without loss of generality, let's assume [tex]\(z_1 = 1\)[/tex] (since the unit circle is symmetric about the origin).

4.Behavior near the pole: Near the pole at [tex]\(z = 1\)[/tex], the function[tex]\(f(z)\)[/tex]can be expanded in a Laurent series, which includes negative powers of [tex]\((z - 1)\)[/tex]. This expansion will have infinitely many terms with negative powers, indicating the singularity at [tex]\(z = 1\)[/tex].

5.Radius of convergence: The radius of convergence of the power series representation of [tex]\(f(z)\)[/tex]is at least the distance from the center of convergence to the nearest singularity. In this case, since the singularity (pole) is on the unit circle, the radius of convergence of the power series cannot exceed 1.

6.Divergence on the unit circle: Since the radius of convergence of the power series representation of [tex]\(f(z)\)[/tex] is at most 1, the power series diverges at every point on the unit circle (except possibly at the point of singularity itself, where it may converge by definition). This divergence occurs because the function has a singularity (pole) on the unit circle.

Therefore, if an analytic function on the unit disk has a pole on the unit circle, its power series representation diverges on the unit circle, as the singularity prevents the power series from converging outside the disk of convergence.

Answer: he will need 28 cups of dough for 3 cups of chocolate.

Step-by-step explanation:

The baker uses 2 1/3 cups of cookie dough and 1/4 cup of chocolate chips to make 10 cookies. Converting 2 1/3 cups of cookie dough into improper fraction, it becomes 7/3 cups of cookie dough.

It means that for every 1/4 cup of chocolate, 7/3 cups of cookie dough is needed.

Let x represent the amount of cookie dough needed for 3 cups of chocolate chips.

If 7/3 dough = 1/4 cup of chocolate

x dough = 3 cups of chocolate

x × 1/4 =7/3 × 3

x/4 = 7

x = 7×4 = 28 cups of dough

The correct equilibrium at one atmosphere pressure associated with its Kelvin temperature is the steam-water equilibrium at 373 K.

The equilibrium at one atmosphere pressure that is correctly associated with the Kelvin temperature at which it occurs is option d. steam-water equilibrium at 373 K. To explain, in the Kelvin temperature scale, the freezing point of water is 273.15 K and the boiling point is 373.15 K, both under standard atmospheric conditions (1 atmosphere pressure). So, at 373 K, the situation would be a steam-water equilibrium, not an ice-water equilibrium as in options a and b. The Kelvin temperature for ice-water equilibrium is 273.15 K and not 0 K and 32 K as stated in options a or b. Similarly, steam-water equilibrium does not occur at 212 K as suggested in option c.

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Option d. steam-water equilibrium at 373 K. The steam-water equilibrium at one atmosphere pressure occurs at 373 K.

The correct answer is:

Let's break down the reasons:

Thus, the correct association is the steam-water equilibrium occurring at 373 K at 1 atmosphere pressure.

Answer:

Step-by-step explanation:

 102+202+302+402+502+602+702+802+902(4518)

+112+212+312+...+ 812+912(4608)

+122+222+322+...+822+922(4698)

+132+232+332+...+932(4788)

..........................................

+192+292+392+...+992(5328)

4518+4608+4698+...+5328

n=10

[tex]s=\frac{10}{2}(4518+5328)\\=5(9846)\\=49230[/tex]

Final answer:

To find the sum of all positive 3-digit numbers ending in 2, we calculate the total for each digit's place and sum them up, resulting in a total sum of 8280.

Explanation:

The problem requires finding the sum of all positive 3-digit numbers with a last digit of 2. To calculate this, we can identify that the first such number is 102 and the last is 992. There are 90 such numbers because they correspond to the tens digit going from 0 to 9 for each of the nine possible hundreds digits (1-9).

Since each number ends in 2, we can think of them as (100x + 10y + 2), where x is the hundreds digit (1 through 9) and y is the tens digit (0 through 9). To find the sum, we calculate the sum of the hundreds digits times their frequency, the sum of the tens digits times their frequency, and add 2 times the number of terms (90). The formula would be:

Sum = (Sum of hundreds values) * 10 * 9 + (Sum of tens values) * 1 * 90 + 2 * 90

The hundreds values are 1 through 9, whose sum is 45, and the tens values are 0 through 9, whose sum is 45 as well. Plugging these values into the formula, we get:

Sum = 45 * 10 * 9 + 45 * 1 * 90 + 2 * 90 = 4050 + 4050 + 180 = 8280.

Complete answer:

Fulfill the requirements for a certain degree, a student can choose to take any 7 out of a list of 20 courses, with the constraint that at least 1 of the 7 courses must be a statistics course. Suppose that 5 of the 20 courses are statistics courses.

(a) How many choices are there for which 7 courses to take?

(b) Explain intuitively why the answer to (a) is not [tex] \binom{5}{1}\binom{19}{6} [/tex]

Answer:

a) 71085 choices

b) See below

Step-by-step explanation:

a) First we're going to calculate in how many ways you can take 7 courses from a list of 20 without the constraint that at least 1 of the 7 courses must be a statistics course, that's simply a combination of elements without repetition so it's: [tex]\binom{20}{7} [/tex], but now we should subtract from that all the possibilities when none of the courses chose are a statistic course, that's is [tex]\binom{15}{7} [/tex] because 15 courses are not statistics and 7 are the ways to arrange them. So finally, the choices for which 7 courses to take with the constraint that at least 1 of the 7 courses must be a statistics course are:

[tex]\binom{20}{7}-\binom{15}{7}=71085 [/tex]

b) It's important to note that the constraint at least 1 of the 7 courses must be a statistics course make the possible events dependent, we can not only fix an statistic course and choose the others willingly ( that is what [tex] \binom{5}{1}\binom{19}{6} [/tex] means) because the selection of one course affect the other choices.

In this exercise, we have to use our knowledge of statistics to calculate how many options can be chosen for a course, so we find that:

a) 169 choices

b) 1 of the 7 courses must be a statistics course, because the selection of one course affect the other choices.

So from the data reported in the exercise we can say that:

a) First we're going to calculate in how many ways you can take 7 courses from a list of 20 without the constraint that at least 1 of the 7 courses must be a statistics course, that's simply a combination of elements without repetition so it's.  So we have that;

[tex]C=\frac{m!}{p!(m-p)!}\\C=\frac{20!}{7!(20-7)!}\\C=169[/tex]

b) It's influential to note that the restraint not completely 1 of the 7 courses must be a enumeration course form the attainable occurrence determined by, we can not only fix a statistic course and select the possible choice gladly because the preference from among choices of individual course influence the added selection.

See more about statistics at brainly.com/question/10951564

Answer:

6

Step-by-step explanation:

Answer:

Step-by-step explanation:

The initial length of Josh's rope was 43 feet. He cuts off a piece that is 10 feet long and gives it to his brother. This means that the length of the remaining rope would be

43 - 10 = 33 feet.

He wants to cut the rest of the rope into pieces measuring exactly 5 feet each. To determine the number of pieces that he can cut, it becomes

33/5 = 6.6

Therefore, only 6 pieces measuring exactly 5 feet each can be cut.

Answer:

a.) y = 3/(2ˣ)

b.) y = 1/(2ˣ)

c.) y = (π^π)ˣ

d.) y = (1/27)(1/√(3))ˣ

e.) y = .002908/(.119025ˣ)

f.) y = .00000004808/(.0413ˣ)

Step-by-step explanation:

Concept need to know is:

Answer:

The data file is included in attached excel file.

Step-by-step explanation:

There are seven variables in the data file for four persons interviewed. Person type, marital status, ethnic group and gender are categorical variables while age, number of pets and GPA are quantitative variables. Person type is classified as A,B,C and D. Marital status consists of category married and single. Age of persons lies between 20-50. Ethic group has three categories that are white, Asian and Hispanic. Gender of persons has two categories male and female. Number of pets for the persons interviewed lies in the range of 2 to 5. Last GPA variable ranges from 2.2 to 3.8.

Answer:

The least possible value is -2, obtained for a = -1 and b = 1.

Step-by-step explanation:

We want the minimum of [tex] \displaystyle\left(\frac{1}{a} \frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right) [/tex] , where

First, lets simplify the expression given, we use common denominator on the second part, using ab as the common denominator. We obtain

[tex] \frac{1}{b} - \frac{1}{a} = \frac{a-b}{ab} [/tex]

As a result

[tex] \displaystyle\left(\frac{1}{a} \frac{1}{b}\right)\left(\frac{1}{b}-\frac{1}{a}\right) = \frac{a-b}{(ab)^2} [/tex]

we need the minimum of the function

[tex] f(a,b) = \frac{a-b}{(ab)^2} [/tex]

with the restrictions [tex] -5 \leq a \ -1, 1 \leq b \leq 3 [/tex]

First, we calculate the gradient of f and find where it takes the zero value.

[tex]\nabla{f} = (f_a,f_b) [/tex]

with

[tex]f_a = \frac{(ab)^2 - (a-b) 2ab^2}{(ab)^2} = -1 + 2 \frac{b}{a}[/tex]

Since it has the reversed sign, we get

[tex]f_b = - (-1 + 2 \frac{a}{b}) =1 - 2 \frac{a}{b}[/tex]

In order for [tex] \nabla{f} [/tex] to be zero, we need both [tex] f_a [/tex] and [tex] f_b [/tex] to be zero, observe that

[tex]f_b = 0 \, \rightarrow 1 -2\frac{a}{b} = 0 \, \rightarrow 1 = 2 \frac{a}{b} \, \rightarrow b = 2a [/tex]

Which is impossible with the given restrictions. Hence, the minimum is realized in the border.

If we fix a value a₀ for a, with a₀ between -5 and -1 the function g(b) = f(a₀,b) wont have a minimum for b in [1,3] because the partial derivate of f over b didnt reach the value 0 in the restrictions given. On the other hand. by making a similar computation that before, we can obtain that the partial derivate of f over the variable a doesnt reach the value 0 either. This means that f doesnt reach the minimum on the sides. As a consecuence, it reach a minimum on the corners.

The 4 possible corner values are (-5,1), (-5,3), (-1,1) and (-1,3)

[tex] f(-5,1) = \frac{-6}{25} =  -0.24 [/tex]

[tex] f(-5,3) = \frac{-8}{225} = -0.0355 [/tex]

[tex] f(-1,1) = \frac{-2}{1} = -2 [/tex]

[tex] f(-1,3) = \frac{-4}{9} = -0.444 [/tex]

Clearly the least possible value between the four corners is -2.

Answer:

2 hours

Step-by-step explanation:

45x + 40x = 170

85x = 170

x = 2

Answer:

40 mph: 4 hours 15 minutes

45 mph: 3 hours 46 minutes 40 seconds

Step-by-step explanation:

The rate of separation = 45 + 40 = 85 mph

speed = distance / time

85 = 170 / t

t= 170/85 hour

Answer:

B) Mean

Step-by-step explanation:

Standard Deviation is the measure of the amount of variation or dispersion of a set of values. Standard Deviation is represented by lower case Greek alphabet sigma σ. Standard Deviation is the square root of variance. Variance is the average of the squared differences or variation or dispersion from the Mean. Therefore, to compute standard deviation, the mean of the given data must be known.

    Standard Deviation, σ =  [tex]\sqrt{Variance}[/tex]

                     σ = [tex]\sqrt({\frac{1}{N-1}}[/tex] ∑[tex]_{i = 1}^{N}(x_{i} - X)^{2})[/tex]

where

[tex]x_{1},x_{2},x_{3}, ...x_{N},[/tex] are the values of the sample observed,

X is the mean value of these observations, and

N is the number of observations in the sample.

Average speed of slower car = 60 mph

Average speed of faster car = 70 mph

Solution:

Given that Two cars enter the Delaware turnpike at shoreline drive at 8 am, each heading for ocean city

Given that One cars average speed is 10 mph faster than the other

Let "x" be the average speed of slower car

Then x + 10 is the average speed of faster car

The faster car arrives at ocean city at 11 am, a half hour before the slower car

Time taken by faster car:

The faster car arrives at ocean city at 11 am but given that they start at 8 a.m

So time taken by faster car = 11 am - 8am = 3 hours

Time taken by slower car:

The faster car arrives at ocean city at 11 am, a half hour before the slower car

So slower car takes half an hour more than faster car

Time taken by slower car = 3 hour + half an hour = [tex]3\frac{1}{2} \text{ hour}[/tex]

Now the distance between Delaware turnpike at shoreline drive and ocean city will be same for both cars

Let us equate the distance and find value of "x"

The distance is given by formula:

[tex]distance = speed \times time[/tex]

Distance covered by faster car:

[tex]distance = (x + 10) \times 3 = 3x + 30[/tex]

Distance covered by slower car:

[tex]distance = x \times 3\frac{1}{2} = x \times \frac{7}{2} = 3.5x[/tex]

Equating both the distance,

3x + 30 = 3.5x

3x - 3.5x = -30

-0.5x = -30

x = 60

Thus average speed of slower car = 60 mph

Average speed of faster car = x + 10 = 60 + 10 = 70 mph