A whole number is 6 more than 2 times another number. The sum of the two numbers is less than 50. This can be written in an inequality as x + 2x + 6 < 50, where x represents the smaller number.

By the integral test, the series is convergent.

The sequence [tex]k^3/e^{k^4}[/tex] is clearly positive and decreasing for [tex]k\in\Bbb N[/tex]; then by the integral test,

[tex]\displaystyle \int_1^\infty \frac{x^3}{e^{x^4}} \, dx \le \sum_{k=1}^\infty \frac{k^3}{e^{k^4}}[/tex]

and

[tex]\displaystyle \int_1^\infty \frac{x^3}{e^{x^4}} \, dx = \frac14 \int_1^\infty e^{-u}\, du = \frac1{4e} < \infty[/tex]

By using the integral test, series converges

What is an integral test for convergence and divergence?

⇒ It is used to prove the divergence or convergence of series. This test is called the integral test, which compares an infinite sum to an improper integral. It is important to note that this test can only be applied when we are considering a series whose terms are all positive.

How to know if a series is converging or diverging?

If the limit exists and is a finite number (a number less than infinity), we say the integral converges. If the limit is ±∞ or does not exist, we say the integral diverges.

let aₓ= f (x) is any function

⇒ In order for the integral test to work The function must be positive it has to be continuous and it has to be decreasing when x≥1

If the integral converges it means the series is also converging

If the integral diverges it means the series is also diverging

The sequence [tex]k^{3} /{e^{k^{4} }[/tex] is clearly positive and decreases for k∈N then by the integral test,

[tex]\int\limits^\infty_1 {x^{3} /e^{x^4} dx[/tex] [tex]\leq\displaystyle \sum^{\infty}_{k = 1} {x^{3} /e^{x^{4}[/tex]

and

[tex]\int\limits^\infty_1 {x^{3} /e^{x^4} dx[/tex][tex]=1/4\int\limits^\infty_1 {x^{3} /e^{-u} du[/tex]

⇒ [tex]1/4 < \infty[/tex]

So it comes with a finite value hence the series converges

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Answer:

[tex]Question 1\\1+i\\-1-i\\Question 2\\\\-1+i\\1-i[/tex]

Step-by-step explanation:

Complex Numbers:

   Complex numbers can generally be expressed in the form: [tex]a+bi[/tex] where a and b are both real numbers, with the a part representing the real part of the complex number, and the bi representing the imaginary part.

   We can also graph these numbers using the complex plane. The complex plane has the real axis where the x-axis would normally be, and the imaginary axis where the y-axis would normally be. So by this definition the "a" is what determines the horizontal position or the position on the real axis and the "b" is what determines the vertical position or the position on the imaginary axis.

I attached a diagram of the complex plane, and it's essentially the same as a normal graph, with a=x, and b=y.

Question 1:

So when a complex number lies above the real-axis, that means the imaginary part is greater than 0. When a complex numbers lies to the right of the imaginary axis, that means the real part is greater than 0.

This means we have the form: [tex]a+bi \text { where } a > 0 \text{ and } b > 0[/tex]. You can literally plug in any number for a and b, so long as it fits this. For example we can just do: [tex]1+i[/tex]

So when a complex number lies below the real-axis, that means the imaginary part is less than 0. when a complex numbers lies to the left of the imaginary axis, the real part is less than 0.

This means we have the form: [tex]a+bi \text { where } a < 0\text{ and }b < 0[/tex]. You can plug in any real number that lies within this restriction. An example would be:

[tex]-1-i[/tex]

Question 2:

Above real axis and to the left of the imaginary axis means: b>0 and a<0. So we can plug in any number into the standard form that fits this restriction. an example would be: [tex]-1+i[/tex]

Below real axis and to the right of the imaginary axis means: b<0 and a>0. So we can plug in any number into the standard form that fits this restriction. An example would be: [tex]1-i[/tex]

Three students want to estimate the mean backpack weight of their schoolmates. To do this, they each randomly chose 8 schoolmates and weighed their backpacks. Then as per the given sample data,

(a) The sample means of the backpacks are: 6.375,6.375,6.625

(b) Range of sample means: 0.25

(c)The true statement is: The closer the range of the sample means is to 0, the less confident they can be in their estimate.

For the first sample, mean= 6.375

For the second sample, mean= 6.375

For the third sample, mean= 6.625

Range of sample means=Maximum Mean- Minimum Mean

                                        = 6.625 - 6.375

                                        = 0.25

The students will estimate the average backpack weight of their classmates using sample means, the true statement is:

The closer the range of the sample means is to 0, the more confident they can be in their estimate.

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The Created cone  with the possible record  of the the values of the radius, and  height is given in the image attached.

A scale factor is known to be a number that is known to often multiplies (doubles or triples, etc.,“scales”) a given quantity.

Since the volume is not attached, it will be manually created here.

Note that the resulting volumes must be in terms of 7.

Hence:

Volume  Formula:   V=V  x  k³

When scale factor is 2, radius  = 4, height  = 8,  then volume will be:

7π x 2³

=  56π

When scale factor is 3, radius  = 6, height  = 18,  then volume will be:

7π x 3³

= 189π

When scale factor is 4, radius  = 8, height  = 24,  then volume will be:

7π x 4³

= 448π

Hence the volume that will be fixed into the scale factor diagram will be:

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Step-by-step explanation:

that is only possible, if we don't need to use a 2 and a 5 (but especially the 2) to create the 1 in the middle (log(10) = log(2×5) or log (5×2) = 1).

in other words, if the 1 in the middle is just a given, and does not use any of our digits for is creation, then a solution is possible.

first of all, the upper left corner has to be log(0⁰). we cannot use the digit 0 for anything else. and 0^n (except for n = 0) is not a valid argument for a logarithm.

log(0⁰) = 0 log(2×1) = 0.3... log(9/4) = 0.35...

log(1×2) = 0.3... 1 log(3×6) = 1.26...

log(7/3) = 0.37... log(4×5) = 1.3... log(6⁹) = 7.0...

but as soon as we need to build the center 1 by log(10), which takes then away one of the 2s (as the only way to build 10 by multiplication is 2×5 or 5×2), there is no possible solution.

Answer:

Please see attachment.

Step-by-step explanation:

There are many possible solutions to this problem.

A few things to note:

There are 8 red boxes and 8 blue boxes, so we do not have to use every digit from 0 to 9.

Place zero in the blue box in the top left of the grid, since any number raised to zero is 1 and [tex]\log_a1=0[/tex].  Leave the red box blank for the moment, and fill it with any number not used at the end.

The value of the product of the red and blue boxes in the top-middle and left-middle of the grid should be less than 10.  It is advisable to place the lowest numbers in these boxes, i.e. 1 and 2.

The value of the product of the red and blue boxes in the bottom-middle and right-middle of the grid should be more than 10.  

It is advisable to place some of the highest numbers in the bottom right corner of the grid, as this is the box with the greatest value.

Please see the attached image for one possible way of filling the grid.  The actual values of the logarithms (to 3 decimal places) have been placed in the bottom right corners of each box of the grid for verification.

The identity for addition for rational numbers is zero which is the option b.

Given addition of rational numbers.

We are required to find the identity for the addition of rational numbers.

Rational numbers are those numbers which can be written in the form of p/q and q cannot be equal to zero because if q is equal to zero the whole fraction will become infinity. The numbers whih cannot be written as p/q are known as irrational numbers.

From the all option zero is the identity for addition for rational numbers. This  means if a is a rational number. Then a+0=0+a=a.

Hence the identity for addition for rational numbers is zero which is the option b.

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❄ Hi there,

let us revise the two rounding rules before getting down to the actual process of rounding.

[tex]\sf{Rules:}[/tex]

Let's see these rules in action!

The number is – [tex]219.2[/tex].

Nearest whole number – [tex]219[/tex], which is followed by a number between 0 and 4, so we just drop that number :

[tex]\sf{219.2- > > 219}[/tex]

❄

The length of the wall of the tank in feet is 62.8 feet.

Circumference is the perimeter of a circular object. Therefore,

circumference = 2πr

where

The fountains pool is enclosed by a circular plastic tank.

The distance from the centre of the pool to the wall of the tank is 10 feet. This is the radius of the circle

Hence,

length of the tank = 2πr

length of the tank = 2 × 3.14 × 10

length of the tank = 20 × 3.14

length of the tank = 62.8 feet.

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The complete statement is no matter what the value of s, √s² is equal to the absolute value of s?

The statement is given as:

No matter what the value of s, √s² is equal to the ______ value of s?

The above statement can be split as follows:

This means that, irrespective of the value of s, what would be the value of the square root of the square of s.

Assume that s is negative (say s = -2), the value of the square root of the square of s would be

√s² = √(-2)²

Evaluate the square

√s² = √4

Evaluate the square root

√s² = 2

See that s = 2 is the positive equivalent or absolute value of s = -2

Now, assume that s is positive (say s = 4), the value of the square root of the square of s would be

√s² = √4²

Evaluate the square

√s² = √16

Evaluate the square root

√s² = 4

See that s = 4 is the positive equivalent or absolute value of s = 4

Hence, the complete statement is no matter what the value of s, √s² is equal to the absolute value of s?

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Complete question

No matter what the value of s, √s² is equal to the ______ value of s?

The probability that a randomly selected adult has an IQ less than 130 is 0.9332

For given question,

We have been given adults IQ scores that are normally distributed with a mean of μ = 100 and a standard deviation σ = 15.

We need to find the probability that a randomly selected adult has an IQ less than 130

Sketch the curve.

The probability that X < 130 is equal to the area under the curve which is less than X = 130

Since μ = 100 and σ = 15 we have:

⇒ P ( X < 130 ) = P ( X- μ < 130 - 100 )

⇒ P ( X < 130 ) = P((X− μ)/ σ < 130 - 100/20 )

Since (X-μ)/σ = Z and (130 - 100)/20 = 1.5 we have:

P (X < 130) = P (Z < 1.5)

Now, we use the standard normal table to conclude that:

P (Z < 1.5) = 0.9332

Therefore, the probability that a randomly selected adult has an IQ less than 130 is 0.9332

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Applying the same-side interior angles theorem,

7. m∠4 = 50°; m∠5 = 130°

8. m∠2 = 15°; m∠8 = 15°.

The same-side interior angles theorem holds that, two interior angles on a side of a transversal are supplementary, that is they add up to 180 degrees.

According to the alternate exterior angles theorem, exterior angles that alternate each other along a transversal are congruent, that is they have equal measures.

7. m∠4 + m∠5 = 180 [same-side interior angles theorem]

Substitute

y + 2y + 30 = 180

3y + 30 = 180

3y = 180 - 30

3y = 150

y = 50

m∠4 = y = 50°

m∠5 = 2y + 30 = 2(50) + 30

m∠5 = 130°

8. m∠2 = m∠8 [alternate exterior angles theorem]

Substitute

x - 30 = 3x - 120

x - 3x = 30 - 120

-2x = -90

x = 45

m∠2 = x - 30 = 45 - 30 = 15°

m∠8 = 3x - 120 = 3(45) - 120 = 15°

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Answer:

Step-by-step explanation:

y=-1/2x+6

the function is increasing from x = 0 to x = 1

when x = 0, y = 2

when x = 1, y = 8

this shows an increase of +6, so we know this statement is true

we can see the rest are false simply because they state a decrease when there is an increase. this question is only confusing because we ask you about the increase and decrease in the function, when really its the same thing as asking about an increase or decrease in the y variable.

in other words, all youre looking for is the variation of the value of the y variable that is attached to the specific x variables.

hope this helps!!

The mass of the sample should be reported as 3.54 g

The amount of matter in an object is expressed in terms of mass.

The most frequent way to determine mass is to weigh something.

The units of mass are grams, kilograms, tonnes (in metric units), or ounces and pounds (US units).

According to the question,

A label on an empty sample container reads 10.000 g

A sample of a compound is added and the mass of the sample container is found to be 13.54 g.

The mass of the sample should thus be reported as,

= 13.54 - 10.000

= 3.54 g

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Answer: 4

Step-by-step explanation:

2.03 / 7 = 0.29

1.16 / 0.29 = 4

Mason will get 4 oranges on spending 1.16$

What is Rate of change?

A rate is anything that can be measured over time. Common rates include velocity (which is the rate of distance traveled), power (rate of work being done), etc. All these have one thing in common which is that they are all divided by the amount of time it took to obtain a certain measurement.

Here it is given that a grocery store sells a bag of 7 oranges for $2.03,

Now firstly let calculate for 1$ how many oranges Mason get by dividing 7 with 2.03$ :

For 1$ we get = 7/2.03 = 3.5 oranges

Therefore to calculate for 1.16 $ amount spend by Mason will be multiplication of 3.5 with 1.16$ : 3.5 x 1.16 = 4 oranges

Therefore, Mason will get 4 oranges on spending 1.16$

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The ball will take 1.15 seconds to hit the floor.

It is given that,

A juggler is performing her act using several balls. The height as a function of time is given by :

[tex]h(t) =-16t^{2} + vt +s[/tex].............(1)

where,

v is the speed of the ball

s is initial height

t is time taken

Putting h(t) = 0 and solving equation (1)

[tex]-16t^{2} + vt +s = 0[/tex]

[tex]-16t^{2} + 15t +4 = 0[/tex]

On solving above equation using graphing calculator and solving for t we get, t = 1.154 second.

Hence, the correct option is (b) " 1.15 seconds ".

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The true and false statement about the mean salary is illustrated below.

The true false statement about the mean salary include:

When Tom's salary has a z-score of 0.5, the amount that he earns will be:

0.5 = (x - 29300)/5400

0.5 × 5400 = x - 29300

2700 = x - 29300

x = 29300 + 2700.

x = 32000

The amount he earns is 32000.

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The given transformation is a reflection across the y-axis that changes the coordinates EFGH into E'F'G'H' as follows:

E(-4, -1) → E'(4, -1)

F(-2, -1) → F'(2, -1)

G(-1, -2) → G'(1, -2)

H(-1, -4) → H'(1, -4)

The transformation rules for reflection are:

Reflection at y-axis: y = f(-x);

Reflection at x-axis: y = -f(x);

The given polygon has four vertices EFGH.

Their coordinates are:

E(-4, -1), F(-2, -1), G(-1, -2), and H(-1, -4)

When the transformation rule: reflection across the y-axis is applied, the coordinates become:

y = f(-x); rule

For E(-4, -1), x-coordinate is -4. So, -(-4) = 4

⇒ E'(4, -1)

For F(-2, -1), x-coordinate is -2. So, -(-2) = 2

⇒ F'(2, -1)

For G(-1, -2), x-coordinate is -1. So, -(-1) = 1

⇒ G'(1, -2)

For H(-1, -4), x-coordinate is -1. So, -(-1) = 1

⇒ H'(1, -4)

Thus, the transformed coordinates are:

E(-4, -1) → E'(4, -1)

F(-2, -1) → F'(2, -1)

G(-1, -2) → G'(1, -2)

H(-1, -4) → H'(1, -4)

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1) a) The volume of the sphere is 100 cubic centimeters.

b) The volume of the cylinder is 200 cubic centimeters.

2) The volume of the cone is 175 cubic centimeters.

Herein we have two case of submerging solids into recipients full of liquid, the volume of the objects is equal to the volume of the displaced liquid. Now we proceed to calculate the volume of each object:

Point 1 - Part A:

0.8 L - 0.5 L = 3 · x

0.3 = 3 · x

x = 0.1 L

x = 100 cm³

The volume of the sphere is 100 cubic centimeters.

Point 1 - Part B:

0.8 L - 0.5 L = x + y

0.3 = x + y

0.3 = 0.1 + y

y = 0.2 L

y = 200 cm³

The volume of the cylinder is 200 cubic centimeters.

Point 2:

350 mL = 2 · z

z = 175 mL

z = 175 cm³

The volume of the cone is 175 cubic centimeters.

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Answer:

  $1876.31

Step-by-step explanation:

The present value can be found using the compound interest formula.

The formula for the future value of an investment of P earning annual interest rate r compounded n times per year for t years is ...

  FV = P(1 +r/n)^(nt)

Filling in the known values gives ...

  4000 = P(1 +0.085/4)^(4·9) = P(1.02125^36)

Then the amount to be invested is ...

  P = 4000/1.02125^36 ≈ 1876.31

The present value is $1876.31.

Considering the given box-and-whisker plot, it is found that the bookstore sold between 2 to 12 books in 20 days.

A box and whisker plot shows three things:

From the plot given in this problem, the have that:

This means that the percentage of days in which between 2 to 12 books were sold is:

P = 75 - 25 = 50%.

Hence, out of 40 days, the amount of days is:

0.5 x 40 = 20 days.

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Answer:

f(x) = -x²

Step-by-step explanation:

Writing f(x) = -x² in vertex form gives f(x)=-(x-0)^2+0 which shows that the vertex (h,k) is at (0,0)

The other equations written in vertex form do not have (h,k) = (0,0)

The simplified equivalent expression exists -1.9b + 7.8.

An equivalent expression exists an expression that contain the same value or worth as another expression, but does not look the same.

Combine the coefficient of the term b, we get

1.3b + 7.8–3.2b

= (1.3−3.2)⋅b + 7.8

simplifying the equation, we get

= (- 1.9) +b + 7.8

= −1.9b + 7.8

The simplified equivalent expression exists -1.9b + 7.8.

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6 possible ways to order 3 couples in a row

Given,

We have three married couples. that is six persons in total.

a, b, c, d, e, f may taken as the six persons.

Considering the statement:

Elder partner on the left, assume a, c and e as elders.

Then we have, (a, b) ------ ( 1)

                        (c, d) ------ (2)

                        (e, f) ------ (3)

There is no change in position between the couples.

So, we get three persons in total.

Possible ways to order these three persons will be like this: 3! = 3×2×1

                                                                                                        = 6

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The most appropriate translation of the English sentence. the amount of time spent driving (which depends on the speed of the car) is 5 hours. is time(speed) = 5, making option C the right choice.

The time spent by a body covering a certain distance at a certain speed is given as time = distance/speed.

In the question, we are asked to translate the English sentence, :"the amount of time spent driving (which depends on the speed of the car) is 5 hours".

In the given sentence, the time spent driving is given to be 5 hours, depending on the speed of the car.

Thus, the time can be shown as a function of speed with a value of 5.

Thus, the translation can be, time(speed) = 5.

Thus, the most appropriate translation of the English sentence. the amount of time spent driving (which depends on the speed of the car) is 5 hours. is time(speed) = 5, making option C the right choice.

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The provided question is incomplete. The complete question is:

"Choose the most appropriate translation of the English sentence. the amount of time spent driving (which depends on the speed of the car) is 5 hours.

a. speed(time) = 5

b. time(5) = 50 mph

c. time(speed) = 5

d. time(speed) > 0"

80d = 400 is the equation that relates the number of days d that an athlete spends training when the athlete trains for 400 min given that athlete trains for 80 min each day for as many days as possible. This can be obtained by converting the given statements into algebraic equation.

Here in the question it is given that,

that is each day the athlete spends 80 min.

So we can write that,

400 min is the time spent by the athlete

400 min  = number of days × time spent by the athlete in one day

400 min  = d × 80

⇒ 80d = 400  

Hence 80d = 400 is the equation that relates the number of days d that an athlete spends training when the athlete trains for 400 min given that athlete trains for 80 min each day for as many days as possible.

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Answer: Mean = 23

Step-by-step explanation:

Given information:

35, 40, 12, 16, 25, 10

Given formula:

[tex]Mean~=~\frac{Sum~of~terms}{Number~of~terms}[/tex]

Substitute values into the formula

[tex]Mean~=~\frac{35~+~40~+~12~+~16~+~25~+~10}{6}[/tex]

Combine like terms

[tex]Mean~=~\frac{75~+~28~+~35}{6}[/tex]

[tex]Mean~=~\frac{103~+~35}{6}[/tex]

[tex]Mean~=~\frac{138}{6}[/tex]

Simplify the fraction

[tex]\Large\boxed{Mean=23}[/tex]

Hope this helps!! :)

Please let me know if you have any questions

Answer:

(a) 22cm^2

Step-by-step explanation:

(a) 6*3=18

2*2=4

4+18=22

The value of the given expression is 105.

To find the value of the given expression:

Follow the PEMDAS order of operations:

Multiply and divide (left to right).

Add and subtract (left to right)

Therefore, the value of the given expression is 105.

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