how do you prove 6^7-6^6+6^5 is divisible by 31

Answer:

y = [tex]\frac{1}{2}[/tex] x + 3

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

given

- 2x + 4y = 12 ( add 2x to both sides )

4y = 2x + 12 ( divide through by 4 )

y = [tex]\frac{2}{4}[/tex] x + [tex]\frac{12}{4}[/tex] , that is

y = [tex]\frac{1}{2}[/tex] x + 3

The roots of the given polynomial using the rational zero theorem are;  2, -4 and 6.

We are given the polynomial;

y = x³ - 4x² - 20x + 48

Since all coefficients are integers, we can apply the rational zeros theorem.

The trailing coefficient (the coefficient of the constant term) is 48

Find its factors (with the plus sign and the minus sign): ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±48.

These are the possible values for p.

The leading coefficient (the coefficient of the term with the highest degree) is 1.

These are the possible rational roots:

±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±48.

Checking the possible roots: if a is a root of the polynomial P(x), the remainder from the division of P(x) by x - a should equal 0 (according to the remainder theorem, this means that P(a)=0

Plugging in those values, the only ones that yield P(a) = 0 are; 2, -4 and 6.

Thus, these are the roots of the given polynomial.

Read more about rational zero theorem at; https://brainly.com/question/17003818

#SPJ1

The expression for the sequence of operations is  3s - u - t .

In the question ,

it is given that ,

the sequence of operation is given as " triple s, subtract u from the result, then subtract t from what you have" .

we have to write an expression for the above operation ,

So ,

first step is : triple s ,

that means the expression is  3s .

next subtract u from the result , that means ⇒ 3s - u .

next we have to subtract t from what we have ,

that means ⇒ 3s - u - t .

Therefore , the final expression is 3s - u - t .

Learn more about Expression here

https://brainly.com/question/19261564

#SPJ4

The Subtraction of the given fraction using the number line is 5/6.

The correct answer option is option A

Fraction refers to a number which consists of a numerator and a denominator.

A numerator is the upper value of a fraction while a denominator is the lower value or bottom value of a fraction.

1 2/3 - 5/6

= 5/3 - 5/6

Find the lowest common multiple of the denominators and divide

= (10-5) / 6

= 5/6

In conclusion, 5/6 is the answer to the subtraction of fraction.

Read more on fraction:

https://brainly.com/question/11562149

#SPJ1

Answer:

(1.5, −0.5)

Step-by-step explanation:

(a) The probability mass function of X will be P(X=n) = 5/(6n)

(b) E[X] =1.2

(c) Probability [ Alice win] will be =0.5

(d) The value of expected winnings will be 4.25

(a) Let 'p' represent the probability of getting the same number on both the dices on a roll of pair of die

And (1-p) = probability of not getting the same number on both the dices on a roll of pair of die

Sample space for the same number appearing on both the dices on a roll of pair of die = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}

As there are a total of 36 outcomes on a roll of pair of dice,

p = 6/36 = 1/6

Let the events be defined as:

Dn : Both Alice and Bob roll the same number on the nth roll

Wn : The game is finished in the nth roll (i.e. Alice and Bob roll different numbers (as the numbers are different, one will be larger))

Then,P(W) = P(W) , n = 1,2,3,4....infinity

Wn takes place when event D takes place in the first (n-1) rolls and Event W takes place in the nth roll

[tex]$\mathrm{P}(\mathrm{W})=\sum P\left(W_n\right), \mathrm{n}=1,2,3,4 \ldots .$[/tex]. infinity

And, W_n takes place when

Event D takes place in the first (n-1) rolls and event W takes place in the [tex]$n^{\text {th }}$[/tex] roll

[tex]P\left(W_n\right)=P\left(D_1\right)^* P\left(D_2\right)^* P\left(D_3\right) \ldots . .{ }^* P\left(D_{n-1}\right) P\left(W_n\right)P\left(D_n\right)\\=(p)P\left(W_n\right)\\=(1-p)[/tex]

Implies

[tex]$$\begin{aligned}& \left.P\left(W_n\right)=(p)^{n-1}\right)^*(1-p) \\& =\left((1 / 6)^{n-1}\right)^*(1-(1 / 6)) \\& =\left((1 / 6)^{n-1}\right)^*(5 / 6) \\& =5 /\left(6^n\right)\end{aligned}$$[/tex]

(b)

[tex]$$\begin{aligned}& \mathrm{E}[\mathrm{X}]=\sum n * P(X=n), \mathrm{n}=1,2,3,4 \ldots . . \text { infinity } \\& =\sum n * \frac{5}{6^n} \\& =5 * \sum n * \frac{1}{6^n}\end{aligned}$$[/tex]

If [tex]$\mathrm{S}=\sum n * \frac{1}{6^n}$[/tex]

Then S is a sum of infinite aritho-geometric series of the form

[tex]$$\sum_{k=1}^{\mathrm{inf}} k * r^k \text {, }$$[/tex]

The sum of which is given as [tex]$S=\frac{r}{(1-r)^2}$[/tex] for 0 < r < 1

Here, r=(1 / 6)

[tex]\Rightarrow & S=\frac{\frac{1}{6}}{\left(\frac{5}{6}\right)^2} \\& =\frac{6}{25}[/tex]

E[X] = (5) x (6/25) = (6/5)= 1.2

(c) Since the dies are fair, and the outcomes are independent, the probability of each player winning is equal

Therefore,

P(Alice WIn) = P(Bob WIn) = 0.5

(d) If X represent the round in which the game ends, then the probability of us winning in that round

P(W|X=n) = 0.5*P(X=n)

= (0.5)*(5/(6n))

P(W|X=1) = (0.5)*(5/(61)) = (5/12)

P(W|X=2,3,4...) = 0.5 - P(W|X=1) = (1/2) - (5/12) = (1/12)

Leth the variable W represent the winnings

(W|X=1) = 10

(W|X=2,3,4...) = 1

E[W] = P(W|X=1)*(W|X=1) + P(W|X=2,3,4...)*(W|X=2,3,4...)

= (10 x (5/12)) + (1 x (1/12))

= (50/12) + (1/12)= (51/12)= 4.25

For more questions on Probability

https://brainly.com/question/29856487

#SPJ4

The value of a in the relative frequency table for the survey results a = 27%

From the given Venn diagram, the total number of players =  11 + 3 + 7 + 5 = 26

The number of players preferred magazines books which are not coupon books = 7

The relative frequency for the players who preferred magazines books which are not coupon books (a)=

magazine books but not coupon books/ total player = 7 / 26 = 0.269

In percent,

7/26 x 100 = 26.9 = 27%

Hence, a = 27%

Therefore, the value of a in the relative frequency table for the survey results a = 27%

To learn more about venn diagram refer here

https://brainly.com/question/2099071

#SPJ4

Answer: 13.55 or 13 11/20.

Step-by-step explanation: First, you have to find a common denominator, in this case, it would be 20. 2 can go into 20 10 times, 4 can go into 20 5 times, and 10 can go into 20 2 times. Then, you would NOW have 3 10/20 + 4 15/20 + 5 6/20, to have all the equal denominators. When you add, you would get 12 31/20, but now you have to simplify. Now, you SHOULD get 13 11/20 after simplifying. As a decimal, it would be 13.55 I hoped this helped!

Answer: x =2, -1

Step-by-step explanation:

Done :) Refer to the photo taken. Let me know if this is what you where looking for. Sorry for the sideways image.

Answer:

C)   (-3,-2)

Step-by-step explanation:

Solutions to the graphed system of inequalities are any points that are contained within the overlapping shaded region.

Plot the points on the given coordinate grid.

The only point that is contained with the overlapping shaded region is:

The x-coordinate for the point  (x, -8), if Y-intercept is 4, and the coordinates of the point is (14, -3), is 24, so option B is correct.

An object having an endless length and no width, depth, or curvature is called a line. Since lines can exist in two, three, or higher-dimensional environments, they are one-dimensional things.

Given:

Y-intercept = 4,

Coordinates of point =  (14, -3), (x, -8)

Calculate the line equation as shown below,

y = m x + 4

Here, m is the slope,

Put point (14, -3) in the equation,

-3 = m × 14 + 4

m = -7 / 14

m = -1 / 2

Hence, the equation of a line is,

y = -1/2 x + 4

Put point (x, -8),

-8 = -1/2 x + 4,

-12 = -1/2x

x = 24

To know more about the line:

https://brainly.com/question/2696693

#SPJ1

The alternative hypothesis for this problem b. HA: µD>0

What is meant by hypothesis?

In math, A hypothesis is an assumption made based on some evidence.

Here we have given that systems analyst is testing the feasibility of using a new computer system. He wants to see if the new system uses less processing time than the old system.

And we need to find the alternative hypothesis for this problem

While we looking into the given question we have identified the following values,

Old System: mean = 27.2 seconds, s = 3.2 seconds, n - 25

New System: mean = 24.3 seconds, s = 2.1 seconds, n = 25

Difference (Old -New): mean = 2.9 seconds, s = 1.4 seconds, n = 25

Basically, while written the hypothesis, we have to follow the order of the following,

=> (Old-New, Larger-Smaller>)

Therefore, the correct option is (B).

To know more about Hypothesis here.

https://brainly.com/question/29576929

#SPJ4

Answer:

59%

Step-by-step explanation:

First, let's find the number of students who play basketball but not baseball: 5 students - 3 students = <<5-3=2>>2 students

Next, let's find the number of students who play baseball but not basketball: 11 students - 3 students = <<11-3=8>>8 students

Now, let's add up the number of students who play basketball but not baseball, the number of students who play baseball but not basketball, and the number of students who play both sports to find the total number of students who play basketball or baseball: 2 students + 8 students + 3 students = <<2+8+3=13>>13 students

Finally, we can divide the total number of students who play basketball or baseball by the total number of students in the class to find the probability that a student chosen at random plays basketball or baseball: 13 students / 22 students = 0.59

Therefore, the probability that a student chosen at random from the class plays basketball or baseball is 0.59.

Answer:

13/22

Step-by-step explanation:

The class has 22 students.

5 play basketball.

11 play baseball.

3 play both.

Break down the 5 who play basketball into:

2 play only basketball

3 play both basketball and baseball

Break down the 11 who play baseball into:

8 play only baseball

3 play both basketball and baseball (These 3 are the same 3 above)

Now we have:

2 play only basketball

8 play only baseball

3 play both

This is a total of 13.

The class has 22, so 9 don't play any sport.

That means out of 22 students, 13 play either sport, and 9 play nothing.

p(basketball or baseball) = 13/22

We observe, that P(V|B)>P(V|A), so the person is more likely to have virus, however is still very small probability (only 15%), so in order to confirm illness, he should make one more test.

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.

P(V/A)  is not 0.95. It is opposite:

P(A/V)=0.95

From the text we can also conclude, that

P(A/∼V)=0.1

P(B/V)=0.9

P(B/∼V)=0.05

P(V)=0.01

P(∼V)=0.99

P(V/A) is not 0.95. It is opposite:

P(A/V)=0.95

From the text we can also conclude, that

P(A/∼V)=0.1

P(B/V)=0.9

P(B/∼V)=0.05

P(V)=0.01

P(∼V)=0.99

What we need to calculate and compare is P(V/A) and P(V/B)

P(V∩A)=P(A)⋅P(V/A)⇒P(V/A)=P(V∩A)/P(A)

P(V∩A) means, that person has a virus and it is detected, so

P(V∩A)=P(V)⋅P(A/V)

=0.01⋅0.95

=0.0095

P(A) is sum of two options: "Person has virus and it is detected" and "Person has no virus, but it was mistakenly detected", therefore:

P(A)=P(V)⋅P(A/V)+P(∼V)⋅P(A/∼V)=0.01⋅0.95+0.99⋅0.1=0.1085

Dividing those two numbers we obtain

P(V/A)=0.0095/0.1085=0.08755760368663594

Analogically,

P(V/B)=P(V∩B)/P(B)=P(V)⋅P(B/V)/P(V)⋅P(B/V)+P(∼V)⋅P(B/∼V)

=0.01⋅0.9/0.01⋅0.9+0.99⋅0.1

=0.1538461538461539

We see that P(V|B)>P(V|A), meaning the individual is more likely to have a virus, but is still very unlikely (only 15%), therefore person should perform another test to confirm illness.

To learn more about probability visit: brainly.com/question/11234923

#SPJ4

Based on this information, we can estimate that a 1,000-person increase in population at the destination, an increase of 441 auto-mode social/recreational trips from the origin during the peak hour time

a 10-mile increase in distance to the destination, and the presence of recreational facilities at the destination are associated with an increase of 725 x (0.43 + (-0.07) + 0.62) = 441 auto-mode social/recreational trips from the origin during the peak hour.

1. Calculate the total increase in auto-mode social/recreational trips from the origin during the peak hour associated with a 1,000-person increase in population at the destination, a 10-mile increase in distance to the destination, and the presence of recreational facilities at the destination by multiplying the coefficients of the logit model by 725 (the total number of trips):

725 x (0.43 + (-0.07) + 0.62) = 441

2. Therefore, we can estimate that a 1,000-person increase in population at the destination, a 10-mile increase in distance to the destination, and the presence of recreational facilities at the destination are associated with an increase of 441 auto-mode social/recreational trips from the origin during the peak hour time.

Learn more about time here

https://brainly.com/question/28050940

#SPJ4

Answer:

Step-by-step explanation:

2sin x=-√3

[tex]sin~x=-\frac{\sqrt{3} }{2} =-sin~\frac{\pi }{3} =sin (\pi +\frac{\pi }{3} ),sin(2\pi -\frac{\pi }{3} )\\=sin(\frac{4\pi }{3} +2k\pi ),sin (\frac{5\pi }{3} +2k\pi )\\=sin(A+Bk\pi ),sin(C+Dk\pi )\\x=A+Bk\pi ,C+Dk\pi \\where~A=\frac{4\pi }{3} ,B=2\\C=\frac{5\pi }{3} ,D=2[/tex]

The solution of the given inequality -x < -x + 7 ( x - 2 ) is given by x > 2.

As given in the question,

Inequality is given by :

-x < -x + 7 ( x - 2 )

Open the parenthesis of the given inequality we have,

⇒ -x < -x + 7x - 14

Collect all the like terms on the same side we get,

⇒ - x < 6x  - 14

Subtract 6x from both the side of the inequality we get,

⇒ - x - 6x < 6x - 6x - 14

⇒ -7x < -14

Divide both the side of the inequality by -7 we get,

⇒ -7x/ -7 > -14 / -7

⇒ x > 2

x has all the value greater than 2.

Therefore, for the given inequality the solution is given by x > 2.

Learn more about inequality here

brainly.com/question/28823603

#SPJ4

Answer:

Step-by-step explanation:

Quite literally you plug in -6 into t. So therefore:
h(t)=6t-2

h(-6)=6(-6)-2

h(-6)=-36-2

h(-6)=-38
-38 is your answer
h can be replaced with f and t can be replaced with x to get your standard f(x) equation if that helps you understand it better. Remember f(x) really just means the y value of the function with an input of value x. So f(x) = x is really just y=x or a linear line.

Answer:

55 1/10 yd

Step-by-step explanation:

26 + 10 + 3/5 + 10 + 1/5 + 8 + 3/10 =

= 54 + 4/5 + 3/10 =

= 54 + (8 + 3)/10 =

= 54 + 11/10 =

= 54 + 10/10 + 1/10 =

= 54 + 1 + 1/10 =

= 55 1/10

Answer:

55[tex]\frac{1}{10}[/tex]

Step-by-step explanation:

Convert all the mixed numbers to fractions with a common denominator (10) then add them up:

26 + 53/5 + 51/5 + 83/10 =

26 + 106/10 + 102/10 + 83/10 =

26 + 291/10 =

26 + 29[tex]\frac{1}{10}[/tex] = 55[tex]\frac{1}{10}[/tex]

Answer:

Step-by-step explanation:

that normally means we want to add our subtract a certain of the 2 equations, so that we are led with one summary equation with only one variable.

the easiest way here is possibly to multiply the second equation by 5 (to get also 10y) and then subtract it from the first equation :

3x + 10y = -4

- 10x + 10y = 10

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

-7x + 0 = -14

7x = 14

x = 2

2x + 2y = 2

gives us then

2×2 + 2y = 2

4 + 2y = 2

2y = -2

y = -1

x = 2

y = -1

Answer:

x = 2

y = -1

Step-by-step explanation:

3x + 10y = -4

- 10x + 10y = 10

7x + 0 = -14

7x = 14

x = 2

2x + 2y = 2

2×2 + 2y = 2

4 + 2y = 2

2y = -2

y = -1

For the given range of theta, cos(theta) is negative, which means that theta is in the third or fourth quadrant. In these quadrants, the sine function is positive, the tangent function is negative, and the cotangent function is negative.

Since cos(theta) = -3/5, we can use the Pythagorean identity to find the value of sin(theta):

sin^2(theta) + cos^2(theta) = 1

sin^2(theta) = 1 - cos^2(theta)

sin^2(theta) = 1 - (-3/5)^2

sin^2(theta) = 1 - 9/25

sin^2(theta) = 16/25

sin(theta) = sqrt(16/25) = 4/5

Therefore, the value of sin(theta) is 4/5.

We can also use the identity cot(theta) = 1/tan(theta) to find the value of cot(theta):

cot(theta) = 1/tan(theta)

cot(theta) = 1/(sin(theta)/cos(theta))

cot(theta) = cos(theta)/sin(theta)

cot(theta) = (-3/5)/(4/5)

cot(theta) = -3/4

Therefore, the value of cot(theta) is -3/4.

The values of the other trigonometric functions can be found using the definitions of these functions:

tan(theta) = sin(theta)/cos(theta) = (4/5)/(-3/5) = -4/3

csc(theta) = 1/sin(theta) = 1/(4/5) = 5/4

sec(theta) = 1/cos(theta) = 1/(-3/5) = -5/3

Therefore, the values of the trigonometric functions for the given range of theta are:

sin(theta) = 4/5

tan(theta) = -4/3

csc(theta) = 5/4

sec(theta) = -5/3

cot(theta) = -3/4

The length of James's rectangular yard, given a fence perimeter of 20 feet and a width of 2 feet, is 8 feet, equivalent to 96 inches.

Given: Perimeter (P) = 20 feet, Width (W) = 2 feet,

Conversion factor: 1 foot = 12 inches

Using the formula for perimeter: P = 2L + 2W

Substitute the given values: 20 = 2L + 2(2)

Simplify: 20 = 2L + 4

Subtract 4 from both sides: 16 = 2L

Divide by 2: L = 8 feet

Convert feet to inches: Length (in inches) = 8 feet * 12 inches/foot = 96 inches

So, the length of the rectangular yard is 96 inches.

Learn more about perimeter of a rectangle on:

https://brainly.com/question/24571594

#SPJ1

To find the length of James' rectangular yard in inches, we can use the perimeter formula P = 2L + 2W. Given that the perimeter is 20 feet and the width is 2 feet, we can substitute these values into the formula and solve for the length in inches. The length of James' rectangular yard is 108 inches.

To find the length of James' rectangular yard in inches, we can use the perimeter formula P = 2L + 2W. Given that the perimeter is 20 feet and the width is 2 feet, we can substitute these values into the formula. However, since we need the length in inches, we will convert the feet to inches using the conversion factor of 1 foot = 12 inches.

Let's solve:

The length of James' rectangular yard is 108 inches.

Learn more about rectangular yard here:

https://brainly.com/question/34773081

#SPJ1

Jacob's body metabolizes caffeine at a rate of 13.5% per hour.

a)Jacob's body will takes 4.77 ~5 hours to metabolize half of the 96 mg of caffeine.

b) If Jacob consumes an energy drink with 212 mg of caffeine in it, Jacob's body will take 4.77948 ~ 5 hours to metabolize half of the 212 mg of caffeine.

c) Jacob consumes a cup of coffee with c mg of caffeine in it. Jacob's body will take 4.77948~ 5 hours to metabolize half of the c mg of caffeine.

The volume will slowly decrease at regular intervals and at a regular rate. This growth reduction is calculated using the exponential decay formula. The general form is y = a(1- r)ᵗ

We have, decay rate r = 13.5 % = 0.135

Initial value , a = 96 mg

plugging the value in formula we get,

y = 96(1 - 0.135)ᵗ --(1)

Now, Half of 96 is 48

so, 48 = 96(0.865)ᵗ --(2)

dividing equation by 96 we get

=> 48/96 = (0.865)ᵗ

taking natrual logarithm both sides,

=> ln(1/2) = ln ((0.865))

=> ln(1/2) = t In (0.865)

=> t = ln(1/2)/In (0.865)

=> t = 4.77948

Since, decay rate is constant so, half never changes for any .

a) 4.779~5 hours, long will it take for Jacob's body to metabolize half of the 96 mg of caffeine.

b) Jacob's body to metabolize half of the 212 mg of caffeine is 4.779~ 5 hours.

c) for Jacob's body to metabolize half of the cmg of caffeine is 4.77948.. So, we have 4.77948 ~ 5 hours is decay constant rate.

To learn more about Decay rate , refer:

https://brainly.com/question/27542728

#SPJ4

Answer:

In 2022, the town's population is 2220.

Step-by-step explanation:

It currently the year 2022. Hence, 22 years have passed since 2000.

2022 − 2000 = 22

To solve for the town's population today, first multiply the years passed since 2000 by the percent decrease each year.

[tex]22 \, \textrm{years} \times \dfrac{3.2 \, \%}{\textrm{year}} = 70.4 \, \%[/tex]

Then, subtract that percent of the population in 2000 from the population in 2000.

[tex]7500-(70.4 \,\% \cdot 7500)[/tex]

[tex]= 7500 - 5280[/tex]

[tex]=2220[/tex]

Answer:

This statement is false

Step-by-step explanation:

This statement is false. The volume of a cylinder with a given height and diameter will always be greater than that of a cone with the same height and diameter if the base of the cone is smaller than the base of the cylinder. However, if the base of the cone is the same size as the base of the cylinder, then the volume of the cone will be equal to that of the cylinder.

Answer:

The shopper received a 75% discount

Step-by-step explanation:

ok so first we turn it into an equation

let x=percentage discount

16(x)=12

x=.75

discount=75%

Answer:

Step-by-step explanation:

x=-4

The value of x that makes sense for the expression is the domain of the expression
The value of x makes sense for √-2x² is 0
The value of x must not exceed 0 for  √(-2x)³ to make sense

What is Angle Sum Property?

The sum of all angles of a triangle is equal to the angle of a straight line i.e. 180°. If we have a triangle ABC, then the Sum of angles A , B, and angle C will be 180 ° and the value of the exterior angle is equal to the sum of two interior opposite angles.

√-2x²

For the above expression to have a defined value, the expression in the bracket must be positive.

However, x² will always be positive, and -2 will ensure that the expression is always negative, except when x = 0

Hence, the value of x makes sense for √-2x² is 0.

√(-2x)³

For the above expression to have a defined value, the expression in the square root must be positive.

However, the power of 3 will ensure that the expression is always positive only when x is less than or equal to 0

Hence, the value of x must not exceed 0 for √(-2x)³ to make sense.

To know more about the Angle sum property visit,

brainly.com/question/21099419

#SPJ1

Answer:

Step-by-step explanation:

the question is asking,  or implying,  when is [tex]t^{2}[/tex] = t   :?  

if you are thinking at 1,  you're correct.   so the ball will hit the ground at 1 second