By the principle of mathematical induction, it has been proved that
[tex]p^3 + 11p[/tex] is divisible by 6
What is Principle of mathematical induction?
Let the statement be given by P(n). At first P(n) is proved to be true for 1. Then it is assumed that P(n) is true for all n. If P(n) is true for (n+1) then by Principle of Mathematical induction, P(n) is true for all natural numbers
The given expression is [tex]p^3 + 11p[/tex]
For p = 1
[tex](1)^3 + 11 \times 1\\12[/tex]
which is a multiple of 6
Let it be true for p = m
[tex]m^3 + 11m[/tex] is a multiple of 6
[tex]m^3 + 11m = 6k[/tex], where k is any integer
For p = m + 1
[tex](m+1)^3 + 11(m+1) \\m^3+3m^2+ 3m +1+11m+11\\m^3+11m +3m^2+3m+12\\m^3+11m + 3m(m+1) + 12[/tex]
Now,
[tex]m^3 + 11m[/tex] is divisible by 6
We know that the product of two consecutive integers are divisible by 6
[tex]3m(m+1)[/tex] is divisible by 6
12 is divisible by 6
[tex](m+1)^3 + 11(m+1)[/tex] is divisible by 6
So it is true for p = m + 1
By the principle of Mathematical induction, it is true for all natural numbers and eventually for all integers
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Find the standard form of the parabola
with vertex: (- 2,5) and directrix: x = 3.
Check the picture below, so the parabola looks more or less like so, the "p" distance is negative, because the horizontal parabola is opening to the left-hand-side, so
[tex]\textit{horizontal parabola vertex form with focus point distance} \\\\ 4p(x- h)=(y- k)^2 \qquad \begin{cases} \stackrel{vertex}{(h,k)}\qquad \stackrel{focus~point}{(h+p,k)}\qquad \stackrel{directrix}{x=h-p}\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix}\\\\ \stackrel{"p"~is~negative}{op ens~\supset}\qquad \stackrel{"p"~is~positive}{op ens~\subset} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\begin{cases} h=-2\\ k=5\\ p=-5 \end{cases}\implies 4(-5)( ~~ x-(-2) ~~ )=(y-5)^2 \implies -20(x+2)=(y-5)^2 \\\\\\ x+2=-\cfrac{1}{20}(y-5)^2\implies {\Large \begin{array}{llll} x=-\cfrac{1}{20}(y-5)^2-2 \end{array}}[/tex]
one serving contains 80 calories; how many calories are in 3 1/4 servings?
Answer: 260 calories
Step-by-step explanation:
If one serving has 80 calories, we will multiply 80 by 3 1/4 to find the total number of calories in 3 1/4 servings.
80 calories * (3 1/4) servings = 260 calories
helppppp <3 i’ll mark brainliest!! thanks in advance :)
Triangle ONM is congruent to Triangle CBA
Create a word problem that can be solved using an equation involving only multiplication or division. Give the equation for the word problem.
Respond to two of your classmates' posts. Solve the equation for their word problem and explain what the solution represents.
Answer:
Belle has 25 apples she shares them equally with her 5 friends, how many apples does each friend get? 25/5=5
Step-by-step explanation:
How does the role of the board of governors compare to the role of the federal open market committee
Answer:
The Board of Governors of the Federal Reserve System is responsible for the discount rate and reserve requirements, and the Federal Open Market Committee is responsible for open market operations.
Step-by-step explanation:
A baseball player makes an annual salary of 3,200,000. and makes $17,260.274 a day.
A charitable organization states that for $120 it can send 1 child to school in a developing country for 1 year . The baseball player decides to donate 10% of his annual salary to this organization .
Calculate 10% of the baseball player’s salary.
Answer:
320,000.00
Step-by-step explanation:
50 points
There are 140 people in a theater watching a movie. The number of children present is 3 times the number of adults. How many children are watching the movie?
12 children
35 children
3 children
105 children
Answer:
The answer is 35 children and 105 adults
Step-by-step explanation:
we don't know the number of kids or adults in the theater , so they are unknown.
x: adults
y: kids
we know that y=3x (1)
we also know that all the people in the theater are 140.
so we also know that x+y= 140 (2)
And now you need to take (1) and put it in (2):
y=3x
x+y= 140 x+3x=140 =} 4x=140=} x=140/4=} x=35 kids. so if we put on (1) the number of the kids we have the number of adults: y=3*35=}y=105 adults
I hope I helped !
What is the inverse of this function? f(x) = -1/2 square root x+3 x> -3.
Answer:
F-1(x) = 4x2 - 3 x<0
ab + bc + ac for a = 2, b = 5, and c = 3.
Answer:
31
Step-by-step explanation:
2⋅5+5⋅3+2⋅3
10+15+6
31
What is the value of 3 + 5x when x
=
3?
Answer:
18
Step-by-step explanation:
What is the value of 3 + 5x when x=3?
3 + 5x =
replace the x with 33 + 5 × 3 =
Remember PEMDAS3 + 15 =
18
---------------
17. Toby is riding his bicycle at 15 m/s. If it
takes him 60 seconds to get to the end of
the street. What was the length of the
street?
18.
met
him
Answer:
900 meters long
Step-by-step explanation:
Toby is riding his bicycle at 15 m/s. If it takes him 60 seconds to get to the end of the street. What was the length of the street?
at 15 meters per second for 60 seconds:
= time * speed
60* 15= 900 meters traveled,
so the street is 900 meters long.
Consider the following equation:
X/4+3= 1/5
The value of a that will make the equation above valid lies between
which of the following numbers?
A) 10 and 15
B) -1 and -5
C) 1 and 5
D) -10 and -15
E) -5 and -10
For the given equation (x/4)+3= 1/5, the value of x which will make the given equation valid is lies between the numbers of ( -10 and -15).
As given in the question,
Given equation is equal to :
(x/4)+3= 1/5
Simplify the given equation (x/4)+3= 1/5 to get the value of x we have,
(x/4)+3= 1/5
Take least common multiple of 4 and 1 we get,
( x + 12) / 4 = 1/5
⇒5( x +12 ) = 4
⇒ 5x + 60 = 4
⇒ 5x = -56
⇒ x= -11.2
-11.2 lies between (-10 and -15).
Therefore, for the given equation (x/4)+3=1/5, the value of x which will make the given equation valid is lies between the numbers of
(-10 and -15).
The complete question is:
Consider the following equation:
(x/4)+3= 1/5
The value of x that will make the equation above valid lies between
which of the following numbers?
A) 10 and 15
B) -1 and -5
C) 1 and 5
D) -10 and -15
E) -5 and -10
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The diagram below shows a circle with centre 0. ACB is a tangent to the circle and DOE is 68* Find the angle EDC. You must show working. Please i beg can someone help me!!!
Answer:
EDC = 56 degrees
Step-by-step explanation:
triangle OED is isosceles : base angles are congruent
2 * EDC + 68 = 180
2 * EDC = 180 - 68
2 * EDC = 112
2/2 EDC = 112/2
EDC = 56 degrees
Step-by-step explanation:
Triangle DOE is isosceles. So OED and ODE has same angle.
Total angle for isosceles triangle is 180.
ODE + OED = 180 - 68 = 112
OED = ODE = 112 /2 = 56
What is an equation of the line that passes through the points (-4, 1) and (4, -1)?
please help me......................!
Answer:
Option 4
Step-by-step explanation:
In integer division or multiplication,
If both numbers are negative or positive, the answer will be positive. If one number is negative and other number is positive, the answer will be negative.1) If a < 0 and b < 0 means a and b are both negative numbers.
Then a/b will be positive. But it is given that a/b < 0. So, it is wrong.
2) If a< 0 and b > 0 means a is negative & b is positive. So, a/b will be negative. But it is given a/b is positive. So, it is wrong.
3) a and b are positive (as a >0 and as b > 0), then a/b > 0 (a\b will be positive). So, it is wrong.
4) a is positive (as a>0) & b is negative (as b < 0), then a/b will be negative So, this is correct.
The Johnson family has decided to create a garden that is in the shape of a parallelogram. The base of the garden is 6 1/2 feet (ft). with a height 2 5/8ft. What is the area of the garden?
The area of the garden with a base of [tex]6\frac{1}{2}[/tex] feet and a height [tex]2\frac{5}{8}[/tex] feet is equal to 17.06 square feet.
What is a Parallelogram?
The word "parallelogram" is a translation of the Greek phrase "parallelogrammon," which means "bounded by parallel lines." As a result, a quadrilateral that is bound by parallel lines is called a parallelogram. It has parallel and equal opposing sides on all sides. Square, rectangle, and rhombus are the three primary varieties of parallelograms, and each one has distinct characteristics.The formula to calculate the area of a parallelogram (A) is given by the equation, [tex]A=b \times h[/tex], where [tex]b[/tex] is the base and [tex]h[/tex] is the height of the parallelogram respectively.In the given question, it is said that the garden decided to be created by the Johnson family is in the shape of a parallelogram.
The base and height of the parallelogram are given as [tex]6\frac{1}{2}[/tex] feet and [tex]2\frac{5}{8}[/tex] feet respectively.
[tex]\implies b=6\frac{1}{2} =\frac{13}{2}[/tex], and
[tex]h=2\frac{5}{8} =\frac{21}{8}[/tex]
The area of a parallelogram is calculated using the formula,
[tex]A=b \times h[/tex]
Substituting the values in the given formula, we get
[tex]A=\frac{13}{2} \times \frac{21}{8} \\\implies A=\frac{273}{16}\\ \implies A=17.06 ft^{2}[/tex]
Therefore, the required area of the garden is equal to 17.06 square feet.
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What’s the answer to 5(2x + 1) = 50 ?
Answer:
x = 4.5
Step-by-step explanation:
5(2x + 1) = 50 Distribute the 5
5(2x) + (5) (1) = 50
10x +5 = 50 Subtract 5 from both sides
10x = 45 Divide both sdies by 10
x = 4.5
NO LINKS!! Find the center and radius of the circle with the given equation
Answer:
[tex]\textsf{center \quad $(x,y)=\left(\; \boxed{6,-2}\; \right)$}[/tex]
[tex]\textsf{radius \quad $r=\boxed{8}$}[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{4 cm}\underline{Equation of a circle}\\\\$(x-a)^2+(y-b)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(a, b)$ is the center. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]
Given equation:
[tex]x^2+y^2-12x+4y-24=0[/tex]
Move the constant to the right side of the equation and collect like variables on the left side of the equation:
[tex]\implies x^2-12x+y^2+4y=24[/tex]
Create perfect square trinomials for the both variables by adding the square of the half the coefficient of x and y to both sides:
[tex]\implies x^2-12x+\left(\dfrac{-12}{2}\right)^2+y^2+4y+\left(\dfrac{4}{2}\right)^2=24+\left(\dfrac{-12}{2}\right)^2+\left(\dfrac{4}{2}\right)^2[/tex]
[tex]\implies x^2-12x+\left(-6\right)^2+y^2+4y+\left(2\right)^2=24+\left(-6\right)^2+\left(2\right)^2[/tex]
[tex]\implies x^2-12x+36+y^2+4y+4=24+36+4[/tex]
[tex]\implies x^2-12x+36+y^2+4y+4=64[/tex]
Factor the two perfect trinomials:
[tex]\implies (x-6)^2+(y+2)^2=64[/tex]
Therefore:
[tex]\textsf{center \quad $(x,y)=\left(\; \boxed{6,-2}\; \right)$}[/tex]
[tex]\textsf{radius \quad $r=\boxed{8}$}[/tex]
Jan is five years older than Tom in three years time will be 3/4 as old is Jan what is Tom's age now
The present age of Tom is 12 years old
Jan is five years older than Tom
The age of Tom = x
The age of Jan = x + 5
After 3 years
The age of Tom = x + 3
The age of Jan = x + 5 + 3
= x + 8
The Tom's age is 3/4 of Jan's age
Then the equation will be
x + 3 = 3/4(x + 8)
Apply distributive property
x + 3 = (3/4)x + 6
x - (3/4)x = 6 - 3
Subtract the terms
1/4 x = 3
Move 1/4 to the right hand side of the equation
x = 3 ÷ 1/4
x = 3 × 4
Multiply the terms
x = 12 years old
Hence, the present age of Tom is 12 years old
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in the spinner below what is the probability of landing on 1
Answer:
1/8Step-by-step explanation:
there are 6 numbers, but 3 and 6 take up double the space, so the probabilities are not 6 but 8
1/8 is your answer
Which statement is true about the solution of the system?
x − y = 4
-x + y = -4
There are no solutions.
There are an infinite number of solutions.
There is one solution.
Answer
Infinite solutions
Step-by-step explanation:
try any example you want it keeps going on and on.
1. If two expressions with exponents are a perfect match except one has a negative exponent and
the other has a positive exponent, like 53 and 5-3, how does the exponent change the meaning
of the expression?
According to the law of exponent, the value of positive exponent is greater than the negative exponent, this defines that they are not equal.
Exponent
Exponent refers the number of times a number is multiplied by itself.
Given,
Here we need to verify that, if two expressions with exponents are a perfect match except one has a negative exponent and the other has a positive exponent, like 5³ and 5⁻³, then how does the exponent change the meaning of the expression.
According to the following law of the exponent, the positive exponent can be written as,
5³ = 5 x 5 x 5 = 125
Similarly, the negative exponent is written as,
5⁻³ = 1/5³
Which can be simplified as,
1/125 = 0.008
Therefore, when we compare these values, the value of positive exponent is greater than the negative exponent.
Thus, defines that the value of positive and negative exponent are not same.
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Need help getting answers.
The solution with a pH of 3.3 has a hydronium ion of 0.000501
pH of a solutionThe pH of a solution is a measure of hydrogen ion concentration, which in turn is a measure of its acidity. Pure water dissociates slightly into equal concentrations of hydrogen and hydroxyl (OH−) ions.
Applying the formula of pH which is given as
pH = -log[H₃O⁺]
But the fruit juice has a pH = 3.3
3.3 = -log[H₃O⁺]
This becomes 10⁻³.³
[H₃O⁺] = 0.000501
The concentration of hydronium ion is 0.000501
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A firm makes circular drink mats , of radius 4.5cm , that are 3mm thick . They want to produce a rectangular box to hold a pile of 12 mats. What are the minimum dimensions of the box.
write answer and working out no spamming
Answer: The minimum is 13.5
Step-by-step explanation:
But if I got to multiply the 12 also it will be 162
The plates that Mom bought are very fragile and expensive. She paid $16.68 per plate and there are eight plates. How much money did she spend on the plates?
Line of best fit
Correlation coefficient
Intercept r
Find y when x=12
Considering the given scatter plot, it is found that:
a. The line of best fit is: y = -0.245x + 5.59.
b. The correlation coefficient is: r = -0.57.
c. The interpretation of the correlation coefficient is of: There is a negative association between y and x.
d. The estimate of y when x = 12 is of: 2.65
What is a scatter plot?A scatter plot is a set of points of a data-set, and these points are used to calculate both the correlation coefficient and the line of best fit.
For this problem, the points are given as follows:
(0,5), (1,7), (2, 6), (3, 4), (4, 3), (5,5), (6,4), (7,2), (8,5), (9,3), (10,4).
Inserting these points into a linear regression calculator, the line of best fit is given as follows:
y = -0.245x + 5.59.
Hence the estimate of y when x = 12 is obtained as follows:
y = -0.245(12) + 5.59 = 2.65.
Inserting these points into a correlation coefficient calculator, the correlation coefficient is given as follows:
r = -0.57.
The interpretation is that there is a negative association between y and x, which is neither strong neither weak, as the absolute value of the coefficient is close to the threshold of 0.6.
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A blue rope is two times as long ad a red rope. A green rope is 8 times as long as the blue rope. If the total length of three ropes is 708.75 meters, what is the length of the blue rope?
Answer: b ≈ 74.605 meters
Step-by-step explanation:
Let b be the length of the blue rope, r be the length of the red rope, and g be the length of the green rope. We will write different equations to show the situation given.
b = 2r
g = 8b
b + r + g = 708.75 meters
Next, we will substitute expressions into the last equation for the variables r and g so we can solve for b.
b + r + g = 708.75 meters
b + ([tex]\frac{b}{2}[/tex]) + (8b) = 708.75 meters
9.5b = 708.75 meters
b ≈ 74.61 meters
b ≈ 74.605 meters
WHOEVER ANSWERS AND SHOW WORK GETS BRAINLIEST AWARD. Show that f(x)= 2x-3/2 and g(x)= 2x+3/2 are inverse functions. Must show fog=x and gof=x.
The functions f(x) = (2x - 3) / 2 and g(x) = (2x + 3) / 2 are the inverse functions because the composite of the given functions equal to x that is fog = x and gof = x.
The conditions for two functions f and g to be inverses: f(g(x)) = x or g(f(x)) = x.
where, f(g(x)) and g(f(x)) is the composite function.
We are given:
f(x) = (2x - 3) / 2
g(x) = (2x + 3) / 2
Let us find the value of f(g(x)) and g(f(x)):
Substitute the value of x with g(x) in f(x), we will get:
f(g(x)) = (2 * g(x) - 3) /2 = (2 * ((2x + 3) / 2) - 3) /2 = (2x + 3 - 3) / 2 = 2x / 2 = x
So, f(g(x)) = x.
Substitute the value of x with f(x) in g(x), we will get:
g(f(x)) = (2 * f(x) + 3) /2 = (2 * ((2x - 3) / 2) + 3) /2 = (2x - 3 + 3) / 2 = 2x / 2 = x
So, g(f(x)) = x.
Thus, the functions f(x) = (2x - 3) / 2 and g(x) = (2x + 3) / 2 are the inverse functions because the composite of the given functions equal to x that is fog = x and gof = x.
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A couple plans to have children until a girl is born, but they will have no more than four children. Assume that each child is equally likely to be a boy or a girl. Let X be the number of girls they have.
The mean of the probability distribution of X is 0.9375.
Given that,
From the given
We have to find the probability distribution of X and mean of X.
Now, We know,
X -- 0 1 2 3 4
Probability of X-- 1/2 1/4 1/8 1/16 1/16
The mean is average of the distribution
Statistics refers to deviation as the difference between the observed value of a data point and the expected value. The average deviation of a data point from the mean, median, or mode of the data collection is known as the mean deviation, sometimes known as the mean absolute deviation.
μₓ=0×1/2+1×1/4+2×1/8+3×1/16+4×1/16
μₓ=1/4+1/4+3/16+1/4
μₓ=15/16
μₓ=0.9375
Therefore, The mean distribution of X is 0.9375.
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Pat and Fran are quality control specialists in a
light bulb factory. Pat inspects every 20 cartons
of bulbs for breakage. Fran inspects every 36
cartons for incorrect labeling. If they start at
the same time, which carton will be the first
that both of them inspect?
Answer:
Carton 180Step-by-step explanation:
This is about the LCM of the numbers 20 and 36.
First find the prime factors of both numbers:
20 = 2*2*536 = 2*2*3*3Now find the LCM:
LCM(20, 36) = 2*2*3*3*5 = 180They will both inspect every 180 cartons.
Answer:
They will both inspect every 180 cartons.
Step-by-step explanation:
Firstly you could find the LCM of 20 and 36.
Multiples of 20 are: 20, 40, 60, 80, 100, 120, 140, 180, etc...
Multiples of 36 are: 36, 72, 108, 144, 180, 216, 252, 288, etc...
Therefore 180 is the answer, all they were asking was what was the LCM. Hope this helped you !!