(a) The inequality follows from the binomial theorem.
[tex](a - b)^2 = a^2 - 2ab + b^2 \ge 0 \implies a^2 + 2ab + b^2 = (a+b)^2 \ge 4ab \\\\ \implies \dfrac{(a+b)^2}4 \ge ab \\\\ \implies \dfrac{a+b}2 \ge \sqrt{ab} ~~~~~~~~ (1)[/tex]
By the same reasoning,
[tex]\dfrac{c+d}2 \ge \sqrt{cd}[/tex]
Adding these results together, we have
[tex]\dfrac{a+b}2 + \dfrac{c+d}2 = \dfrac{a+b+c+d}2 \ge \sqrt{ab} + \sqrt{cd}[/tex]
and since [tex]\sqrt{ab}[/tex] and [tex]\sqrt{cd}[/tex] are both positive, they also satisfy the AM-GM inequality,
[tex]\dfrac{\sqrt{ab} + \sqrt{cd}}2 \ge \sqrt{\sqrt{ab}\times\sqrt{cd}} = \sqrt[4]{abcd}[/tex]
and the 4-variable result follows,
[tex]\dfrac{a+b+c+d}2 \ge 2\times\dfrac{\sqrt{ab}+\sqrt{cd}}2 \ge 2 \sqrt[4]{abcd} \\\\ \implies \dfrac{a+b+c+d}4 \ge \sqrt[4]{abcd} ~~~~~~~~ (2)[/tex]
(b.i) With [tex]m=\frac{a+b+c}3[/tex], we have
[tex]\dfrac{3m}4 = \dfrac{a+b+c}4 \implies m - \dfrac m4 = \dfrac{a+b+c}4 \\\\ \implies m = \dfrac{a+b+c+m}4[/tex]
(b.ii) From (2) it follows that
[tex]\dfrac{a+b+c+m}4 \ge \sqrt[4]{abcm} = \sqrt[4]{abc\times\dfrac{a+b+c+m}4}[/tex]
Using the result from (b.i), this is equivalent to
[tex]\dfrac{a+b+c}3 \ge \sqrt[4]{abc\times\dfrac{a+b+c}3}[/tex]
Take 4th powers on both sides.
[tex]\left(\dfrac{a+b+c}3\right)^4 \ge abc \times \dfrac{a+b+c}3[/tex]
Divide both sides by [tex]\frac{a+b+c}3[/tex].
[tex]\left(\dfrac{a+b+c}3\right)^3 \ge abc[/tex]
Finally, take the cube root of both sides.
[tex]\dfrac{a+b+c}3 \ge \sqrt[3]{abc}[/tex]
as required.
Of the
50
5050 U.S. states,
4
44 have names that start with the letter
W
Wstart text, W, end text.
What percentage of U.S. states have names that start with the letter
W
Wstart text, W, end text?
The 8% of us states have names that start with the letter w the answer is 8%.
What is the percentage?It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol.
The complete question is:
Of the 50 us states 4 have names that start with the letter w what percentage of us states have names that start with the letter w.
The percentage of the US states that have names that start with the letter W can be calculated:
Percentage = (No of US states starting with W/Total no of US states)×100
Percentage = (4/50)×100
Percentage = 8%
Thus, the 8% of us states have names that start with the letter w the answer is 8%.
Learn more about the percentage here:
brainly.com/question/8011401
#SPJ1
The sum of three numbers is 12. If the second number is subtracted from the sum of the first and third numbers, the result is 2. If the third number is subtracted from the sum of the first and second numbers, the result is 4. Find the three numbers.
Answer:
the numbers are 3, 5, 4
Step-by-step explanation:
The given relations can be written as a system of equations. These are conveniently solved using elimination.
SetupLet the three numbers be represented by x, y, z. The given relations are ...
x +y +z = 12
x -y +z = 2
x +y -z = 4
SolutionSubtracting the second equation from the first gives ...
(x +y +z) -(x -y +z) = (12) -(2)
2y = 10 . . . . . . . . . simplify
y = 5 . . . . . . . . divide by 2
Subtracting the third equation from the first gives ...
(x +y +z) -(x +y -z) = (12) -(4)
2z = 8 . . . . . . . . . simplify
z = 4 . . . . . . . . divide by 2
Using these values of y and z in the first equation lets us find x:
x +5 +4 = 12
x = 3 . . . . . . . . . subtract 9
The numbers, in order, are (3, 5, 4).
Write 18,25% as a common fraction in its simplest form.
Step-by-step explanation:
1825%
and percentage means 100
so 1825/100=73/4
Below are the weekly salaries, in dollars, of nine recent college graduates. Utilize the data to calculate the midrange.
833 692 1057 955 749 1113 857 946 907
The midrange of the weekly salaries is $902.50.
What is the midrange?Midrange is a measure of variation of a data set. Midrange is the mean of the lowest and highest salary.
Midrange + (lowest salary + highest salary) / 2
(692 + 1113) / 2 = $902.50
To learn more about range, please check: https://brainly.com/question/12372689
#SPJ1
simplify the expression by using the distributive property and combining like terms
6(3y + 7) + 4(9+9y)
Answer:
54y+78
Step-by-step explanation:
6*3y+ 18y
6*7=42
4*9=36
4*9y=36y
18y+36y+42+36= 54y+78
Juan got 4 questions incorrect out of the 80 questions on his exam. Express the number of questions that he got correct as a fraction of the total number of questions on the exam, and reduce the fraction to the lowest terms
Answer:
19/20
Step-by-step explanation:
He got 4 incorrect, meaning he got 80-4 correct.
80-4 = 76.
Fraction = 76/80 = 38/40 = 19/20.
Answer:
19/20
Step-by-step explanation:
[tex]\frac{80-4}{80} =\frac{76}{80} =\frac{19}{20}[/tex]
Hope this helps
La empresa ABC S.A.A tiene como saldo al 31 de marzo de 2021 en Servicios Contratados por Anticipado S/ 16 000 por cuatro meses de alquiler. A la fecha ya ha recibido el servicio de dos meses, los cuales aún no han sido registrados.
ABC S.A.A. debe registra el siguiente ajuste contable al 31 de marzo de 2021.
Answer:
its 20, because I said it
Step-by-step explanation:
so 5+5=10
30-10=20
Can anyone help me solve this question?
Answer:
corresponding angles
Step-by-step explanation:
i dont cap
Simplify: (−12az2−2z3)+(−2a3+7a2z)−(−4a2z+8az2)
Answer:
[tex] \large{ \boxed{ \tt{ - 20a {z}^{2} - 2 {z}^{3} - 2 {a}^{3} + 11 {a}^{2} z }}}[/tex]
- Please see the attached picture for full solution!:)
----------------- HappY LearninG <3 --------------
A friend asks to borrow $500 from you. She wishes to pay you back at the end of six months and agrees to pay 6% simple interest. How much interest will you earn? Round your answer to the nearest dollar. Do NOT round until you have calculated your final answer.
The interest you will earn from the loan of 500 dollars at rate of 6% and 6 months is $15.
How to find interest?He borrowed $500 and she will pay at the end of 6 months.
The interest rate is 6%.
Therefore,
I = prt / 100
where
p = principalr = ratet = time in yearsTherefore,
I = 500 × 6 × 0.5 / 100
I = 1500 / 100
I = $15
Therefore, the interest you will earn from the loan is $15.
learn more on interest here: https://brainly.com/question/25845758
#SPJ1
From the diagram below, if AD = 5, and DB = 10, what is the length of CD?
Answer:
20
Step-by-step explanation:
According to Euclidian theorem:
h^2 = AD*CD (h is the height, DB in this case)
rewrite the equation by using given values
10^2 = 5*CD find the square value of 10
100 = 5*CD divide both sides by 5
20 = CD
In the parallelogram above what is the value of x?Please explain for brainlest
E.36
F.18
G.6
H.5
I need help setting up my problem
The measure of angle A is 144.3 degrees and the angle to cut the molding is 54.3 degrees
How to solve for angle A?Start by solving the acute part of angle A using the following sine function
sin(Ax) = (30 - 4)/32
Evaluate the quotient
sin(Ax) = 0.8125
Take the arc sin of both sides
Ax = 54.3
The measure of angle A is then calculated as:
A = 90 + Ax
This gives
A = 90 + 54.3
Evaluate
A = 144.3
Hence, the measure of angle A is 144.3 degrees
The angle to cut the moldingIn (a), we have:
Ax = 54.3
This represents the angle where the molding would be cut
Hence, the angle to cut the molding is 54.3 degrees
Read more about angles at:
https://brainly.com/question/1592456
#SPJ1
In a bag of 60 candies, 36 are green and 45 have caramel in them. If the events of
picking a green candy and picking a candy with caramel are independent, what
percent of the green candies have caramel in them?
Answer:
9/20
Step-by-step explanation:
Total number of candies =60
Number of Green candies, n(G)=36
Number of candies with caramel, n(C)=45
Since the events are independent, the probability of selecting two candies where the first is green and the second has caramel is given by:
P(GC)=P(G) X P(C)
\begin{gathered}=\dfrac{n(G)}{n(S)} X \dfrac{n(C)}{n(S)} \\=\dfrac{36}{60} X \dfrac{45}{60}\\ =\dfrac{9}{20}\end{gathered}=n(S)n(G)Xn(S)n(C)=6036X6045=209
The probability in its lowest form is 9/20.
A parallelogram has sides measuring 200mm by 150mm. what would be the length of side of a square having the same perimeter?
Answer:
175mmStep-by-step explanation:
To find the total perimeter of the parallelogram we must add 200+200+150+150
This equals 700. Then, since squares have side lengths that are all equal, we have to divide by 4.
700 / 4 = 175
So, the length of a side of a square having this perimeter would be 175mm
Hope this helps!
If you know this please help
1) given
2) given
3) Definition of linear pair
4) If alternate interior angles are congruent, then the lines are parallel.
Complete the following, using ordinary interest. (Use Days in a year table.) (Do not round intermediate calculations. Round the "Interest" and "Maturity value" to the nearest cent.)
Principal: 1,000
Interest rate: 8%
Date borrowed: March 8
Date repaid: June 9
Exact time:
Interest:
Maturity value:
Answer:
75
Step-by-step explanation:
Graph F(x)=-6|x| how to graph this
See the graph in the attached image.
Which of the following is the graph of the inequality y-2>-2(x-1)?
A
B
C
(2.1)
(0, -3)
ACE
AD
(2.1)
(4,-1)
UN
(1.2)
(3,-2)
14TH
D
(1.2)
(3,-2)
The graph is shown in the attached image.
A decorator wants to line the bottom with through George with paper if the bottom each drawer measures 36“ x 20“ how many square inches of paper are needed
The amount of paper that are needed to line the bottom of 3 drawers with paper is equal to 2,160 square inches.
How to determine the amount of paper?First of all, we would calculate the area of the bottom of each drawer as follows:
Area = length × width
Area = 36 × 20
Area = 720 square inches.
Next, we would multiply this area by 3:
Total area = 720 × 3
Total area = 2,160 square inches.
Read more on area here: https://brainly.com/question/12940992
#SPJ1
Complete Question:
A decorator wants to line the bottom of 3 drawers with paper. If the bottom of each drawer measures 36 inches by 20 inches, how many square inches of paper are needed?
A. 1,040
B. 1,080
C. 2,040
D. 2,160
ASAP PLEASE Given that
Jim weighs 30 pounds less than Tom, and together they weigh 210 pounds. Let n = Tom's weight in pounds.
(n - 30) + n = 210
(n + 30) + n = 180
(n + 30) + n = 210
(n - 30) + n = 180
The equation that represents the given problem is (n-30) + n = 210. The correct option is the first option (n-30) + n = 210
Linear equationFrom the question, we are to write an equation for the given problem
From the given information,
n = Tom's weight in pounds
If Jim weighs 30 pounds less than Tom
Then,
Jim weighs (n - 30)
Thus,
The sum of their weight is
(n-30) + n
∴ (n-30) + n = 210
Hence, the equation that represents the given problem is (n-30) + n = 210. The correct option is the first option (n-30) + n = 210
Learn more on Linear equations here: https://brainly.com/question/4681624
#SPJ1
Find a real number c such that the line cx-5y+8=0 has x-intercept 4
Using the x-intercept concept, the function has x-intercept of 4 when c = -2.
What are the x-intercepts of a function?The x-intercepts of a function are the values of x when y = 0.
In this problem, the function is:
cx - 5y + 8 = 0.
When y = 0, we have that:
cx + 8 = 0
x = 4, hence:
4c = -8
c = -2.
More can be learned about x-intercepts at https://brainly.com/question/24388356
#SPJ1
if you borrow 200$ for 12 years at an annual interest rate of 2%,how much will you pay altogether!!!!!!HURRY
Answer:
248.00
Step-by-step explanation:
A = P(1 + rt)
A = the future value
P = the initial principal
r = annual interest rate (decimal)
t = the time in years
A=200(1+0.2*12)
Find the derivative of each function. Then,
determine whether the function has any local
extrema.
Using the quotient rule, the derivative of [tex]y[/tex] is
[tex]y' = \dfrac{(x^2-1) x' - x(x^2-1)'}{(x^2-1)^2} = \dfrac{(x^2-1) - x(2x)}{(x^2-1)^2} = -\dfrac{x^2+1}{(x^2-1)^2}[/tex]
as you're shown in the picture.
If [tex]y[/tex] has any local extrema, they occur at critical points where [tex]y'=0[/tex] or possibly when [tex]y'[/tex] is undefined.
In this case, [tex]y'[/tex] is undefined when [tex]x=1[/tex], but this is outside the domain of [tex]y[/tex] anyway, so we can ignore that.
That leaves us with the zero-case. For [tex]x\neq1[/tex], the denominator is always positive, so the numerator must be zero. But this never happens, since
[tex]-(x^2+1) = 0 \implies x^2 = -1[/tex]
but [tex]x^2\ge0[/tex] for all real [tex]x[/tex]. Therefore [tex]y[/tex] has no critical points and thus no extrema.
The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is twice the measure of the first angle. The third angle is fourteen more than the second. Find the measures (in degrees) of the three angles.
A line of best fit is f(x) ≈ −0.86x + 13.5 for the set of points in the table.
A 2-column table with 7 rows. The first column is labeled x with entries 2, 3, 5, 6, 7, 8, 9. The second column is labeled f(x) with entries 12, 10, 10, 8, 9, 5, 6.
Using the equation for the line of best fit, what is a good approximation for the value of the function, f(x), when x = 18?
−5
−2
3
12
The required value of function F(x), when x=18 is -2
A line of best fit is f(x) ≈ −0.86x + 13.5 for the set of points in the table
x = 2 3 5 6 7 8 9
F(x) = 21 10 10 8 9 5 6
What is line?
line, is curve showing the shortest distance between 2 points.
Now, putting x = 18 in f(x) ≈ −0.86x + 13.5
f(x) ≈ −0.86(18) + 13.5
= -1.98
≅ -2 (approximation)
Thus, the required values of x=18 in a line of best fit is -2.
learn more about lines here:
brainly.com/question/2696693
#SPJ1
The region under the curve with equation y = 1/(2x-1) is rotated through four right angles about the x-axis to form a solid. Find the volume of the solid between x=3 and x = 8.
Using the disk method, the volume of the solid is
[tex]\displaystyle \pi \int_3^8 \left(\dfrac1{2x-1}\right)^2 \, dx = \pi \int_3^8 \frac{dx}{(2x-1)^2}[/tex]
Substitute [tex]u = 2x-1[/tex] and [tex]du=2\,dx[/tex].
[tex]\displaystyle \pi \int_3^8 \frac{dx}{(2x-1)^2} = \frac\pi2 \int_5^{15} \frac{du}{u^2} \\\\ ~~~~~~~~ = -\frac\pi2 \frac1u \bigg|_5^{15} \\\\ ~~~~~~~~ = -\frac\pi2 \left(\frac1{15} - \frac15\right) = \boxed{\frac\pi{15}}[/tex]
Which expression is
equivalent to the following complex fraction?
Answer:
(y - x)/(y + x)
Step-by-step explanation:
(1/x - 1/y)/(1/x + 1/y) =
= (1/x - 1/y)/(1/x + 1/y) × (xy)/(xy)
= (y - x)/(y + x)
it is customary to give a 15% tip for a meal at a restaurant. what is 15% of $64.40
Answer:
$9.66 isb15% of $64.40
Step-by-step explanation:
64.40×15%