Answer:
See below
Step-by-step explanation:
we want to prove that A.M, G.M. and H.M between any two unequal positive numbers satisfy the following relations.
(G.M)²= (A.M)×(H.M) A.M>G.M>H.Mwell, to do so let the two unequal positive numbers be [tex]\text{$x_1$ and $x_2$}[/tex] where:
[tex] x_{1} > x_{2}[/tex]the AM,GM and HM of [tex]x_1[/tex] and[tex] x_2[/tex] is given by the following table:
[tex]\begin{array}{ |c |c|c | } \hline AM& GM& HM\\ \hline \dfrac{x_{1} + x_{2}}{2} & \sqrt{x_{1} x_{2}} & \dfrac{2}{ \frac{1}{x_{1} } + \frac{1}{x_{2}} } \\ \hline\end{array}[/tex]
Proof of I:[tex] \displaystyle \rm AM \times HM = \frac{x_{1} + x_{2}}{2} \times \frac{2}{ \frac{1}{x_{1} } + \frac{1}{x_{2}} } [/tex]
simplify addition:
[tex] \displaystyle \frac{x_{1} + x_{2}}{2} \times \frac{2}{ \dfrac{x_{1} + x_{2}}{x_{1} x_{2}} } [/tex]
reduce fraction:
[tex] \displaystyle x_{1} + x_{2} \times \frac{1}{ \dfrac{x_{1} + x_{2}}{x_{1} x_{2}} } [/tex]
simplify complex fraction:
[tex] \displaystyle x_{1} + x_{2} \times \frac{x_{1} x_{2}}{x_{1} + x_{2}} [/tex]
reduce fraction:
[tex] \displaystyle x_{1} x_{2}[/tex]
rewrite:
[tex] \displaystyle (\sqrt{x_{1} x_{2}} {)}^{2} [/tex]
[tex] \displaystyle AM \times HM = (GM{)}^{2} [/tex]
hence, PROVEN
Proof of II:[tex] \displaystyle x_{1} > x_{2}[/tex]
square root both sides:
[tex] \displaystyle \sqrt{x_{1} }> \sqrt{ x_{2}}[/tex]
isolate right hand side expression to left hand side and change its sign:
[tex]\displaystyle\sqrt{x_{1} } - \sqrt{ x_{2}} > 0[/tex]
square both sides:
[tex]\displaystyle(\sqrt{x_{1} } - \sqrt{ x_{2}} {)}^{2} > 0[/tex]
expand using (a-b)²=a²-2ab+b²:
[tex]\displaystyle x_{1} -2\sqrt{x_{1} }\sqrt{ x_{2}} + x_{2} > 0[/tex]
move -2√x_1√x_2 to right hand side and change its sign:
[tex]\displaystyle x_{1} + x_{2} > 2 \sqrt{x_{1} } \sqrt{ x_{2}}[/tex]
divide both sides by 2:
[tex]\displaystyle \frac{x_{1} + x_{2}}{2} > \sqrt{x_{1} x_{2}}[/tex]
[tex]\displaystyle \boxed{ AM>GM}[/tex]
again,
[tex]\displaystyle \bigg( \frac{1}{\sqrt{x_{1} }} - \frac{1}{\sqrt{ x_{2}}} { \bigg)}^{2} > 0[/tex]
expand:
[tex]\displaystyle \frac{1}{x_{1}} - \frac{2}{\sqrt{x_{1} x_{2}} } + \frac{1}{x_{2} }> 0[/tex]
move the middle expression to right hand side and change its sign:
[tex]\displaystyle \frac{1}{x_{1}} + \frac{1}{x_{2} }> \frac{2}{\sqrt{x_{1} x_{2}} }[/tex]
[tex]\displaystyle \frac{\frac{1}{x_{1}} + \frac{1}{x_{2} }}{2}> \frac{1}{\sqrt{x_{1} x_{2}} }[/tex]
[tex]\displaystyle \rm \frac{1}{ HM} > \frac{1}{GM} [/tex]
cross multiplication:
[tex]\displaystyle \rm \boxed{ GM >HM}[/tex]
hence,
[tex]\displaystyle \rm A.M>G.M>H.M[/tex]
PROVEN
What is the purpose for post tests?
Answer:
The real reason of post test is to measure it's result in comparison to a pre test and determine d how much student has progressed over a term of instruction.
Determine the value of x
Answer:
[tex]6\sqrt{3}[/tex]
Step-by-step explanation:
using the sine rule,
[tex]\frac{sin 60}{x} = \frac{sin 30}{6} \\x=\frac{sin60 * 6 }{sin30} \\x=6\sqrt{3}[/tex]
Find the ÷98 and place a point on the # line
Answer:
Draw a number line starting with -100, -95, -90, -85, -80, -75, -70, -65, -60, -55, -50, all the way to 0. Then put a point mark at where the -98 would be.
Help please!??!!?!?
9514 1404 393
Answer:
a) CP = SP/1.1
b) CP = $59.50
c) GST = $5.95
Step-by-step explanation:
a) Divide by the coefficient of CP.
SP = 1.1×CP
CP = SP/1.1
__
b) Use the formula with the given value.
CP = $65.45/1.1 = $59.50
__
c) You can do this two ways: subtract CP from SP, or multiply CP by 0.1.
GST = SP -CP = $65.45 -59.50 = $5.95
GST = CP×0.10 = $59.50 × 0.10 = $5.95
I need help Plz help
The intersection of plane A and plane S will be
The intersection of lines n and k is
Point X is the intersection of
Answer:
bjbjbfbdvdjbfv
Step-by-step explanation:
fdbfdvbdjv
For the given annual rate of change, find the corresponding growth or decay factor. + 300%
Answer:
50% growth would would be (1 + .5)^ n
and 100% growth would be (1+ 1)^n
I assume that the answer would be (1+3)^n
for 300% growth the factor would be 3
Step-by-step explanation:
in a classroom of 15 men and 12 women, 2 different homework papers were selected at random. find the probability that both papers belonged to women
Answer:
24 papers belongs to women
The retail of a price of an LCD TV was $7000 what was the original price before the GST of 10% was added?
Answer:
$636.36
Step-by-step explanation:
7000 = 110% of total cost
Try to get it to 100 percent
700/11 = 63.(63)
63.(63)*10= 636.36
look at the image below forr the question plz
Answer:
[tex]308 \ m^3\\[/tex]
Step-by-step explanation:
The volume of a three-dimensional shape is the amount of space the figure takes up. This can be found by multiplying all of the dimensions of a figure together. In essence, the following formula can be used to find the volume.
[tex]A=l*w*h[/tex]
Where (l) represents the figure's length; (w) is the width: and (h) the height. Substitute the given values into the formula and solve for the volume.
[tex]A=l*w*h[/tex]
[tex]A=7*4*11[/tex]
Simplify,
[tex]A=7*4*11[/tex]
[tex]A=28*11[/tex]
[tex]A=308[/tex]
Lilly makes $6.35 using only 5 cemts coins.
How many 5 cent coins does she need?
Answer:
127
Step-by-step explanation:
turn $ 6.35 into 635 and divided by 5
What is the domain of this graph?
Answer:
the domain is all real numbers
find the derivation of the given equation f (x) =7
pllz batao??
Answer:
0
Step-by-step explanation:
the derivative is zero since it has no variable
Answer:
0
Step by step explanation:
f'(x)=lim f(x+h)-f(x)/h
now since 7 is constant with no variable
f'(x)= lim(7-7)/h
f'(x)=0
1 simplify 6x64 ÷ 16 +7-21
Answer:
10
Step-by-step explanation:
Rectangle with the dimensions of 2 2/5 wide and 4 1/8 long what is the area
Answer:
10 5/16
Step-by-step explanation:
2 2/5 x 4 1/8 = 10 5/16
I need help on the decimal estimate can you tell me step by step please
Ms Perez khẳng định rằng: Có khả năng công ty của cô ấy sẽ mở chi nhánh ở Phoenix (thành phố thủ phủ của tiểu bang Arizona, Mỹ). Khi đó, cô ấy có khả năng sẽ được bổ nhiệm là giám đốc ở chi nhánh mới. Tìm khả năng để Ms Perez trở thành giám đốc ở Phoenix
Answer:
Could you ask your question in English?
Step-by-step explanation:
The bar graph shows the z-score results of four students on two different mathematics tests. The students took Test 1 and then, a month later, took Test 2. Which student had the lowest score on Test 2? Euan Felicia Dave Carla
Answer:
euan had lowest score on test 2
The student with lowest score on test 2 is Euan.
What is bar graph ?Bar graph is used for the graphical representation of data or quantities by using bars or strips.
Here,
The z-score results of four students on two different mathematics tests is represented by the given bar graph.
Calculating the scores of each students for the two tests respectively.
1) Carla
Test 1: 0.75
Test 2: -0.5
2) Dave
Test 1: -0.5
Test 2: 1
3) Euan
Test 1: 0.25
Test 2: -1
4) Felicia
Test 1: 1.25
Test 2: 1.5
Hence,
The student with lowest score on test 2 is Euan.
To learn more about bar graph, click:
https://brainly.com/question/17343218
#SPJ3
I need to know the answer and explaining how to do it please
Answer:
Its 0.11
Step-by-step explanation:
When you divide any number by 10, you just move the decimal place one to the left, as for this number 1.1, you move it one place to the left which makes it 0.11. Mathmatically, to check your work you can do 0.11 x 10 to get 1.1
Mark brainliest and helpful
What is the length of the diagonal of a rectangle with the base of 4 and an altitude of 3
Answer:
5
Step-by-step explanation:
Use the Pythagorean Theorem
C^2=a^2 + b^2
c^2= 3^2 + 4^2
c^2= 9+16
c^2= 25
The take the square root
c= √25
c = 5
On the first day of a two-day meeting, 10 coffees and 10 doughnuts were purchased for a total of $20.00. Since nobody drank the coffee and all the doughnuts were eaten, the next day only 2 coffees and 14 doughnuts were purchased for a total of $13.00. How much did each coffee and each doughnut cost?
Answer:
1.25 dollars- the value of each coffee
0.75 dollars- the value of each doughnut
Step-by-step explanation:
Suppose that value of one coffee is x dollars, when one doughnut costs y dollars. The value o 10 coffee is 10x, when 10 doughnuts cost 10y. The sum is 10x+10y and it is 20.
10x+10y= 20 (we can divide each part by 10)
x+y=2
2coffee cost 2x, 14 doughnuts cost 14y
2x+14y=13 (it can be divided by two)
x+7y=6.5
We have the system of equations x+y=2, x+7y=6.5
Subtract the first equation from the second one (the left side of the first equation from the left side
x+7y - (x+y)= 6.5-2
6y=4.5
y=4.5/6= 3/4 = 0.75 dollars- the value of each doughnut
x=2-0.75=1.25 dollars- the value of each coffee
if 25 liters of milk can make 8 tin of cheese how many liters of milk would b needed to make 11 tin of cheese?
Answer:
Step-by-step explanation:
34.375 Liters of milk make 11 tins of cheese. Assuming that milk is available in whole liters only, 35 Liters of milk are needed.
Please help me out with this geometry
Answer:
[tex](x-[-7])^{2} +(y-[9])=[25][/tex]
Step-by-step explanation:
Center (h,k) = (-7, 9)
Radius, r = 5
Equation: [tex](x-h)^{2} +(y-k)^{2} =r^{2}[/tex]
[tex](x-(-7))^{2} +(y-9)^{2} =(5)^{2}[/tex]
[tex]5^{2} =25[/tex]
So, [tex](x-(-7))^{2} +(y-9)=25[/tex]
OAmalOHopeO
A team wishes to purchase 10 shirts of the same color. A store sells shirts in 3 different colors. What must the inventory of the store be in order to conclude that there are at least 10 shirts in one of the three colors?
Answer:
30
Step-by-step explanation:
fill in the missing. 1:1000=1cm:_?
Answer:
meter
10 meters
Or
a dekameter
A pollster wants to minimize the effect the order of the questions has on a person's response to a survey. How many different surveys are required to cover all possible arrangements if there are 8 questions on the survey
Answer:
40320 different surveys would be necessary ...
Step-by-step explanation:
let's start small :
how many are required for 2 questions ?
well, 2 (we can either ask question 1 first and then question 2, or question 2 first and then question 1).
how many for 3 questions ?
well, we have a choice of 3 questions to be asked first. but then for the second and third question we are back in the previous problem of 2 (remaining) questions.
so, we multiply the 3 initial options with the 2 following options. 3×2 = 6
how many for 4 questions ?
well, we have a choice of 4 questions as first question, and then fit the rest we are back in the previous problem of 3 questions. 4×3×2 = 24
you see the pattern ?
yes, that continues as far as we want to go.
for our example the problem was about 8 questions.
so,
8×7×6×5×4×3×2 = 8! = 40320
Can anyone help? I am terrible at math!!
The number line shows the record low temperatures for four states.. what is the difference, between the low temperature in Hawaii and the low temperatures In south Dakota
A -70°F
B -46°F
C 46°F
D 70°F
Answer:
D 70°F
Step-by-step explanation:
Difference between temperature in Hawaii and in South Dakota is 12°F - (-58°F) = 12°F + 58°F = 70 °F
1) -9 + X = -4
Helpppppppp
Answer:
5
Step-by-step explanation:
-9 + x = -4
( move +x to right and -4 to left)
-9 +4= -x
[sign changes when number is moved]
-5= -x
[negative signs cancel out each other/ divide both sides by negative sign]
x = 5
How many skirt-blouse outfits can a woman choose from if she has 5 skirts and 6 blouses?
