Expanding the cube, we have
[tex]\dfrac1{(a+b+c)^3} = \left(\dfrac1{a+b+c}\right)^3 \\\\ = \dfrac1{a^3} + \dfrac1{b^3} + \dfrac1{c^3} + 3 \left(\dfrac1{a^2b} + \dfrac1{a^2c} + \dfrac1{ab^2} + \dfrac1{b^2c} + \dfrac1{ac^2} + \dfrac1{bc^2}\right) + \dfrac6{abc}[/tex]
so it remains to be shown that
[tex]3 \left(\dfrac1{a^2b} + \dfrac1{a^2c} + \dfrac1{ab^2} + \dfrac1{b^2c} + \dfrac1{ac^2} + \dfrac1{bc^2}\right) + \dfrac6{abc} = 0[/tex]
Factorize the grouped sum on the left as
[tex]\dfrac1{a^2b} + \dfrac1{a^2c} + \dfrac1{ab^2} + \dfrac1{b^2c} + \dfrac1{ac^2} + \dfrac1{bc^2} = \dfrac1{abc} \left(\dfrac ca + \dfrac ba + \dfrac cb + \dfrac ab + \dfrac bc + \dfrac ac\right)[/tex]
so that with simplification, it remains to be shown that
[tex]\dfrac ca + \dfrac ba + \dfrac cb + \dfrac ab + \dfrac bc + \dfrac ac + 2 = 0[/tex]
With a little more manipulation, we have
[tex]\dfrac ba + \dfrac ca = \dfrac{a+b+c}a - 1[/tex]
[tex]\dfrac cb + \dfrac ab = \dfrac{a+b+c}b - 1[/tex]
[tex]\dfrac bc + \dfrac ac = \dfrac{a+b+c}c - 1[/tex]
so that our equation simplifies to
[tex]\dfrac{a+b+c}a + \dfrac{a+b+c}b + \dfrac{a+b+c}c - 1 = 0[/tex]
which we can factorize as
[tex](a+b+c)\left(\dfrac1a+\dfrac1b+\dfrac1c\right) - 1 = 0[/tex]
Finish up by using the hypothesis:
[tex]\left(\dfrac1{\frac1{a+b+c}}\right)\left(\dfrac1a+\dfrac1b+\dfrac1c\right) - 1 = 0[/tex]
[tex]\underbrace{\left(\dfrac1{\frac1a+\frac1b+\frac1c}\right)\left(\dfrac1a+\dfrac1b+\dfrac1c\right)}_{=1} - 1 = 0[/tex]
and the conclusion follows.
Find the area of the triangle.
Answer:
20 m²
Step-by-step explanation:
Formula (Area of triangle):
[tex]\text{A (Triangle)} = \dfrac{1}{2} \times \text{Base} \times \text{Height}[/tex]
From the triangle, we can see that the base of the triangle is "7 + 3" m and the height of the triangle is "4" m.
[tex]\implies \text{A (Triangle)} = \dfrac{1}{2} \times (7 + 3) \times (4)[/tex]
Simplify the right-hand-side as needed to evaluate the area.
[tex]\implies \text{A (Triangle)} = \dfrac{1}{2} \times (10) \times (4)[/tex]
[tex]\implies \text{A (Triangle)} = \dfrac{40}{2} = 20 \ \text{m}^{2}[/tex]
Therefore, the area of the provided triangle is 20 m².
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22. A small-appliance manufacturer finds that it costs $9000 to produce 1000 toaster ovens a week and $12,000 to produce 1500 toaster ovens a week.
(b) What is the slope of the graph and what does it represent?
A small-appliance manufacturer finds that it costs $9000 to produce 1000 toaster ovens a week and $12,000 to produce 1500 toaster ovens a week.
station at any time?
OA. 100
OB. 80
OC. 40
OD. 20
Step-by-step explanation:
Station at any time A 100
Performance Task A. Directions: Solve the problem and answer the following questions. Renz' family attended a pool party. The pool is circular in shape. He wants to find out the area of the pool. If the radius is six meter, what could be the area? Fill in the blanks. 1. The radius of the pool is m 2. The diameter of the pool is __m 3. The value of pi is 4. Formula in finding the area of a circle 5. Area of the swimming pool X m²
Answer: 144 meters
Step-by-step explanation: area of circle = r^2
6^2 = 36 = 144 Meters given = 4
radius of pool is 6 meters
diameter of pool is 12 meters
= 4
area of circle = r^2
area of swimming pool is 144 meters
if your expenses total $5,600 per month. you charge$25 per hour and have contracts for study work that total 45 hours per week for 52 weeks of the year. How does this rate cover your expenses?
Answer:
$8700
Step-by-step explanation:
since there are 12 months in a year:
5600 x 12 = $67200
pay per week: 25 x 45 = $1125
pay per year: 1125 x 52 = $58500
67200 - 58500 = $8700 left over
not too sure by what you meant with the question (How does this rate cover your expenses?) but i hoped this helped in some way :)
Bob has $600 in a savings account that earns' 5% annually. The interest is not compounded
How much interest will he earn in 1 year?
Use the formula /= prt, where I is the interest earned, p is the principal (starting amount),
is the interest rate expressed as a decimal, and t is the time in years.
[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$600\\ r=rate\to 5\%\to \frac{5}{100}\dotfill &0.05\\ t=years\dotfill &1 \end{cases} \\\\\\ I = (600)(0.05)(1)\implies I=30[/tex]
Answer:
30
Step-by-step explanation:
You would frist start by mutiplying 600 you principal by .05 your rate which equals 30. The you would multiply that by you time which is 1.
If it take 8 cups of flour to make 4 Loaves of bread, how many loaves of bread can you make from 40 cups of flour
Answer:
20
Step-by-step explanation:
Since 8 x 5 is 40, we can multiply 4 x 5 to get 20.
How many solutions exist for the given equation?
3x + 13 = 3(x + 6) + 1
Answer:
0
Step-by-step explanation:
3x+13 = 3(x+6)+1
3x+13=3x+19
13=19
This is never true so there are 0 solutions
If the x's didnt cancel out, you could use the fundemental theorom of algebra, which states that in the simplest form(nothing will cancel out anymore), the highest degree(highest power) on a variable is the number of solutions.
What is the kinetic energy of a 50-kg child running to catch the school bus at 2 m/s
Answer:
100 J
Step-by-step explanation:
The formula for kinetic energy is as follows:
[tex]\boxed{KE= \frac{1}{2} mv²}[/tex]
where
KE= kinetic energy (J)
m= mass (kg)
v= velocity (m/s)
Given: m= 50 kg, v= 2 m/s
Substituting the given information into the formula:
KE
= ½(50)(2²)
= ½(50)(4)
= 100 J
Consider the following graph, which details one bookstore’s sales over the course of one day, divided by genre.
A graph titled Book Sales by Genre has genre on the x-axis and Number sold on the y-axis. Romance, 52; horror, 55; Fantasy, 60; Biograph, 57; Humor, 58; Cookbook, 54.
By percent, how much greater do sales of horror books appear to be than sales of cookbooks? How much greater are they in reality?
The percent by which the sales of horror books is greater than sales of cookbook is 1.85% approx.
How to find the percentage from the total value?Suppose the value of which a thing is expressed in percentage is "a'
Suppose the percent that considered thing is of "a" is b%
Then since percent shows per 100 (since cent means 100), thus we will first divide the whole part in 100 parts and then we multiply it with b so that we collect b items per 100 items(that is exactly what b per cent means).
Thus, that thing in number is
[tex]\dfrac{a}{100} \times b[/tex]
Given that:
Genre number sold
Romance 52Horror 55Fantasy 60Biography 57Humor 58Cookbook 54Let we have:
Number of horror books sold = P% greater than number of cookbooks sold
55 = P% greater than 54
55 = 54 + P% of 54
1 = P% of 54
[tex]1 = \dfrac{P}{100} \times 54\\\\P = \dfrac{100}{54} \approx 1.85[/tex]
Thus, the percent by which the sales of horror books is greater than sales of cookbook is 1.85% approx.
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Answer:
Say that you put the set of tiles shown below into a bag and conduct a series of trials in which you draw one tile randomly from the bag, record the color, and replace it. The table below shows the number of times in each set of trials that you drew a blue tile.
Round tiles: 5 purple, 6 orange, 2 yellow, 4 red, 1 white, 10 blue, 5 black, 3 green.
Trial
1
2
3
4
5
# of Draws
54
36
80
22
75
# of Blues
18
11
20
4
32
Between the theoretical probability that you will draw a blue tile and the experimental probability that you will draw a blue tile, which is greater, and how much greater is it? Express all probabilities as percentages to two decimal places, and express differences by number of percentage points (for example, 12% is 2 percentage points greater than 10%).
a.
The theoretical probability is 2.78 percentage points greater than the experimental probability.
b.
The theoretical probability is 1.49 percentage points greater than the experimental probability.
c.
The experimental probability is 4.06 percentage points greater than the theoretical probability.
d.The experimental probability is 2.17 percentage points greater than the theoretical probability. <<<CORRECTStep-by-step explanation:
Edge 2022
How many four- digit numbers can be formed from the digit 1,3,4,5,6&7 where the digit is used at most once?( repetition are not allowed)
a)if the number must be odd
b)if the number must be even
Answer:
A. 24
B. I don't think you can make any since you can't repeat numbers and there is only two even numbers
Also, if you use all six numbers, it's 120 combos
If the vertex angle of an isosceles triangle is 39°, what is the measure of each base angles?
A. 71°
B. 70.5°
C. 81°
D. 80.5°
Other:----------------------------------
Step-by-step explanation:
Total =180°
Top vertex=39°
2base angles=180-39
2base angles=141
base anglez=141/2
base angle=70.5°MARK ME AS BRAINLISTWhat are the possible transformations that could have been used to create the red figure A'B'C'D' from the blue
figure ABCD?
A
D'
-5-4-3-2-1. B
3 A 5 x
-2
a reflection across
the y axis followed by a translation of 5 units up and 5 units left
O a reflection across the y axis followed by a 90-degree rotation counterclockwise about the origin
a rotation of 90 degrees counterclockwise about the origin followed by a reflection across the y axis
A
5
4
3
2
B
BB
Ni
& &
2
Answer:
Second option.
Step-by-step explanation:
A reflection across the y axis followed by a counterclockwise rotation around the origin.
How do you convert joules to kJ
Answer:
Divide by 1000.
Step-by-step explanation:
For example, 1 joule is 0.001 kilo joule.
1 divided by 1000 is 0.001.
If there were 30 joules, there would be 0.032 kilo joules.
A group of 200 people who exercise regularly reported the following exercise habits :
5 people run
65 people swim or run
30 people swim and run
If a person from this group is selected at random , what is the probability that the person swims ?
Answer:
9/20
Step-by-step explanation:
65 people swim or run
BUT
5 people just run
so
60 people swim or run
+
30 people swim and run
90/200 =
9/20
Evaluate 3x+ y when x equals 1/3 and y equals -7/4. Write your answer as a fraction in simplest form.
Please help!! Thanks!!
Answer:
-3/4
Step-by-step explanation:
Put in the values for 'x' and 'y'
3 * (1/3) + (-7/4)
1 - 7/4
4/4 - 7/4
(4-7)/4 = -3/4
What are the coordinates of point J’ after the rotation?
Step-by-step explanation:
(5,4) is the coordinate of the point j after rotation
Marian is 29 and her base premium for auto insurance is $1239.
What does she pay in total premiums?
O$1239.00
O$1379.80
O $1551.60
O $1610.70
Answer: $1239.00
Step-by-step explanation:
I had to recheck on this.
For females that are ages 25-29, they have base auto insurance rate, meaning that the rate is set at 1. So multiply the base premium and base rate, 1239x1, and obviously it's the same 1239.
Hope this helped.
PLEASE HELP ASAP WILL MARK BRAINLIEST!
Solve and show your work for each question.
Note: Remember per the rule of 9s if you have a repeating and non-repeating component you need
to handle them separately.
(a) What is 0.47 expressed as a fraction in simplest form?
(b) What is 0.4-7 expressed as a fraction in simplest form?
(c) What is 0. -47 expressed as a fraction in simplest form?
There are lines above the seven in decimal B and the line in decimal C is above both numbers. pleaseeeee help me
Answer:
See below
Step-by-step explanation:
a) 47/100
b) .4777777..... =x
then 10x = 4.77777....
100x = 47.777777 .... now subtract 10x to get
90x = 43
x = 43/90
c) .4747474747...... = x
100x = 47.474747 now subtract x to get
99x = 47
x = 47/99
A recipe uses 3 cups of sugar to make 48 pieces of taffy. How much sugar would it take to make 72 pieces of taffy
Answer:
4.5 cups of sugar
Step-by-step explanation:
This is how i solved it...
3 cups sugar = 48 pc. taffy
??? cups sugar = 72 pc. taffy
48 x 1.5 = 72
that means 3 x 1.5 should get you how many cups!
3 x 1.5 = 4.5
4.5 cups of sugar! hope this helps!!!
Answer:
3 1/2 cups of sugar
Step-by-step explanation:
3=48
3 1/2= 72
Easy math what one a b c d e
Answer:
problem 14 = E
Step-by-step explanation:
well that's all ther is to it, it's E because y or x being 0 would make the equation undefined
2) What is the area of the parallelogram?
A. 16 square centimeters
B. 30 square centimeters
C. 60 square centimeters
D. 120 square centimeters
(20 points)
height= 6 cm
base= 10 cm
to find:the area of the parallelogram.
solution:[tex]formula = bh[/tex]
[tex]a = 6 \times 10[/tex]
[tex]a = 60 {cm}^{2} [/tex]
therefore, the area of the parallelogram is 60 square centimeters.
answer= option c.
Ollie drank 1/2 of a 16 fluid ounce bottle of juice how many cups of juice are left in the bottle explain
Answer:
Step-by-step explanation:
1 cup is left because 16 ounces is 2 cups so half would be one
When designing a training plan, what do you need to think about?
A. Frequency, rest, progression
B. Moderation, progression and overtraining
C. Rest, goals, overload
D. Frequency, overtraining, time
The product of a number and 7 less than the number is 30. Find the number.
Answer:
-3 or 10
Step-by-step explanation:
[tex]x(x - 7) = 30[/tex]
[tex] {x}^{2} - 7x = 30[/tex]
[tex] {x}^{2} - 7x - 30 = 0[/tex]
[tex](x - 10)(x + 3) = 0[/tex]
x = -3 or 10
(18) (3, -2) lies on terminal side of angle (θ).
Find the EXACT value of sec (θ)
Show steps
By applying fundamental and derivate trigonometric expressions it is to be concluded that the secant of the line segment whose terminal side is (3, - 2) is approximately 1.202.
How to apply trigonometric expressions
Trigonometric expressions are trascendent expressions derived from relationships between sides of the triangle contained in a goniometrical diagram.
There are at least six trigonometric expressions: sine, cosine, tangent, cotangent, secant, cosecant. The first two expressions are fundamental, whereas the rest are derived from them.
We can derive an expression for the secant of the angle by applying trigonometric expressions:
[tex]\sec \theta = \frac{1}{\cos \theta} = \frac{\sqrt{x^{2}+y^{2}}}{x}[/tex] (1)
If we know that x = 3 and y = - 2, then the secant of the angle is:
[tex]\sec \theta = \frac{\sqrt{3^{2}+(-2)^{2}}}{3}[/tex]
sec θ ≈ 1.202
By applying fundamental and derivate trigonometric expressions it is to be concluded that the secant of the line segment whose terminal side is (3, - 2) is approximately 1.202.
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Manufacturers of tires report that tires should be able to last an average of 50,000 miles. A new tire company produces a different type of tread and tests 100 randomly selected tires. This sample of 100 tires lasted an average of 52,000 miles. Assuming the new type of tread does not improve the mileage of the tire, 200 sample means were simulated and displayed on the dotplot.
Using the dotplot and the sample mean mileage, is there convincing evidence that the new type of tread improves the mileage?
Yes, because a mean mileage of 52,000 or more occurred only 7 out of 200 times, the mean mileage is statistically significant. There is convincing evidence the new type of tire tread improves mileage of the tire.
Yes, because a mean mileage of 52,000 or less occurred 193 out of 200 times, the mean mileage is statistically significant. There is convincing evidence the new type of tire tread improves mileage of the tire.
No, because a mean mileage of 52,000 or less occurred 193 out of 200 times, the mean mileage is not statistically significant. There is not convincing evidence the new type of tire tread improves mileage of the tire.
No, because a mean mileage of 52,000 or more occurred 7 out of 200 times, the mean mileage is not statistically significant. There is not convincing evidence the new type of tire tread improves mileage of the tire.
The convincing evidence is Yes, because a mean mileage of 52,000 or more occurred only 7 out of 200 times, the mean mileage is statistically significant.
What is a dot plot?A dot plot is known to have the same characteristics as a bar graph as both are made up of different height of “bar” that consist of dots which is said to be equal to the number of items in any of the stages or categories.
Conclusively, when a person uses a dot plot and the sample mean mileage, one will be able to find a convincing evidence that the new type of tread improves the mileage because out of mean mileage of 52,000 or more, only 7 out of 200 times, took place and this shows that the mean mileage is statistically significant and that the new type of tire tread boast the mileage of the tire.
Therefore, option A is correct.
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Answer:
Theres another question that looks like this one, the awnser to that is
^ ^ ^ ^ ^
No, because a mean mileage of 51,500 or more occurred 14 out of 200 times, the mean mileage is not statistically significant. There is not convincing evidence the new type of tire tread improves mileage of the tire.
Step-by-step explanation:
but the guy above me is correct
Order the numbers from least to greatest -1 -1/4 -1 1/2 3/4 and 1/8
Answer:
-11/2
-1
-1/4
1/8
3/4
Step-by-step explanation:
-11/2 = -5 1/2
-1 = -1
-1/4 = -1/4
1/8 = 1/8
3/4 = 6/8
Look at this function:
x y
−3 −8
−2 −6
−1 −4
0 0
Matt said that the function is linear.
Justine said that the function is nonlinear.
Which of the following explains who is correct?
Matt, because for every 1-unit increase in x, there is an increase in y by 2
Matt, because for every 2-unit increase in x, there is an increase in y by 1
Justine, because the point (−3, −8) does not lie on the straight line that contains the other points
Justine, because the point (0, 0) does not lie on the straight line that contains the other points
Justine, because the point (0, 0) does not lie on the straight line that contains the other points.
Explanation:
See the graph below. The following coordinates are not in a stright line.
Hence, it is a non linear function. Justine is right.
When drawn a straight line. the coordinates: (0,0) does not lie on the line.
Subtract 89.768 from 201.316
show your work
Answer:
111.548
-------------------------------
Subtract 201 from 89.
[tex]201 - 89 = 112[/tex]
we can estimate the answer will be close to 112
Next, work with the decimals. Note that the more you subtract, the bigger the decimal places get.
[tex]0.768 - 0.316 = 0.452[/tex]
Round 0.452 up from the hundredths place to nearest tenths place.
[tex]0.452 = 0.5[/tex]
Since the decimal place is bigger, estimate 0.55 instead.
Subtract 0.22
[tex]0.55 - 0.2 = 0.548[/tex]
Add this number to 112
[tex]112 + 0.548 = 112.548[/tex]
Subtract 1 because of the estimate.
[tex]112 - 1 = 111[/tex]
Therefore, we have 111.548 based off of subtracting and estimating in a specific method.