a² + b² + ab = 7
b² + c² + bc = 21
a² + c² + ac = 28 ab + bc + ac = ?

Answers

Answer 1

Answer:

ab+bc+ac = 7+21 + 28 =56

so the answer is 56

Answer 2

Observe that

[tex]a^2 + b^2 + ab = 7 \iff a^2 + b^2 - 2\cos(120^\circ) = \left(\sqrt 7\right)^2[/tex]

[tex]b^2 + c^2 + bc = 21 \iff b^2 + c^2 - 2\cos(120^\circ) = \left(\sqrt{21}\right)^2[/tex]

[tex]a^2 + c^2 + ac = 28 \iff a^2 + c^2 - 2\cos(120^\circ) = \left(\sqrt{28}\right)^2[/tex]

Drawing a comparison to the law of cosines, we see the first equation defines a triangle with sides [tex]a[/tex], [tex]b[/tex], and [tex]\sqrt 7[/tex]; the second equation defines one with sides [tex]b[/tex], [tex]c[/tex], and [tex]\sqrt{21}[/tex]; and the third equation defines another one with sides [tex]a[/tex], [tex]c[/tex], and [tex]\sqrt{28}[/tex]. We can then join these triangles at a point D to form a new triangle ABC with sides [tex]\sqrt7,\sqrt{21},\sqrt{28}[/tex]. (See attached sketch)

Now, the area of ABC is the sum of the areas of ABD, BCD, and ACD. The area of any of these triangles is equal to half the area of a parallelogram spanned by the sides involving the vertex D. The area of a parallelogram with adjacent sides [tex]x[/tex] and [tex]y[/tex] and with angle [tex]\theta[/tex] between them is [tex]ab\sin(\theta)[/tex].

For example, triangle ABD has area

[tex]\mathrm{area}_{\Delta ABD} = \dfrac12 \times ab \sin(120^\circ)[/tex]

Then the total area of triangle ABC is

[tex]\mathrm{area}_{\Delta ABC} = \dfrac12 \sin(120^\circ) (ab + ac + bc)[/tex]

We use Heron's formula to find the area of ABC. If [tex]s[/tex] is its semiperimeter, i.e.

[tex]s = \dfrac{\sqrt7 +\sqrt{21} + \sqrt{28}}2[/tex]

then, with a lot of algebraic simplification,

[tex]\mathrm{area}_{\Delta ABC} = \sqrt{s (s-\sqrt7) (s - \sqrt{21}) (s - \sqrt{28})} = \dfrac{7\sqrt3}2[/tex]

Finally, we find

[tex]\dfrac{7\sqrt3}2 = \dfrac12 \sin(120^\circ) (ab + ac + bc) \implies ab + ac + bc = \dfrac{\frac{7\sqrt3}2}{\frac12\times\frac{\sqrt3}2} = \boxed{14}[/tex]

A + B + Ab = 7b + C + Bc = 21a + C + Ac = 28 Ab + Bc + Ac = ?

Related Questions

Find f ° g and g ° f, if they exist. f={(-4,-5),(0,3),(1,6)} and g={(-5,0),(3,-4),(6,1)}.

Answers

The composite function of f o g are (-4,-5),(0,3),(1,6). The composite function of g o f are (-5,0),(3,-4),(6,1).

What is a function?

A function is defined as a relation between the set of inputs having exactly one output each.

Given;

f = {(-4,-5),(0,3),(1,6)}

g = {(-5,0),(3,-4),(6,1)}.

In order to find f o g and g o f,

f o g  = f(g)

The composite function of f o g

f(-5,0) = (-4,-5)

f(3,-4) = (0,3)

f(6,1) = (1,6)

The composite function of g o f

g(-4,-5) = (-5,0)

g(0,3) = (3,-4)

g (1,6) = (6,1)

Learn more about function sum;

https://brainly.com/question/14438192

#SPJ1

Help I have no idea to do this!! I’m stuck!

Answers

Answer: 7

Step-by-step explanation:

[tex]A \cap C[/tex] denotes the intersection of A and C, or in other words, how many elements are shared between them.

The number of elements is 2 + 5 = 7

the answer is 7
hope that helps !

How much would it cost to buy a piece of candy for 50 cents plus tax

Answers

Answer:

it depends on which state your in or in your even in the us usually it is either 6% or 4% some times more or less % so im going to do both

50*0.06=53

50*0.04=52

Hope This Helps!!!

solve for x: 5x-4y=24

Answers

Answer:

x=4y+24/5

Step-by-step explanation:

add 4y

Divide by 5

Consider figures 1 and 2 shown in the coordinate plane. Figure 1 has been transformed to produce figure 2.

Answers

Answer:

0 and -6

(0,-6)

Step-by-step explanation:

Can't really explain, but this is a guess on the actual question.

Which expression is equivalent to...

Answers

Answer:

A is correct yes

Step-by-step explanation:

ndnanndnqnjed

determined about the triangle formed by the ground,
wall, and ladder? Check all that apply.
The triangle is isosceles.
The leg-to-hypotenuse
ratio is 1:√√2.
The leg-to-hypotenuse ratio is 1:√2.
The nonright angles are congruent.
The ladder represents the longest length in the
triangle.
(A,B,D,E)

Answers

Answer:

a² b² c² equals mc² and that is the answer

a plane intersects a sphere 5 cm away from the center of the sphere. the cross section is a circle whose diameter is 12cm. determine the radius of the sphere

Answers

Answer:

7.81 cm

Step-by-step explanation:

diameter = 12, 12/2 = 6

(r+5)(r-5) = 6 * 6

r^2 - 25 = 36

r^2 = 61

r = 7.81

radius = 7.81 cm

Answer:

26

Step-by-step explanation:

By selling a table for sh.56, gain is as much as percent as its cost in dollars. What is the cost price? By selling a table for sh.56 , gain is as much as percent as its cost in dollars . What is the cost price ?​

Answers

The cost price of the table is $40.

What is Gain ?

Gain is the amount gain by selling the product at a higher price than its cost.

Let the cost of the table is $ x

The percentage gain is x%  (as given in the question)

Cost price = ?

It is known that

Step 1 : Gain = ( selling Price - Cost Price) * 100 / Cost Price

Selling price = 56

Cost Price = $ x

Therefore substituting the value

x = (56 - x) * 100 / x

x² = 5600 - 100x

x² +100x -5600 = 0

Step 2 : Factorizing

x² + 140x - 40 x -5600 = 0

x( x+14 ) -40( x +14) = 0

( x - 40)(x +14) = 0

x = $40

Therefore the cost price of the table is $40.

To know more about Gain

https://brainly.com/question/23385214

#SPJ1

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Answers

[tex]\quad \huge \quad \quad \boxed{ \tt \:Answer }[/tex]

[tex]\qquad \tt \rightarrow \: a = -1[/tex]

____________________________________

[tex] \large \tt Solution \: : [/tex]

For a function f(x) = y,

x represents the value ( x - coordinate ) at which the function has a value y.

[tex]\qquad \tt \rightarrow \: f(x) = a {x}^{2} + 2[/tex]

for point (-1 , 1) to satisfy the equation, we have to calculate the value of a

[tex]\qquad \tt \rightarrow \: f( - 1) = a( - 1) {}^{2} + 2[/tex]

[tex]\qquad \tt \rightarrow \: 1 = a + 2[/tex]

[tex]\qquad \tt \rightarrow \: a = 1 - 2[/tex]

[tex]\qquad \tt \rightarrow \: a = - 1[/tex]

Answered by : ❝ AǫᴜᴀWɪᴢ ❞

Substitute -1 into x

Substitute 1 into y

f(x)= a (-1) ^2+2=1

       a=-1


ING DROPS
1. The total volume to be given is 30 ml.
The time over which this is to be given is 30 minutes.
The drop factor is 15.
How many drops per minute will be delivered ?

Answers

Answer:

15 drops per minute

Step-by-step explanation:

(drops per minute) flow/drip rate: volume ÷ time × drop factor

note:

volume (mL)

time (minutes)\

30 / 30 = 1

1 * 15 = 15

so, 15 drops per minute will be delivered

hope this helps!! have a lovely day :)

Answer:

15 drops/minute

Step-by-step explanation:

30 ÷ 30 = 1

1 · 15 = 15

hope this is helpful :)

What negative Z-score (standard score) would the area under the curve be 0.8621

z* = _______​

Answers

0.8721 = z because its have a asymptote there

paul asked to order snacks for an office meeting of 180 people. she decides to order desert squares which come in boxes of 24. she estimates she will need 3.5 snacks per person. how many boxes should she buy?

Answers

Answer:

27 boxses

Step-by-step explanation:

calculations is shown in the picture

QUESTION 8 - 1 POINT
The median monthly rent in 1994 was $499 and $633 in 2001. What was the percent increase in the
median monthly rent prices from 1994 to 2001? Round to the nearest tenth of a percent.

Answers

Answer:

26.85%

Step-by-step explanation:

first you find the difference in price increase

633-499=134

*percentage increase=increase value/original value x100

so;=134/499x100=28.85%

hope it helps you.please don't forget to give me a thanks

Find the domain and range

Answers

Answer:

Domain: (-∞, ∞)
Range: [-2, ∞)

Step-by-step explanation:

The domain of any parabola is (-∞, ∞)
The range of this parabola is [-2, ∞) because the vertex is at (-2, -2).


A line / passes through point (-2, 5) and has the same y-intercept as
the line y-3x-4. What is the equation for the line /?

Answers

I'm assuming you meant to write y = 3x-4

We know that the y-intercept of the given line is (0,-4), so we need to find the equation of the line passing through (-2,5) and (0,-4).

The slope is

[tex]\frac{-4-5}{0-(-2)}=-\frac{9}{2}[/tex]

So, this means the equation is [tex]\boxed{y=-\frac{9}{2}x-4}[/tex]

Find the numeral preceding and succeeding each of the following numerals. Use T as the digit for 10.
a. TTO eleven
b. 100000 nine
c. 222 three

Answers

The numeral preceding and succeeding each of the following numerals.

Now, what happens when you add 1 or subtract 1 from 100 in the familiar base 10.

Adding adds 1 to the unit digit.

Subtracting 1 gives 99 because 9 is the largest digit in base 10.

So A. 100 base 8

The largest digit in base 8 is 7, so

..., 77, 100, 101,...

B. 10000 base 7

What is the largest digit in base 7?

The largest digit in base 7 is 6, so

..., 6666, 10000, 10001, ...

Next what happens when you add 1 to 509 in base 10. 9 is the largest digit in base 9, so adding 1 makes the units digit 0, and you carry a 1 to the next digit to the left:

509 + 1 = 510.

C. 305 base 6

5 is the largest digit in base 6, so

..., 304, 305, 310, ...

To learn more about the numeral preceding visit:

https://brainly.com/question/251701

#SPJ1

Translate the​ phrase; then simplify. Find the sum of −45​, −13​, and 37.

Answers

Answer:

-21

Step-by-step explanation:

37 - 45 - 13 is translated

-8 - 13

-21

The solution set to 6 + 2n > 12 is n > 3. Which are correct representations of this solution? Select two options.

{n | n < 3}
{n | n ≥ 3}
A number line going from negative 5 to positive 5. An open circle appears at positive 3. The number line is shaded from positive 3 to positive 5.
A number line going from negative 5 to positive 5. An open circle appears at positive 3. The number line is shaded from positive 3 to negative 5.
(3, ∞)

Answers

The correct representations of this solution are

(d) A number line going from negative 5 to positive 5. An open circle appears at positive 3. The number line is shaded from positive 3 to negative 5.(e) (3, ∞)

How to determine the solution set?

The inequality is given as:

6 + 2n > 12

The solution is given as:

n > 3

When the above inequality is represented as an interval, we have:

(3, ∞)

Also, the inequality symbol > is represented using an open circle.

This means that the number line would use an open circle and the arrow would point right

Hence, the correct representations of this solution are (d) and (e)

Read more about inequalities at:

https://brainly.com/question/25275758

#SPJ1

What is the value of the expression 24 + 3²? What is the value of the expression 24 + 3² ? ​

Answers

Answer:

33

Step-by-step explanation:

3² = 3.3 = 9 => 24 + 9 = 33

Samantha opened a savings account and deposited $8192. The account earns 10% in interest annually. She makes no further deposits and does not withdraw any money. In t years, she has $25,710 in this account. Write an equation in terms of t that models the situation.

Answers

The equation in terms of t that models the situation is given by

[tex]8192(1+\frac{0.01}{12} )^{12t} = 25710[/tex].

We know that the interest on a loan or deposit that is accrued on both the initial principal and the total interest from prior periods is known as compound interest and it can be calculated as

[tex]A = P(1+\frac{r}{n})^{nt}[/tex],

where P is the initial principal, r is the rate of interest, n is the number of times interest was charged for each time period, t is the amount of time that has passed and A is the final amount.

Here, P = 8192, r = 10% = 10/100 = 0.01, n = 12 and A = 25,710

So, we can write

[tex]8192(1+\frac{0.01}{12} )^{12t} = 25710[/tex]

Therefore the required equation is

[tex]8192(1+\frac{0.01}{12} )^{12t} = 25710[/tex]

Know more about compound interest here -

https://brainly.com/question/13870442

#SPJ10

Basically look at the picture I don’t even know what to say

Answers

Answer:

C

Step-by-step explanation:

x = [tex]\frac{2}{3}[/tex] πr³ ( multiply both sides by 3 to clear the fraction )

3x = 2πr³ ( divide both sides by 2π )

[tex]\frac{3x}{2\pi }[/tex] = r³ ( take cube root of both sides )

[tex]\sqrt[3]{\frac{3x}{2\pi } }[/tex] = r

Answer:

[tex]r=\sqrt[3]{\frac{3x}{2\pi }}[/tex]

Step-by-step explanation:

This question make look scary, but don't judge its book by its cover.

[tex]x=\frac{2}{3} \pi r^{3} \\\frac{2}{3} \pi r^{3}=x\\r^{3} =x / \frac{2\pi }{3} \\r^{3} =x*\frac{3}{2\pi } \\r^{3} =\frac{3x}{2\pi } \\r=\sqrt[3]{\frac{3x}{2\pi }}[/tex]

Which of the numbers below are whole numbers? Check all that apply.
A. 6830
B. 66
C. 717.59
D. 808
E. 42.3
F. 1/10

Answers

Step-by-step explanation:

Anything which has a decimal point is not a whole number

6830 is

66 is

717.59 is not

808 is

42.3 is not

1/10 = 0.1 ergo is not

A, B, D ARE ALL WHOLE NUMEBRS

Migdalia is a consultant. She works 3 days a week in the company's headquarters and 2
week at a production site.When she works at the company headquarters, she takes a bus
from work. When she goes to production sites, she uses her car and travels 52 ms
cast for using the bus is $48.00 for 10 one-way trips. The cost of using her car she
30 per mile Migdalia budgets $50 per week for travel expenses to and from work whether or not she has set aside enough money.

Answers

Since she has set aside only $50 for the trip, she hasn't set aside enough money for the trip

How to find the expenses and know if it's enough?

She works 3 days in a week in the headquarter and 2 days a week at the production site.

When she works at the company headquarters, she takes a bus to and from work. The cost for using the bus is $48.00 for 10 one-way trips.

Therefore,

each one way trip = 48 / 10 = $4.8

To and from the headquarters = 4.8 × 2 = $9.6

Hence,

Therefore,

three days cost = 9.6 × 3 = $28.8

The cost of using her car she estimates at 30 cent per mile. When she goes to production sites, she uses her car and travels 52 miles per day.  Therefore,

cost of travelling to production site in a week = 2 × (52 × 0.3) = $31.2

Hence,

cost of expenses per week = 31.2 + 28.8 = $60

Since she has set aside only $50 for the trip, she hasn't set aside enough money for the trip

learn more on cost here: https://brainly.com/question/19416362

#SPJ1

A researcher conducts an between-subjects study examining the
effectiveness of a memory enhancement program at an assisted living
facility for elderly adults. One group of residents is selected to
participate in the program, and a second group of participants serves as
a control group. After 6 weeks, the researcher reports a combined score
measuring memory for each individual. The data are:
Control
Exercise
n = 10
M = 12
SS = 120.5

n=15
M= 15.5
SS = 190.0
a) Does the exercise program have a significant effect? Use a two
tailed test with an alpha level of .05
b) Compute Cohen's d to measure the size of the treatment effect

Answers

H0: µ1 = µ2; The exercise program do not have significant effect on physical fitness, H1: µ1 =/ µ2; the exercise program had a significant effect on physical fitness and the  Estimated Cohen's d is 0.82.

Hypothesis

A. Significant effect

H0: µ1 = µ2

Based on the above hypothesis the exercise program does not have significant effect on physical fitness

H1: µ1 =/ µ2

Based on the above hypothesis the exercise program had a significant effect on physical fitness


Pooled variance: sp2 = (190 + 120.5/(15-1) + (10-1)

Pooled variance: sp2 = 310.5/23

Pooled variance: sp2 =13.5

Estimated standard error ( sM1-M2):

sM1-M2 =√13.5/15 +13.5/10

sM1-M2 =√2.25

sM1-M2 = 1.5

Tailed test (t)

t = (15.5 - 12)/1.5

t=3.5/1.5

t= 2.33

B. Estimated Cohen's d:

Estimated Cohen's d  = 3/√13.5

Estimated Cohen's d = 3/3.67

Estimated Cohen's d= 0.817

Estimated Cohen's d= 0.82 (Approximately); this is a large effect.

Conclusion demonstrating how the outcome of the hypothesis test would tend to appear in a research report is:  (Reject H0). Because the  group exercise program tend to have a significant effect on the physical fitness of elderly adults, t = 2.33, p <.05, d = 0.82.

Learn more about hypothesis here:https://brainly.com/question/15980493

https://brainly.com/question/14042255

#SPJ1

From this diagram, select the
pair of lines that must be
parallel if angle 1 is congruent to angle 8. If there is no pair of lines, select "none".

Answers

Answer:

Line L is parallel to N

angle 1 is equal to angle 4 ( vertically opposite angles)

if angle 1 is congruent to angle 8

then angle 4 is also congruent to angle 8

then slope of line L and n is equal

that is L is parallel to n

Find the solutions to the equation below. check all that apply. x^2-2x-15=0

Answers

The student did not make any mistake, the solution is correct. Then the correct option is A.

What is a factorization?

It is a method for dividing a polynomial into pieces that will be multiplied together. At this moment, the polynomial's value will be zero.

A student perform the following steps to find the solution of the equation x² – 2x – 15 = 0.

The factor of the equation will be

Step 1 :  x² – 2x – 15 = (x – 5)(x + 3)

Step 2 : (x – 5) = 0 and (x + 3) = 0

Step 3 : x = 5 and – 3

The student did not make any mistake, the solution is correct.

Then the correct option is A.

More about the factorization link is given below.

https://brainly.com/question/6810544

#SPJ1

Which choices are equivalent to the expression below? Check all that apply. √8 × √8

A. √ 8

B. 8

C. 64

D. 8

E. 16

F. √64​

Answers

Answer:

the answers are B and F since √8 x √8 = √64 which simplifies to 8.

evaluate this exponential expression 5 • (3 + 6)^2 - 6^2

Answers

The answer of this expression is 369

We know that to solve a numerical expression, we will calculate with property base form.

When there are symbols in an expression, we first solve the numbers in order that are between:

Parentheses ( ) Brackets [ ] Braces { }

The math signs calculated first in order are;

Power - square root Multiplication - divisionAddition - subtraction

Given the expression, the resolution will be:

5 * ( 3 + 6 )² - 6²

5 * 9² - 6²

5 * 81 - 36

406 - 36

369

So, the value of this numeric expression is 369

NEED HELP ASAP WILL MARK BRAINLIEST!

Answers

Answer:

log_{b} (5b) = 2.609

Given that:

log_{b} (3) = 1.099

log_{b} (5) = 1.609

Evaluating with that:

[tex]\sf log_b (5b) = log_b\left(5\right)+log_b\left(b\right)[/tex]

            [tex]= \sf 1.609 + 1[/tex]

             [tex]= \sf 2.609[/tex]

Rule Used:

[tex]\rm \log _a\left(xy\right)=\log _a\left(x\right)+\log _a\left(y\right)[/tex]

Other Questions
The number N of tree of a given species per acre is approximated by the modelN = 68(10)-0.042above the ground.Use the model to approximate the average diameter of the trees in a test plot when N = 21.7,5 x 40 where x is the average diameter of the trees (in inches) 3 feet A chamber within a bone normally filled with air is a: Need help to understand how to do this. PLEASE HELP (not good at math) Determine whether each ordered pair is a solution to the inequality 7xy>2. Select the correct answer.What is the monthly rent of the apartment that the following sentence refers to?Le loyer mensuel pour cet apartment est 380 .OA trols cent quarante eurosOB.trois cent quatre-vingt eurosOC. trois cent soixante-dix eurosOD. trois-cent quatre-vingt-dix euros Gabriel quiere comprar un carro que cuesta 286,000. La agencia le ofrece un crdito a una tasa de 9% anual, con pagos trimestrales durante un ao. Elabora la tabla de amortizacin del crdito Someone pls help I dont really understand graphing what is the percent yield of sulfur dioxide if the burning of 25.0 g of carbon disulfide produces 40.5 g of sulfur dioxide? Find the sum of all the evennumbers between 23 and 29. Which of these describes one function of a household in the economy? The two-way frequency table below shows the preferred communication method of employees at a company Please help worth my math i need help with this one please which patient is most likely to experience primary airway compromise? The ratio of the number of picture books encyclopedias, and fairy tale books Annette had is 3:4:5 She have half of her encyclopedias to her brother and he gave her 5 books of fairytales Now she has 14 more books of fairy tales than encyclopedias How many picture books does she have what HTMLwhat is CSSHw to create website page Which muscle contributes to external rotation of the glenohumeral joint? Solve: 0.06+7.6p Assume that the notation (a,r,s,t) means multiply q and r, then add the product to s and (2,6,4,8)+(4,7,2,6)? Please show work and choose one of the choices like e f g or h