a positive number is 5 times less than another positive number. Six times the lesser number plus 3 times the greater is 3. Find the two positive numbers

Answer:     Arithmetic , Divergent

Step-by-step explanation:

Given sequence is  

{aₙ} =  {  4 , 10/3  , 8/ 3 , 2 , . . .   }

To check whether the sequence is geometric or not , we divide second term by first term  to find the common ratio . Then we again divide third term by second term to get common ratio .

The common ratio we get would same , then it is geometric .

         10/3               10            5

r₁  =  -----------  =     ------    =   ------

             4                12             6

        8/3                8                  3                4

r₂=   ----------   =  ----------  *   ------------   =     -----

           10/ 3             3              10                  5

Thus the common ratio are not same . So the sequence is not geometric .

Now we check for arithmetic .

We take difference of second and first term and then difference of third and second term . If it will be same then it is arithmetic . This is called common difference , d .

             10                       -  2

d₁   =    ------     - 4   =       -------

              3                          3

                8            10             - 2

d₂   =    ------    -   --------    =   ---------

               3             3                 3

Thus the common difference is same .

So the given sequence is arithmetic .

To find whether it is convergent or divergent , we need to write sum of n terms first .

Formula for finding sum of n terms of arithmetic sequence is

            n

sₙ =     -----  [ 2a + ( n - 1 ) d]

             2

We have a = 4 , d = - 2/3 .

Plug  in this formula we get

              n                                                    n                2             2

sₙ  =     ------- [ 2 * 4 + ( n - 1 ) ( -2/3) ]  =     ------ [  8 -   -----  n  +   ------  ]

              2                                                    2                3               3

           n          26              2

sₙ  =   ------  [    ------    -    ------- n  ]

           2           3                3

To check whether it is convergent or divergent , we take limit sₙ approaches to infinity .

                               n       26       2

lim   sₙ   =     lim    { ---   [  ---  -  ------ n ] } =     - ∞

n → ∞           n→∞      2        3       3

As the sequence diverge , thus the series is divergent .

Thus given series is arithmetic , divergent .

Second is the  correct option .

Answer:

b^2-4ac >0  the roots are real

Step-by-step explanation:

b^2-4ac >0  the roots are real

b^2-4ac = 0 there is 1 root

b^2 -4ac <0 the root are complex

Answer:

[tex]A.\ a_n=n^2+1[/tex]

Step-by-step explanation:

[tex]Check:[/tex]

[tex]a_n=n^2+1\\\\a_1=1^2+1=1+1=2\qquad CORRECT\\\\a_2=2^2+1=4+1=5\qquad CORRECT\\\\a_3=3^2+1=9+1=10\qquad CORRECT\\\\a_4=4^2+1=16+1=17\qquad CORRECT\\\\a_5=5^2+1=25+1=26\qquad CORRECT[/tex]

Used PEMDAS:

P Parentheses first

E Exponents (ie Powers and Square Roots, etc.)

MD Multiplication and Division (left-to-right)

AS Addition and Subtraction (left-to-right)

First Power, next Addition

Answer: A.

Step-by-step explanation:

2, 5, 10, 17, 26

First, check the difference between each term:

2 → 5 = +3

5 → 10 = +5

10 → 17 = +7

17 → 26 = +9

Since the difference (d) is not the same, this is not an arithmetic sequence.

Now, check the second tier {3, 5, 7, 9}

3 → 5 = 2

5 → 7 = 2

7 → 9 = 2

The difference (d) of the second tier is the same, so it is an exponential sequence.

--> the only option that is an exponential sequence is option A.

Answer:

They have 1,690 miles left.

Step-by-step explanation:

12 x 30 = 360 miles

2050 - 360 = 1690 miles

First question:

By definition of sine and cosine, we have

[tex]\sin30^\circ=\dfrac h{\sqrt5}\implies h=\dfrac{\sqrt5}2[/tex]

[tex]\cos30^\circ=\dfrac g{\sqrt5}\implies g=\dfrac{\sqrt{15}}2[/tex]

Second question:

By definition of tangent, we have

[tex]\tan60^\circ=\dfrac{5\sqrt2}x\implies x=\dfrac{5\sqrt2}{\sqrt3}=\dfrac{5\sqrt6}3[/tex]

Then by Pythagoras' theorem,

[tex](5\sqrt2)^2+x^2=y^2\implies y=\sqrt{(5\sqrt2)^2+\left(\dfrac{5\sqrt6}3\right)^2}=\sqrt{50+\dfrac{50}3}=\sqrt{\dfrac{200}3}=\dfrac{10\sqrt6}3[/tex]

Answers: -4, 5

Step-by-step explanation:

[tex]e^{x^{2}}=e^{x}*e^{20}[/tex]

[tex]e^{x^{2}}=e^{x+20}[/tex]

x² = x + 20

x² - x - 20 = 0

(x + 4)(x - 5) = 0

x + 4 = 0      x - 5 = 0

 x = -4            x = 5

**************************************************

Answers:

Step-by-step explanation:

domain: there are no restrictions on the x-value so x = All Real Numbers

range: since the new graph has a reflection then y < 0

y-intercept is when x = 0:

horizontal asymptote: y ≠ 0 so the H.A. is y = 0

graph:  The parent graph is: y = 2ˣ

The new graph of y = -2ˣ⁻³ has the following transformations:

***************************************************************************

graph: see attachment

(0, 0) and (-1/3, 1)

(-1, ∞)

x = -1

Answer:

[(x+3)(x+2)] /[(2x+3)(x-3)]  

Step-by-step explanation:

 [(2x²+x-6)/(x²-x-6)]/[(4x²-9)/(x²+5x+6)]

= (2x²+x-6)/(x²-x-6) × (x²+5x+6) /(4x²-9)                           Factor

= [(2x-3)(x+2)]/[(x-3)(x+2] × [(x+3)(x+2)] /[(2x+3)(2x-3)]     Cancel terms

= (2x-3)/(x-3) × [(x+3)(x+2)] /[(2x+3)(2x-3)]                        Cancel terms

= [(x+3)(x+2)] /[(2x+3)(x-3)]

[tex]\dfrac{\frac{2x^2+x-6}{x^2-x-6}}{\frac{4x^2-9}{x^2+5x+6}}=(*)\\---------------------\\2x^2+x-6=2x^2+4x-3x-6=2x(x+2)-3(x+2)=(x+2)(2x-3)\\-----------\\x^2-x-6=x^2+2x-3x-6=x(x+2)-3(x+2)=(x+2)(x-3)\\-----------\\x^2+5x+6=x^2+2x+3x+6=x(x+2)+3(x+2)=(x+2)(x+3)\\-----------\\4x^2-9=(2x)^2-3^2=(2x-3)(2x+3)\\----------------------[/tex]

[tex](*)=\dfrac{(x+2)(2x-3)}{(x+2)(x-3)}\cdot\dfrac{(x+2)(x+3)}{(2x-3)(2x+3)}=(**)\\----------------------\\(x+2)-canceled\\(2x-3)-canceled\\----------------------\\(**)=\dfrac{(x+2)(x+3)}{(x-3)(2x+3)}\\\\Answer:\ \boxed{\boxed{\dfrac{(x+2)(x+3)}{(2x+3)(x-3)}}}[/tex]

Answer:

Money  club have after the show is  $358 .

Step-by-step explanation:

As given

The art club at a school raised $248 at a car wash.

They spent $206 to set up an art show, and raised $316 at the art show by selling artwork the members had made.

Than

Money club have after the show =  school raised money at a car wash +  Raised money at the art show  -  spent money to set up an art show.

Money club have after the show =  248 +  316 - 206

Money club have after the show =  564 - 206

Money club have after the show =  $358

Therefore  money  club have after the show is  $358 .

Answer:

It is 358.

Step-by-step explanation:

The club started with $248, and then spent $206.

248−206=42

The club had $42, left after they spent for the art show.

Then, they raised another $316, at the show.

42 + 316 = 358

Answer: 358

Answer:

2, and 10

Step-by-step explanation:

Let's work through each one.

First, plug -10 into the equation:

-10² + 20 = 12 x -10?

-10² = 100

100 + 20 = 120

12 x -10 = -120

120 = -120?    False!

But, using these same calculations, we can see that 10 is a correct answer.

Because everything's the same up to 12 x 10 = 120.

120 = 120

Same with 2 and -2

-2 does not work, but 2 does.

2² + 20 = 12 x 2?

4 + 20 = 24?

24 = 24?    True!

The solution of the given quadratic equation is x = 2 and x = 10.

A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared.

Given a quadratic equation, x²+20 = -12x

x²-12x+20 = 0

On factorizing the equation,

x²-10x-2x+20 = 0

x(x-10)-2(x-10) = 0

(x-10)(x-2) = 0

x = 10 or x = 2

Hence, the solution of the given quadratic equation is x = 2 and x = 10.

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Answer:

239 ft³

Step-by-step explanation:

Multiply the squared radius by π:

2*2*3.14=12.56 ft².

Multiplying the height:

12.56*19≅239 ft³.

Answer:

200

Step-by-step explanation:

150/3=50 so she sorts 50 items per minute, then 50*4=200

Answer:

Number of recyclable items sort in 4 minutes by Faye = 200

Step-by-step explanation:

Given that Faye can sort 150 recyclable items in 3 minutes.

   Number of recyclable items sort in 3 minutes = 150

   [tex]\texttt{ Number of recyclable items sort in 1 minute =}\frac{150}{3}=50[/tex]

Number of recyclable items sort in 4 minutes = 4 x Number of recyclable items sort in 1 minute

Number of recyclable items sort in 4 minutes = 4 x 50 = 200

Number of recyclable items sort in 4 minutes by Faye = 200

[tex]\heartsuit[/tex]  If x = 4 is the Only Solution of the Equation x² + bx + 16 = 0

[tex]\heartsuit[/tex]  Then the Value x = 4 should satisfy the Given Equation

⇒ (4)² + b(4) + 16 = 0

⇒ 16 + 4b + 16 = 0

⇒ 4b + 32 = 0

⇒ 4b = -32

⇒ b = -8

Answer:

Given the coordinates of rectangle ABCD are;

A = (3, 5)

B = (5,5)

C = (5, 1) and

D = (3, 1)

The rule of rotation 90 degree counter-clockwise around the origin is given by;

[tex](x, y) \rightarrow (-y, x)[/tex]

Now, the coordinates of each vertex when rectangle ABCD is rotated 90 degree counter-clockwise around the origin are;

[tex]A(3, 5) \rightarrow A'(-5, 3)[/tex]

[tex]B(5, 5) \rightarrow B'(-5, 5)[/tex]

[tex]C(5, 1) \rightarrow C'(-1, 5)[/tex]

[tex]D(3, 1) \rightarrow D'(-1, 3)[/tex]

So firstly, we need to find how much the entire trip will cost. To do this, add the product of 16 and 35 (since tickets are 35 per student, and there are 16 students) with 960:

[tex](35*16)+960\\560+960\\\\\$ 1520[/tex]

The total cost of the trip is $1520. Next, we know that the sweatshirts will be making a profit of $19 per article sold so divide 1520 by 19:

[tex]1520\div 19=80[/tex]

The students need to sell 80 sweatshirts total to get enough money for the trip. Now, to know how many sweatshirts each student needs to sell individually, divide 80 by 16 (since there are 16 students):

[tex]80\div 16=5[/tex]

In short, each student must sell 5 sweatshirts to pay for the entire trip.

Answer:

(B) ASA

Step-by-step explanation:

we will verify each options

option-A:

we can see that there is no consecutive two angles

so, this postulates can not be true

so, this is FALSE

option-B:

We can see that

one angle and then side and another angle are equivalents

so, this is TRUE

option-C:

We can see that

one side and two angles are equivalents

so, this is FALSE

option-D:

we can see that

one side and two angles are equivalents

so, this is FALSE

Answer:

B

Step-by-step explanation:

Answer

3/7 - three sevenths

Step-by-step explanation:

Multiply both numerators and both denominators

4*3=12 and 7*4=28

first one is your numerator

second one is your denominator

12/28

Then simplify

12/4=3 and 28/4=7

3/7

Let the Distance traveled by the tourist by walking be : D miles

Given : Tourist walks at a rate of 4 Miles per Hour

[tex]\mathsf{\implies Tourist\;travels\;D\;Miles\;in : \frac{D}{4}\;Hours}[/tex]

Given : The Total Distance of Mountain Path = 31 Miles

Distance traveled by the tourist by hiring a Taxi is : (31 - D) Miles

Given : The Taxi travels at a constant speed of 50 miles per hour

[tex]\mathsf{\implies Tourist\;travels\;(31 - D)\;Miles\;in : (\frac{31 - D}{50})\;Hours}[/tex]

Given : The Tourist reaches the destination 2 Hours after he started

Time taken by Walking + Time Taken by travelling in Taxi = 2 Hours

[tex]\mathsf{\implies \frac{D}{4} + \frac{31 - D}{50} = 2}[/tex]

[tex]\mathsf{\implies \frac{D}{2} + \frac{31 - D}{25} = 4}[/tex]

[tex]\mathsf{\implies (\frac{25D + 2(31 - D)}{50}) = 4}[/tex]

[tex]\mathsf{\implies (\frac{25D + 62 - 2D}{50}) = 4}[/tex]

[tex]\mathsf{\implies ({23D + 62}) = 200}[/tex]

[tex]\mathsf{\implies 23D = 200 - 62}[/tex]

[tex]\mathsf{\implies 23D = 138}[/tex]

[tex]\mathsf{\implies D = 6}[/tex]

⇒ Distance traveled by the tourist by walking = 6 Miles

⇒ Distance traveled by the tourist by hiring a taxi = (31 - 6) = 25 Miles

⇒ The Distance for which tourist has to pay for cab driver is 25 Miles

The distance traveled by taxi = 25 miles

Linear motion consists of 2: constant velocity motion with constant velocity and uniformly accelerated motion with constant acceleration

• At constant velocity motion:

the speed of vo = v = constant

acceleration = a = 0

An equation of constant velocity motion

[tex]\large {\boxed {\bold {x = v \times \: t}}}[/tex]

x = distance = m

v = speed = m / s

t = time = seconds

x1, x2 = distance traveled

t1, t2 = travel time

1 = tourist, 2 = taxi

[tex]\rm x1=v1.t1=4t1\\\\x2=v2.t2=50.t2\\\\t1+t2=2~hours\\\\t1=2-t2[/tex]

Total distance = x1 + x2

[tex]\rm 31~miles=4t1+50t2\\\\31=4(2-t2)+50t2\\\\31=8-4t2+50t2\\\\23=46t2\\\\t2=0.5[/tex]

then:

[tex]\rm x2=v2.t2\\\\x2=50\times 0.5\\\\x2=\boxed{\bold{25~miles}}[/tex]

The distance of the elevator

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resultant velocity

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the average velocity

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6+4x=y or y=4x+6

Plug in to second equation: -5x-(4x+6)=21

-9x-6=21

-9x=27

x=-3

Plugging in, you get that y=-12+6 or -6.

Therefore, the solution is (-3,-6)

Answer:

Short-response question = 20.

Step-by-step explanation:

Let x be number of short response questions and y be the number of essay questions.

We have been given that there are 22 questions that are worth 100 points in total. Each short-response question is worth 4 points, and each essay question is worth 10 points. Using our given information we can form a system of equations as:

[tex]x+y=22...(1)[/tex]  

[tex]4x+10y=100...(2)[/tex]

Now we will solve our system of equations using substitution method.

From equation 1 we will get,

[tex]y=22-x[/tex]

Upon substituting [tex]y=22-x[/tex] in equation 2 we will get,

[tex]4x+10(22-x)=100[/tex]

[tex]4x+220-10x=100[/tex]

[tex]4x-10x=100-220[/tex]

[tex]-6x=-120[/tex]

[tex]x=\frac{-120}{-6}[/tex]    

[tex]x=20[/tex]  

Therefore, there are 20 short-response questions on the exam.

Answer:

D. 20

Step-by-step explanation:

I got it right on prepworks

Answer:

C = $100+$35x

Step-by-step explanation:

Given :

A company spends $100 per day on fixed expenses such as electricity, rent, and the like.

It also spends $35 per day on each employee working that day.

To Find : a linear equation relating the total daily cost, C, with the number of employees working on any particular day, x.

Solution :

A company spend  $100 per day on fixed expenses

No. of employees working on a particular day = x

Company spend on each employee  per day = $35

It spends on x employee = $35x

Since total daily cost = C

Thus C = $100+$35x

Hence , a linear equation relating the total daily cost, C, with the number of employees working on any particular day, x. :  C = $100+$35x

Answer

Given the statement: In parallelogram ABCD, diagonals AC and BD intersects at point E.

Given that ABCD is a parallelogram.  

As, we know that in a parallelogram diagonals bisect each other.

Since AC and BD intersect at E, and we get E is the mid point of both diagonals AC and BD.

⇒ BE = DE                  .....[1]

Substitute the given values of [tex]BE = 2x^2 -3x[/tex] and [tex]DE = x^2+10[/tex] in [1] we have;

[tex]2x^2-3x = x^2+10[/tex]

Subtract [tex]x^2[/tex] on both sides we get;

[tex]2x^2-3x-x^2 = x^2+10-x^2[/tex]

Simplify:

[tex]x^2-3x =10[/tex]

Subtract 10 on both sides we get;

[tex]x^2-3x-10 =0[/tex]

or

[tex]x^2-5x+2x-10 =0[/tex]

[tex]x(x-5)+2(x-5)=0[/tex]

[tex](x-5)(x+2)=0[/tex]

equate these factors equal to zero we get;

(x-5) = 0  and (x+2) = 0

we have;

x = 5 and x = -2

Since, x cannot be negative.

So, x =5

[tex]BE = DE = x^2+10 = (5)^2+10 = 25+10 = 35 units[/tex]

Diagonals BD = BE + DE = 35 + 35 =70 units.

Therefore, the value of BD = 70 units.

Answer:

45 %

Step-by-step explanation:

Time left until Christmas = 2 weeks

                                         = 2 * 7 days                [since 1 week = 7 days]

                                         = 14 days

Number of days in December = 31

Percentage of the month left before Christmas = [tex]\frac{Time left until Christmas}{Number of days in December} * 100[/tex]

= [tex]\frac{14}{31} * 100[/tex]

= 0.45 * 100

= 45 %

Answer:

what the other person said.

Step-by-step explanation:

Answer:

(D)68

Step-by-step explanation:

If 1 litre =4.23 cups

and 1 cup = 8 ounces

Therefore: 1 litre = 4.23 X 8 Ounces = 33.84 Ounces

2 litres = 2 x 33.84 Ounces

           =67.68 Ounces

           =68 Ounces (To the nearest whole number)

The number of ounces in 2 liters will be 68 ounces. Thus, the correct option is D.

Given that:

1 liter = 4.23 cups

1 cup = 8 ounces

Unit modification is the process of converting the measurement of a given amount between various units, often by multiplicative constants that alter the value of the calculated quantity without altering its impacts.

The number of cups in 2 liters is calculated as,

⇒ 2 x 4.23

⇒ 8.46 cups

The number of ounces in 8.46 cups is calculated as,

⇒ 8.46 x 8

⇒ 67.68 or 68 ounces

Thus, the correct option is D.

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Lets hope i'm right.

Triangle formula: 1 / 2 b x h

y:

Half one of them. Lets do 4 since 5 would result in a decimal.

(4) = 2 x 5 = 10.

y (10) + 1 = 11

x:

Half one.

(2) = 1 x 3 = 3

Answers:

y = 11

x = 3

Hope this helps! Wait for another answer to make sure i'm right. ;) ~Pooch ♥

The correct answer is A.

Hope this helps,

Davinia.

The inverse of the function g(x)=-4/3x+2 is

[tex]g^{-1}(x) = -\frac{3}{4}x+\frac{3}{2}[/tex]

The given function is:

[tex]g(x) = -\frac{4}{3}x + 2[/tex]

To find the inverse of the function g(x), follow the steps below

Make x the subject of the formula

[tex]g(x) = -\frac{4}{3}x+2\\\\ -\frac{4}{3}x = g(x) - 2\\\\-4x=3g(x)-6\\\\x = -\frac{3}{4}g(x)+\frac{6}{4} \\\\x = -\frac{3}{4}g(x)+\frac{3}{2}[/tex]

Replace x by [tex]g^{-1}(x)[/tex] and replace g(x) by x

The inverse function therefore becomes:

[tex]g^{-1}(x) = -\frac{3}{4}x+\frac{3}{2}[/tex]

The inverse of the function g(x)=-4/3x+2 is [tex]g^{-1}(x) = -\frac{3}{4}x+\frac{3}{2}[/tex]

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Answer: width = 4 ft, length = 8 ft, height = 4 ft

Step-by-step explanation:

width (w): w

length (L): 2w

height (h): w

Volume (V) = w * L * h

128 = w(2w)(w)

128 = 2w³

64 = w³

∛64 = ∛w³

4 = w

width (w): w = 4

length (L): 2w = 2(4) = 8

height (h): w = 4

Answer:

42° and 48°

Step-by-step explanation:

Complementary angles sum to 90°, hence

x + 10 + x + 4 = 90

2x + 14 = 90 ( subtract 14 from both sides )

2x = 76 ( divide both sides by 2 )

x = 38

x + 10 = 38 + 10 = 48° and x + 4 = 38 + 4 = 42°

Answer:

x = 5

Step-by-step explanation:

in ΔABC , AB = BC , hence triangle is isosceles, thus

∠A = ∠C = 73° ( base angles are equal )

the sum of the 3 angles in a triangle = 180°

subtract the sum of the base angles from 180 for ∠B

∠B = 180° - (73 + 73)° = 34°

⇒ 6x + 4 = 34 ( subtract 4 from both sides )

6x = 30 ( divide both sides by 6 )

x = 5