exponential growth and decay real-world word problems
​

Answer:

2

Step-by-step explanation:

just because all three numbers are even

Hope this helps

The greatest common factor of 30, 22, and 8 will be 2.

The Highest Common Factor (HCF) of two numbers is the highest possible number that is divisible by both numbers.

In other words, the highest common factor is the common factor between the two numbers but it should be the highest among all common factors.

For example in 6 and 12 6 is the highest common factor.

The factor of 30 ⇒ 2,3,5

The factor of 22 ⇒ 2,11

The factor of 8 ⇒ 2,4

The only common factor among 30,22 and 8 is 2 so it will be the greatest common factor.

Hence "The greatest common factor of 30, 22, and 8 will be 2".

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[tex]\cos(x)[/tex] is an even function, while [tex]\sin(x)[/tex] is odd. This means

[tex]\cos(-x) = \cos(x) \text{ and } \sin(-x) = -\sin(x)[/tex]

[tex]\cot(x)[/tex] is defined by

[tex]\cot(x) = \dfrac{\cos(x)}{\sin(x)}[/tex]

so it is an odd function, since

[tex]\cot(-x) = \dfrac{\cos(-x)}{\sin(-x)} = \dfrac{\cos(x)}{-\sin(x)} = -\cot(x)[/tex]

Putting everything together, it follows that

[tex]\cot(-x) \cos(-x) \sin(-x) = (-\cot(x)) \cos(x) (-\sin(x)) \\\\~~~~~~~~= \cot(x) \cos(x) \sin(x) \\\\ ~~~~~~~~= \cos^2(x)[/tex]

Answer:

10 degrees

Step-by-step explanation:

8x = 2x + 60

6x = 60

x = 10

Answer: B, C, E

Step-by-step explanation: a is wrong 32 tens is 320.  B is right. 321 tenths is 32.1 and 48 thousandths is 0.048 so C is correct. 32,148 hundredths is too much but 32,148 thousandths is 32.148

4710

4709.94107458

The nearest whole number for 4709.94107458 is 4710 since the number 9 after the decimal is greater than 5. Therefore, if you add 1 to 9 before the decimal, you get 10.

Hope this helps :)

Using the t-distribution, it is found that since the test statistic is between -1.7341 and 1.7341, we do not reject(accept) the null hypothesis, hence there is not enough evidence to conclude that the true mean discharge differs from 7 ounces.

At the null hypothesis, it is tested if the mean is of 7 ounces, that is:

[tex]H_0: \mu = 7[/tex]

At the alternative hypothesis, it is tested if the mean is different of 7 ounces, that is:

[tex]H_1: \mu \neq 7[/tex]

The test statistic is given by:

[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]

The parameters are:

The values of the parameters are given by:

[tex]\overline{x} = 7.04, \mu = 7, s = 0.17, n = 18[/tex]

Hence the value of the test statistic is found as follows:

[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]

[tex]t = \frac{7.04 - 7}{\frac{0.17}{\sqrt{18}}}[/tex]

t = 1

Considering a two-tailed test, as we are testing if the mean is different of a value, with 18 - 1 = 17 df and a significance level of 0.1, the critical value is of [tex]t^{\ast} = 1.7341[/tex]

Since the test statistic is between -1.7341 and 1.7341, we do not reject(accept) the null hypothesis, hence there is not enough evidence to conclude that the true mean discharge differs from 7 ounces.

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The conditional probability, in a single roll of two fair 6 sided dice, that the sum is greater than 6, given that neither die is a two : 17/25

We know that the conditional probability is given by,

P(B | A) = probability of occurrence of event B, given that event A has

               occurred

           = P(A ∩ B) / P(A)

Here, P(A ∩ B)  means the probability of happening two events A and B at the same time.

We also know that  if  P (B | A ) = P(B)   i.e.,   P(A ∩ B) = P(A) × P(B)  the two events A and B are independent of each other.

For this question, let the dice D1 and D2 are rolled once.  

Let the numbers displayed on the dice be d1 and d2 respectively.  

The dice D1 and D2 are independent.

We need to find the conditional probability that the sum is greater than 6, given that neither die is a two.

Let S represents the sum of the numbers displayed on the dice.

S = d1 + d2

The sum is even, if d1 = d2 is odd OR  if d1 = d2 is even

P(d1 = even) = 3/6

                    =1/2

P(d2=even) = 1/2

P(d1 = odd) = 1/2

P(d2 = odd) = 1/2

So, P(S = even) = [P(d1=even) × P(d2 = even)] + [P(d1= odd) × P(d2=odd)]

                          = [1/2 × 1/2] + [1/2 × 1/2]

                          = 1/2

So, we can say that, the sum is either even or odd which are equally likely and hence its probability is 1/2.

First we find the probability for the sum is greater than 6 i.e., P(S > 6)

The possible combination of d1 and d2 for the sum greater than 6 would be,

{(1,6), (2,5), (2, 6), (3, 4), (3, 5), (3, 6), (4,3), (4,4), (4,5), (4,6), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

⇒ n(S > 6) = 21

The number of all possible outcomes = 36

So, P(S > 6) = 21/36

                   = 7/12

Now we find the probability that neither die is a two

⇒ P(neither die is a two) = [P(1) ∪ P(3 ≤ d1 ≤ 6)]  AND  [P(1) ∪ P(3 ≤ d1 ≤ 6)]

⇒ P(neither die is a two) = 5/6 × 5/6

⇒ P(neither die is a two) = 25/36

Now, we find the probability that the sum S > 6 AND  neither die is a two.

The possible combination for  the sum S > 6 AND  neither die is a two would be,

{(1,6), (3, 4), (3, 5), (3, 6), (4,3), (4,4), (4,5), (4,6), (5,3), (5,4), (5,5), (5,6), (6,1), (6,3), (6,4), (6,5), (6,6)}

⇒ n(S > 6 AND neither die is a two) = 17

So, P(S > 6 AND neither die is a two) = 17/36

Now we find the conditional probability P(S > 6 | neither die is a two)

⇒ P(S > 6 | neither die is a two) = P(S > 6 AND neither die is a two) ÷

                                                           (neither die is a two)

⇒ P(S > 6 | neither die is a two) = (17/36) / (25/36)

⇒ P(S > 6 | neither die is a two) = 17/25

Therefore, the conditional probability, in a single roll of two fair 6 sided dice, that the sum is greater than 6, given that neither die is a two : 17/25

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For the given data set

Minimum = 48

Lower quartile = 55

Median = 63.5

Upper quartile = 74

Maximum = 80

Interquartile range = 19

From the question, we are to determine the minimum, lower quartile, median, upper quartile, maximum, and interquartile range of the given data set

The given data set is

48, 50, 54, 56, 63, 63, 64, 68, 74, 74, 79, 80

Minimum = 48

Lower quartile = (54+56)/2

Lower quartile = 110/2

Lower quartile = 55

Median = (63+64)/2

Median = 127/2

Median = 63.5

Upper quartile = (74+74)/2

Upper quartile = 148/2

Upper quartile = 74

Maximum = 80

Interquartile range = Upper quartile - Lower quartile

Interquartile range = 74 - 55

Interquartile range = 19

Hence, for the given data set

Minimum = 48

Lower quartile = 55

Median = 63.5

Upper quartile = 74

Maximum = 80

Interquartile range = 19

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Answer: 104 degrees farenheit

Step-by-step explanation:    H =  record high temperature. -5 + 109 = H. -5+109 = 109 + (-5) = 109-5 = 104. H = 104.

Answer:

104

Step-by-step explanation:

Let x = record high and y = record low temperature in Charlotte. The difference between the records high and low, x and y, is 109 degrees Fahrenheit, so x - y = 109. Record low is -5, so x - (-5) = 109.

x + 5 = 109

x = 104

Answer:

8x³ - 2x² - 51x - 45

Step-by-step explanation:

(x - 3)(2x + 3)(4x + 5) ← expand the 2nd/3rd factors using FOIL

= (x - 3)(8x² + 10x + 12x + 15)

= (x - 3)(8x² + 22x + 15)

multiply each term in the second factor by each term in the first factor.

x(8x² + 22x + 15) - 3(8x² + 22x + 15) ← distribute parenthesis

= 8x³ + 22x² + 15x - 24x² - 66x - 45 ← collect like terms

= 8x³ - 2x²- 51x - 45

Expand first 2 bracket first to get:

2x^2 + 3x - 6x - 9 & simplify, then expand with last bracket.

2x^2 - 3x - 9 (4x + 5)

2x^2 x 4x = 8x^4

2x^2 x 5 = 10x^2

Repeat for the next two numbers next to the bracket.

You get => 8x^3 + 10x^2 - 12x^2 - 15x - 36x - 45

Final simplified answer of:

8x^3 - 2x^2 - 51x - 45

Hope this helps!

1/2 value can be used as the common ratio in an explicit formula that represents the sequence.

Definition of sequence -

A sequence that progresses geometrically has the first term as 12.

Each number is the previous number divided by 2.

so the sequence will be 12, 6, 3, 1.5...........

Explicit formula of a geometric sequence is given by

[tex]T_{n} = ar^{n-1}[/tex]

Where [tex]T_{n}[/tex]  = nth term of the sequence

a = first term

r = common ratio

and n = number of term

In this sequence common ratio = [tex]\frac{Successive term }{previous term}[/tex]

            = 6/12

            =  1/2

Therefore, common ratio will be 1/2.

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Step-by-step explanation:

In a Mathematics lab. There are some cubes and cuboids of following measurements

(i) One cube of side 4 cm

(ii) One cube of side 6 cm

(iii) 3 cuboids each of dimensions 4cm ×4 cm ×6cm

(iv) 3 cuboids each of dimensions 4cm ×6 cm ×6cm

A student wants to arrange these cubes and cuboids in the form of big cube. Is it

possible to arrange them in the form of big cube? If yes, then find the length of side of

Answer:

D. 2

Step-by-step explanation:

soh cah toa rule

sine equals opposite over hypotenuse,

cosine equals adjacent over hypotenuse, and

tangent equals opposite over adjacent

csc is 1/sin which is hypotenuse over opposite

if csc is -√5/2 then

hypotenuse is -√5

opposite is 2

3rd quadrant on a graph is bottom left

both x and y values are negative

hypotenuse means its a triangle

pythagorean theorem

a^2 + b^2 = c^2

a^2 + 2^2 = (-√5)^2

a^2 + 4 = 5

a = 1 = adjacent

tangent equals opposite over adjacent

tan is 2 / 1 which is 2

third quadrant means it's still positive

because tan is positive in quadrant 3

intmathcom

revisionmathscom

mathworldwolframcom

Answer:

C

Step-by-step explanation:

For any function x, inverse will = 1/x. so the answer is C

Answer:

C. [tex]f^{-1}(x) = \frac{1}{5} x[/tex]

Step-by-step explanation:

The question is asking to find [tex]f^{-1}(x)[/tex] of f(x)=5x, which is the same as finding the inverse of the function.

To find the inverse of a function, we first need to replace f(x) with y.

The function will therefore be:

y = 5x

Now, we solve the equation for x.

So divide both sides by 5.

y = 5x

÷5  ÷5

_________

[tex]\frac{y}{5} = x[/tex]

Now, we replace x with y and y with x.

[tex]\frac{x}{5} = y[/tex]

Finally, we replace y with [tex]f^{-1}(x)[/tex].

[tex]\frac{x}{5} = f^{-1}(x)[/tex]

We can re-write this though, to make it easier to read.

We can write [tex]f^{-1}(x)[/tex] first, and rewrite [tex]\frac{x}{5}[/tex] as [tex]\frac{1}{5}x[/tex].

[tex]f^{-1}(x) = \frac{1}{5} x[/tex]

Therefore, the answer is C.

Debido a restricciones de extensión y la características del ejercicio, recomendamos leer la explicación de esta pregunta para mayores detalles sobre la adición de números enteros.

En este ejercicio tenemos un grupo de sumas con números enteros positivos y negativos, en las cuales se prueba la capacidad del estudiante para realizar varias operaciones en serie (adición, sustracción) y comprender las diferencias entre números positivos, negativos y neutros. Ahora procedemos a determinar el resultado de cada una de las expresiones:

20 + 50 + 30 + 7 = 107

30 + 5 + 2 = 37

- 200 - 50 - 70 - 8 = - 328

- 500 + 100 - 20 + 50 = - 370

10 - 5 = 5

20 + 50 - 25 - 10 = 35

- 100 + 20 = - 80

- 30 + 5 + 4 - 20 + 8 = - 33

- 258 + 8 = - 250

- 10 + 20 + 520 - 100 + 8 = 438

- 20 - 5 - 42 + 3 = - 64

1000 - 200 + 50 + 30 - 45 + 75 - 87 + 90 + 50 - 100 + 50 - 10 = 903

- 400 + 500 - 200 - 50 + 48 + 8 - 47 - 50 = - 191

300 + 20 - 50 + 30 - 84 + 35 - 7 + 20 - 40 + 10 - 45 + 65 + 8 - 55 = 207

800 + 50 - 69 + 8 - 35 + 85 - 54 + 40 + 85 + 74 - 32 - 8 + 65 - 27 = 982

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Answer:

31. (98,765 x 9) + 3 = 888,888

32. (12,345 x 9) + 6 + 111,111

33. 3367 x 15 + 50,505

34. 15,873 x 28 = 555,555

35. 33,334 x 33,334 = 1,111,155,556

36. 11,111 x 11,111 = 123,454,321

41. 3 + 9 + 27 + 81 + 243 = 3(243 -1) / 2

42.  1/1x2 + 1/2x3 + 1/3x4 + 1/4x5 + 1/5x6 = 5/6

hope this helps

Answer:

see the step-by-step explanation

Step-by-step explanation:

31. (98,765 x 9) + 3 = = 888,888

32. (12,345 x 9) + 6 + 111,111

33. 3367 x 15 + 50,505

34. 15,873 x 28 = 555,555

35. 33,334 x 33,334 = 1,111,155,556

36. 11,111 x 11,111 = 123,454,321

41.3+9+27+81 +243 = 3(243-1)/2

42. 1/1x2 + 1/2x3 + 1/3x4 + 1/4x5 + 1/5x6 = 5/6

If machine a can make 350 widgets in 1 hour and machine b can make 250 widgets in 1 hour then by forming equation we will get that both machines have to work for one hour and 40 minutes to make 1000 widgets.

Given that machine a can make 350 widgets in 1 hour, and machine b can make 250 widgets in 1 hour.

How much time will both machines take to make 1000 widgets?

Suppose the time taken by both machines be x hours. Time is equal because both the machines need to work together.

According to the question the equation will be as under:

350x+250x=1000

600x=1000

x=1000/600

x=10/6

x=5/3

x=1.67

Converting 0.67 to minutes 0.67*60=40.2

Adding will result 1 hour and 40 minutes.

Hence if machine a can make 350 widgets in 1 hour and machine b can make 250 widgets in 1 hour then by forming equation we will get that both machines have to work for one hour and 40 minutes to make 1000 widgets.

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A major advantage of a direct implementation is that people whose roles span modules do not have to switch back and forth between old and new modules.

With this technique, the machine is implemented and tested to make certain it plays well. Then the antique machine is removed and the brand new one is put in its area without any overlap or restricted rollout.

There are three predominant methods used: phased implementation, direct changeover, and parallel going for walks. Phased implementation: A staged technique whereby one part of the overall system that desires change is changed.

Structures implementation is the process of defining how the data device needs to be built (i.e., physical machine design), ensuring that the records gadget is operational and used, and ensuring that the information device meets greatly preferred i.e. nice warranty.

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Answer:

c

Step-by-step explanation:

if you take the triangle off of the left side and attach it to the left side then you would have a 6 by 4 rectangle that has an area of 24 units squared.

Answer:

A=113 cm²

Step-by-step explanation:

If the length of the rope is 6 cm

then the radius of the circle we draw is 6 cm.

Area of the circle is A = π·r²

A = π·6² ≈ 3.14· 36 ≈ 113cm²

The first term of the arithmetic progression exists at 10 and the common difference is 2.

let the nth term be named x, and the value of the term y, then there exists a function y = ax + b this formula exists also utilized for straight lines.

We just require a and b. we already got two data points. we can just plug the known x/y pairs into the formula

The 9th and the 12th term of an arithmetic progression exist at 50 and 65 respectively.

9th term = 50

a + 8d = 50 ...............(1)

12th term = 65

a + 11d = 65 ...............(2)

subtract them, (2) - (1), we get

3d = 15

d = 5

If a + 8d = 50 then substitute the value of d = 5, we get

a + 8 [tex]*[/tex] 5 = 50

a + 40 = 50

a = 50 - 40

a = 10.

Therefore, the first term is 10 and the common difference is 2.

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Check the picture below.

to get the equation of any straight line, we simply need two points off of it, so let's use those in the picture

[tex](\stackrel{x_1}{-2}~,~\stackrel{y_1}{1})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{6})~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{6}-\stackrel{y1}{1}}}{\underset{run} {\underset{x_2}{4}-\underset{x_1}{(-2)}}} \implies \cfrac{6 -1}{4 +2} \implies \cfrac{ 5 }{ 6 }[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{1}=\stackrel{m}{\cfrac{5}{6}}(x-\stackrel{x_1}{(-2)})\implies y-1=\cfrac{5}{6}(x+2) \\\\\\ y-1=\cfrac{5}{6}x+\cfrac{5}{3}\implies y=\cfrac{5}{6}x+\cfrac{5}{3}+1\implies y=\cfrac{5}{6}x+\cfrac{8}{3}[/tex]

Answer:

A

Step-by-step explanation:

the answer is A because it is absolute so it would actually be x + 3

The minimum value of data is 24,lower quartile is 29,median is 41, upper quartile is 50 and maximum value is 56 and the interquartile range is 21.

Given a data about ages of 13 history teacher as under:

24, 27, 29, 29, 35, 39, 43, 45, 46, 49, 51, 51, 56.

We are required to find the minimum value, lower quartile,median,upper quartile,maximum value, interquartile range.

The minmum value is 24.

Lower quartile=(n+1)/4 th term

=(13+1)/4

=7/2

=3.5

Lower quartile=(29+29)/2

=29

Median=(n/2)th term

=13/2 th term

=6.5 th term

Median=(39+43)/2

=82/2

=41

Upper quartile=3(n+1)/4   th term

=3(13+1)/4

=3*14/4

=10.5 th term

Upper quartile=(49+51)/2=100/2=50

Inter quartile range=Upper quartile- lower quartile

=50-29

=21

Hence the minimum value of data is 24,lower quartile is 29,median is 41, upper quartile is 50 and maximum value is 56 and the interquartile range is 21.

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The angle between the two extreme vectors of Sofia bats is 74.60 degrees.

In this question,

The angle between two vectors will be deferred by a single point, which is called as the shortest angle at which we have to turn around one of the vectors to the position of co-directional with another vector.

The extreme vectors of Sofia bats are  [tex]\vec u = < -8, 12 >[/tex] and [tex]\vec v = < -8, 12 >[/tex]

The angle between the vectors is

[tex]\theta = cos^{-1} \frac{\vec u.\vec v}{||\vec u|| ||\vec v||}[/tex]

The dot product is calculated as

[tex]\vec u. \vec v = (-8)(13)+(12)(15)[/tex]

⇒ [tex]-104+180[/tex]

⇒ 76

The magnitude can be calculated as

[tex]||\vec u|| = \sqrt{(-8 )^{2} +(12)^{2} }[/tex]

⇒ [tex]\sqrt{64+144}[/tex]

⇒ [tex]\sqrt{208}[/tex]

[tex]||\vec v|| = \sqrt{(13 )^{2} +(15)^{2} }[/tex]

⇒ [tex]\sqrt{169+225}[/tex]

⇒ [tex]\sqrt{394}[/tex]

Thus the angle between the vectors is

[tex]\theta = cos^{-1} \frac{76}{(\sqrt{208} )(\sqrt{394} )}[/tex]

⇒ [tex]\theta = cos^{-1} \frac{76}{\sqrt{81952} }[/tex]

⇒ [tex]\theta = cos^{-1} \frac{76}{286.27 }[/tex]

⇒ [tex]\theta = cos^{-1} (0.2654)[/tex]

⇒ [tex]\theta=74.60[/tex]

Hence we can conclude that the angle between the two extreme vectors of Sofia bats is 74.60 degrees.

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If the function is [tex](x-3)^{-2}[/tex] and  f(4) − f(1) = f '(c)(4 − 1) then there is not any answer.

Given function is [tex](x-3)^{-2}[/tex] and  f(4) − f(1) = f '(c)(4 − 1).

In this question we have to apply the mean value theorem, which says that given a secant line between points A and B, there is at least a point C that belongs to the curve and the derivative of that curve exists.

We begin by calculating f(2) and f(5):

f(2)=[tex](2-3)^{-2}[/tex]

f(2)=1

f(5)=[tex](5-3)^{-2}[/tex]

f(5)=1

And the slope of the function:

[tex]f^{1}[/tex](x)=[tex]f(5)-f(2)/(5-2)[/tex]

[tex]f^{1}[/tex](c)=0

Now,

[tex]f^{1} (x)=-2*(x-3)^{-3}[/tex]

=-2[tex](x-3)^{-3}[/tex]

=0

-2 is not equal to 0. So there is not any answer.

Hence if the function is [tex](x-3)^{-2}[/tex] and  f(4) − f(1) = f '(c)(4 − 1) then there is not any answer.

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The speed of the wind is 50 miles per hour.

The term speed is defined as the ratio of the distance to the time taken. Now we can see that the movement of the plane and the wind were once in the same direction and then in opposite direction. This could be used to obtain a pair of simultaneous equations that could be used to solve the problem.

Hence;

300 = (250+s)* t = 250t + st ----- (1)

200 = (250-s)* t = 250t   - st  ------- (2)

Adding equations (1) and (2)

500 = 500t

t = 1 hour

To obtain the speed of the wind;

300 =250t + st

300 = 250(1) + (s * 1)

300 = 250 + s

300 - 250 = s

s = 50 miles per hour

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Missing parts;

A plane has a cruising speed of 250 miles per hour when there is no wind. At this speed, the plane flew 300 miles with the wind in the same amount of time it flew 200 miles against the wind. Find the speed of the wind.

Since the product of AB and BC is -1, hence the ABC is a right triangle and the length AC is the hypotenuse

A right angles triangle is a triangle that has 3 sides and angles with one of its angles to be 90 degrees.

For the given coordinates to be a right angled triangle, one of the line must be perpendicular to the other and for two lines to be perpendicular, the product of its slope must be -1.

Find the slopes of AB, BC and AC

Slope of AB = 5-11/1-6
Slope of AB = -6/-5 = 6/5

Slope of BC = 0-5/7-1
Slope of BC = -5/6

Slope of AC = 0-11/7-6
Slope of AC = -11/1. = -11

Take the product of AB and BC
AB * BC = 6/5 * -5/6
AB * BC = -1

Since the product of AB and BC is -1, hence the ABC is a right triangle and the length AC is the hypotenuse

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The number of additional days she wants to rent the car for the two specials are equivalent is 5 days.

Compact car:

y = 50 + 6x

Another special:

y = 40 + 8x

let

x = number of additional days

50 + 6x = 40 + 8x

50 - 40 = 8x - 6x

10 = 2x

x = 10/2

x = 5 days

Therefore, the number of additional days she wants to rent the car for, the two specials are equivalent is 5 days.

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