The sample space S= [E1, E2, E3, E4, E5, E6, E7, E8, E9, E10]
Given A = [E1, E3, E6, E9], define Ȃ
Given A = [E1, E3, E7, E9] and B = [E2. E3, E, E9]
a) What is A intersection B?
b) What is the union of A & B?
c) Is the union of A & B collectively exhaustive?

One example of real-world situation that uses logarithms is for amount of substance that are decaying over time.

An amount of a decaying substance after t years is modeled by the following exponential equation:

[tex]P(t) = P(0)e^{-kt}[/tex]

In which:

The simplest example of the use of logarithms is to find the half-life of the substance, that is, the year t for which P(t) = 0.5P(0).

Then:

[tex]0.5P(0) = P(0)e^{-kt}[/tex]

[tex]e^{-kt} = 0.5[/tex]

The natural logarithm is used to find the half life, as it is the inverse of the exponential.

[tex]\ln{e^{-kt}} = \ln{0.5}[/tex]

[tex]-kt = \ln{0.5}[/tex]

[tex]t = -\frac{\ln{0.5}}{k}[/tex]

Hence we showed another application of logarithms in a real-word model.

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The bathroom must be tiled with 49 tiles, each of which has a surface area of 1 square meter and measures 7 meters by 7 meters.

How can I figure out how many tiles I need to tile the bathroom?

The bathroom's area is,

A= 7m x 7m

A= 49m^2

likewise the area of each tile = 1 square meter

The total amount of tiles needed to tile the bathroom is thus,

49/1 = 49

Therefore, the bathroom must be tiled with 49 tiles.

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[tex](\stackrel{x_1}{0}~,~\stackrel{y_1}{9})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{1}{5} \\\\\\ \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}{9}=\stackrel{m}{ \cfrac{1}{5}}(x-\stackrel{x_1}{0})\implies {\Large \begin{array}{llll} y=\cfrac{1}{5}x+9 \end{array}}[/tex]

The expression divide r by 8 when doubled is r/4

An algebraic expression can be seen or described as a mathematical or arithmetic expressions that is composed of arithmetic terms, factors, constants, variables, and coefficients.

These expressions are also known to be composed of mathematical operations, such as;

From the information given, we have;

divide r by 8

This is expressed as;

r/8

In doubling the expression, we get;

2(r/8)

expand the bracket

2r/8

Simplify further

r/4

Hence, the expression is r/4

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Answer:

8.97 x 10^6

Step-by-step explanation:

To add in scientific notation, the exponents need to be the same for the bases of 10.  You can either make them both to the exponent of 6 or 5.  It does not matter.  I made them both to the power of 5

(8.6 x10^6 ) + (3.7 x10^5 )

(86 x 10^5) + (3.7x10^5)  

(86 x 3.7) x 10^5

89.7 x 10^5 Rewrite in scientific notation

8.97 x 10^6

Based on the number of trucks needed and the snowfall per inch, the relationship between snowfall and trucks is proportional.

A relationship is said to be proportional when the two variables increase at a set rate. In other words, proportional relationships will see variables increasing at the same rate.

The rate of snowfall per trucks for 6 inches of snowfall is:

= 15 / 6

= 2.5 trucks per inch of snowfall

The rate of snowfall per trucks for 12 inches of snowfall is:

= 30 / 12

= 2.5 trucks per inch of snowfall

The rate is therefore the same across the table so the relationship between snowfall and the number of trucks is proportional.

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Answer:

x = 37

Step-by-step explanation:

∡CAE = 180°

Then:

∡ABC = 180° - (10+8X)

∡ABC = 180 - 10 -8X

∡ABC = 170 -8X

The sum of the internal angles of a triangle results im 180°:

70 + (6x+14) + (170-8x) = 180

70 + 14 + 170 + 6x - 8x = 180

254 - 2x = 180

254 - 180 = 2x

74 = 2x

x = 74/2

x = 37

Explanation:

y% of some value, x:

Convert y to a decimal by dividing y by 100.

Then multiply that decimal value to x.

The equation that involves division and addition and has a solution of x=-25 will be 3x - 40 = x + 90

It should be noted that an equation is used to show the relationship between the variables that are given.

The equation that involves division and addition and has a solution of x = -25 will be:

3x - 40 = x - 90

Collect like terms

3x - x = -90 + 40

2x = -50

Divide

x = -50 / 2

x = -25

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A.

The length of the diagonal is given by the Pythagorean theorem therefore

The length of the diagonal is ) units

B.

The area of the square is given by the next formula

where s is the side

s=4

The area of the square is 16 units^2

C.

For the area of the triangle we will use the next formula

where b is the base and h is the height

b=4 units

h=4units

The area of the triangle formed by a diagonal and two of the sides is 8 units^2

D.

For the area of this triangle, we will use the same formula that we use in C. but in this case

b=sqrt(32)/2

h=sqrt(32)/2

We substitute the values

The area of one of these triangles is 4 units^2

ANSWER

A. sqrt(32) units

B.16 units^2

C. 8 units^2

D. 4 units^2

The original function and its expression derived from rigid transformations are defined below:

In this problem we must use rigid transformations to change the definition of a series of functions. Rigid transformations are transformations applied on functions such that its form is not altered. There are the following transformations, which are used in the five cases:

Horizontal translation

g(x) = f(x - k), translation is to the right for k > 0.

Vertical translation

g(x) = f(x) + k, translation is upward for k > 0

Reflection over the x-axis

g(x) = - f(x)

Reflection over the y-axis

g(x) = f(- x)

Now we proceed to present the image of each function after using a sequence of rigid transformations:

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If guests staying at Marada Inn were asked to rate the quality of their accommodations as being excellent (E), above average (AA), average (A), below average (BA), or poor (P). The frequency is : 1, 8, 6, 3, 2.

Analysis

E = E =1

AA= AA +AA + AA + AA + AA+ AA + AA +AA = 8

A= A + A+ A+ A+ A+ A = 6

BA = BA + BA + BA =3

P = P + P =2

Hence,

Class     frequency

E                     1

AA                  8

A                     6

BA                   3

P                      2

Total               20

Therefore  1, 8, 6, 3, 2 is the frequency .

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Note that:

[tex]\int f(x) \text{ } dx=-e^{-x/5}+C[/tex]

Part (b)

[tex]\int^{1}_{0} f(x) \text{ } dx =-e^{-1/5}+e^{0} \approx 0.1813[/tex]

Part (c)

[tex]\int^{\infty}_{0} f(x) \text{ } dx =1.0000[/tex]

Part (e)

[tex]\int^{6}_{1} f(x) \text{ } dx \approx 0.5175[/tex]

The function, [tex]\displaystyle { p(x) = \frac{ {cos}^{2}x }{sin(2 \cdot x)} }[/tex] has a limit that is undefined (does not exist) as x approaches 0. The correct option therefore option b;

b. As x approaches 0, the limit of p(x) does not exist

The given function is presented as follows;

[tex]\displaystyle { p(x) = \frac{ {cos}^{2}x }{sin(2 \cdot x)} }[/tex]

Required;

The limit of the function as x approaches (zero) 0

Solution;

The limit of a function at a given point within the domain of the function is the value of the function as the function's argument approaches a.

Therefore, the limit of the given function at the point x = 0 is given by the function's value as the argument of the function, x approaches 0.

cos²(0) = 1

sin(2×0) = sin(0) = 0

Therefore;

[tex]\displaystyle { p(x) = \frac{ {cos}^{2}0 }{sin(2 \times 0)} = \frac{ 1 }{0} = \infty}[/tex]

Therefore, the limit of the function does not exist as x approaches 0

The correct option is therefore, option b

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In still air, the rate of the jet is Distance traveled/ time taken

With the jetstream, the rate of jetstream is

The warmest recorded temperature than the coldest recorded temperature is 54oF

The given parameters are:

Coldest recorded temperature = 80°F, in Alaska

Warmest recorded temperature = 134°F, in California

Next, we calculate the difference between these temperatures

Difference = Warmest recorded temperature - Coldest recorded temperature

Substitute the known values in the above equation

Difference = 134 - 80

Evaluate

Difference = 54

Hence, the amount of temperature difference is 54oF

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The equation that shows a relationship between t (time in minutes) and d (distance in miles) is d = 0.075t

An equation is an expression that shows the relationship between two or more numbers and variables.

Equations are classified based on degree (value of highest exponents) as linear, quadratic, cubic and so on. Variables can be dependent or independent. Dependent variables depend on other variable while an independent variable do not depend.

The standard form for linear equation is:

y = mx + b

Where m is the slope and b is the y intercept

Let d represent the runners distance after t minutes.

From the table, using the points (12, 0.9) and (16, 1.20):

d - 0.9 = [(1.2 - 0.9)/(16 - 12)](t - 12)

d - 0.9 = 0.075(t - 12)

d = 0.075t

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 ( -4)¹⁹  =  -  274877906944 is standard form of 4 to power -19.

What are power and exponent?

a)  ( -8 )⁰   =   1               ( positive )

b)  ( -4 )⁶    = - 4× -4 × -4 × -4 × -4 × -4 = 4096  ( Positive )

c)  ( -5 )¹²  =  244,1406,25           ( positive )

d)  ( -4)¹⁹  =  -  274877906944   ( Negative)

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The population proportion when the standard error of the sampling distribution of the sample proportion is 0.0337 for samples of size 200 must be D. 0.35 or 0.65.

Based on the information, it should be noted that the formula for the standard error for the population will be:

✓p(1 - p)/n

where

p = population proportion

n = sample size

Therefore, this will be illustrated as:

✓p(1 - p)/n = 0.0337

Remove the square root

p( 1 - p)/200 = 0.00113569

Cross multiply

p(1 - p) = 0.227138

p - p² = 0.227138

Equate to 0

p² - p + 0.227138 = 0

Using almighty formula, P1 = 0.6512 and P2 = 0.3488

Therefore, the correct option is D.

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If the standard error of the sampling distribution of the sample proportion is .0337 for samples of size 200, then the population proportion must be:

A) 0.25

B) 0.75

C) 0.20 or 0.80

D) 0.35 or 0.65

Answer:

56

Step-by-step explanation:

[tex]\binom{8}{3}=\frac{8!}{5!3!}=56[/tex]

Given:

(12, 8) and (X, -4 )

Required:

To calculate the value of x

Explanation:

Determine the parameter through distance formula

Required answer:

x=7 or x=17

Answer:

The triangle shown is an isosceles triangle. It has two equal sides and the base angels are equal.

5x - 7 = 8x - 55

3x = 48

x = 16

The nearest half of 7.68 is 7.5  

Given,

A number is given by:

7.68

and, to find the rounded to the nearest half

Now, The number 7.68 is  between 7.5 and 8.

Let's find which of these two numbers 7.68 is closer too.

We can do this by subtracting 7.68 from each of those numbers:

8 - 7.68 = 0.32

7.5 - 7.68 = -0.18

So, 7.68 is 0.32 away from 8, but it's only 0.18 away from 7.5

Hence, The nearest half of 7.68 is 7.5  

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Using transformations, more specifically a translation, the functions that could represent the one graphed are given as follows:

We first consider the parent square root function, defined as follows:

y = sqrt(x).

It's domain, that is, the possible values that the input x can assume, is given as follows:

x ≥ 0.

The graphed function has the same domain as the parent square root function, just with a different range, as it was shifted down, assuming the following format:

y = sqrt(x) - a.

In a translation, the function keeps the same format and orientation, just changing it's position, which is what happened in this case.

The possible translations are:

The other functions would change the domain of the function, hence they are not possible.

The information to solve this problem is given by the image at the end of the answer.

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The proper subset relationship of the given set A, B and C are in option;

For the given question;

The set are given as;

A = { x | x = 2n + 1, n ∈ R }

This, can be written as for the real number say, n = 0, 1, 2, 3....

A = {1, 5, 7, 9 , 11 ..}

B = { x | x = 4n + 1, n ∈ R }

This, can be written as for the real number say, n = 0, 1, 2, 3....

B = {1, 5, 9, ....}

C = { x | x = 8n + 1, n ∈ R }

This, can be written as for the real number say, n = 0, 1, 2, 3....

C = {1, 9, 17...}

Thus, it is clear from the above relation that,

C contain B and A while B contain some element of A.

Thus, C ⊂ A,  C ⊂ B and B ⊂ A are the correct relationship of the proper subset.

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An estimate of the quotient is in improper fraction [tex]\frac{133}{60}[/tex]. An estimate of the quotient is in mixed fraction [tex]2 \frac{13}{60}$$[/tex]

[tex]6 \frac{1}{3} \div 2 \frac{6}{7}$$[/tex]

Convert mixed numbers to improper fractions: [tex]$\quad 6 \frac{1}{3}=\frac{19}{3}$[/tex]

[tex]=\frac{19}{3} \div 2 \frac{6}{7}$$[/tex]

Convert mixed numbers to improper fractions: [tex]$2 \frac{6}{7}=\frac{20}{7}[/tex]

[tex]$=\frac{19}{3} \div \frac{20}{7}$[/tex]

Apply the fraction rule: [tex]$\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \times \frac{d}{c}$[/tex]

[tex]=\frac{19}{3} \times \frac{7}{20}$$[/tex]

Multiply fractions: [tex]$\frac{a}{b} \times \frac{c}{d}=\frac{a \times c}{b \times d}$[/tex]

[tex]=\frac{19 \times 7}{3 \times 20}$$[/tex]

Multiply fractions:[tex]\frac{a}{b} \times \frac{c}{d}=\frac{a \times c}{b \times d}$[/tex]

[tex]=\frac{19 \times 7}{3 \times 20}$$[/tex]

Multiply the numbers: [tex]$19 \times 7=133$[/tex]

[tex]=\frac{133}{3 \times 20}$$[/tex]

Multiply the numbers: [tex]$3 \times 20=60$[/tex]

[tex]=\frac{133}{60}$$[/tex]

Convert improper fractions to mixed numbers: [tex]$\frac{133}{60}=2 \frac{13}{60}$[/tex]

[tex]=2 \frac{13}{60}$$[/tex]

When it comes to adding Mixed or Improper fractions, we can have either the same denominators for both the fractions to be added or the denominators can differ too.

A mixed fraction is one that is represented by both its quotient and remainder. A mixed fraction is, for instance, 2 1/3, where 2 is the quotient and 1 is the remainder. An amalgam of a whole number and a legal fraction is a mixed fraction.

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1.

[tex] = (65 \times 12) - 92 + 13 \\ =780 - 92 + 13 \\ = (780 - 92) + 13 \\ = 688 + 13 \\ = 701 [/tex]

2.

[tex] = \frac{4}{2} + 19 -(5 \times 11) \\ = (2 + 19 )- 55 \\ = 21 - 55 \\ = - 34[/tex]

ATTACHED ARE THE SOLUTIONS