Susie is three times as older as her sister Jenny. Paula is 7 years younger
than Jenny.

Write an expression in simplest form that represents the sum of their ages.

Answer:

Step-by-step explanation:

If the worker drove a total of 134 miles on Monday, and received $32.13 for the work-related miles, then we can assume that the work-related miles pay rate is constant, and calculate the total pay for all the miles:

Total pay for all miles = Total miles * Pay rate per mile

Since the worker received $32.13 for the work-related miles, and we assume a constant pay rate, we can find the pay rate per mile:

Pay rate per mile = Total pay for all miles / Total miles

= $32.13 / Total work-related miles

Now we can use this pay rate per mile to calculate the percentage of work-related miles:

Percentage of work-related miles = (Pay for work-related miles / Pay rate per mile) / Total miles * 100

Plugging in the given values, we get:

Percentage of work-related miles = (32.13 / (134 - 32.13)) / (134) * 100

= 0.2874 * 100

= 28.74%

Therefore, approximately 28.74% of the worker's total miles driven on Monday were work-related.

The given events A and B are two independent events.

In probability, independent events are those events where the occurrence of one event does not affect the probability of the other event occurring.  

On the other hand, non-independent events are those events where the occurrence of one event affects the probability of the other event occurring.

If A and B are two independent events then

Here we have

A and B are two events

where P(A) = 0.7, P(B) = 0.8, and P(A and B) = 0.56

Using the above formula,

=> P(A and B) = (0.7) (0.8) = 0.56

=> P(A and B) = 0.56

From the data, P(A and B) = 0.56

Therefore,

The given events A and B are two independent events.

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Megan's biking speed was 6.4 mph and her jogging speed was 1.6 mph

A variable is a quantity that may change within the context of a mathematical problem.

Let's use the following variables to represent the unknown speed :

We know that Megan biked for a certain amount of time, and then jogged for one hour longer. Let's use the variable t to represent Megan's biking time, in hours. Then her jogging time is t + 1 hour.

We can use the formula:

distance = rate x time

To create two equations based on the given information.

For the biking distance of 10 miles, we have:

10 = v_bike * t

For the jogging distance of 8 miles, we have:

8 = v_jog * (t + 1)

We also know that Megan's biking speed was 4 times her jogging speed, so we have:

v_bike = 4 * v_jog

We can substitute this equation into the first equation to get:

10 = 4 * v_jog * t

Simplifying this equation, we get:

t = 2.5 * v_jog

We can substitute this equation into the second equation to get:

8 = v_jog * (2.5 * v_jog + 1)

Simplifying this equation, we get:

8 = 2.5 * v_jog^2 + v_jog

Rearranging the terms, we get:

2.5 * v_jog^2 + v_jog - 8 = 0

We can solve this quadratic equation using the quadratic formula:

Finally, we can use the equation v_bike = 4 * v_jog to find Megan's biking speed:

v_bike = 4 * 1.6 mph = 6.4 mph

Therefore, Megan's biking speed was 6.4 mph and her jogging speed was 1.6 mph

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The arc angle QPS is 258 degrees.

A cyclic quadrilateral is a four sided shape(quadrilateral) that can be inscribed into a circle. The sum of angles in a cyclic quadrilateral is 360 degrees.

The opposite angles of a cyclic quadrilateral have a total of. 180 ° . In other words, the opposite angles of a cyclic quadrilateral is supplementary angles.

Therefore,

m∠P = 4x - 1

m∠R = 12x - 27

Hence,

4x - 1 + 12x - 27 = 180

16x - 28 = 180

16x = 180 + 28

16x = 208

x = 208 / 16

x = 13

m∠R = 12(13) - 27

m∠R = 156 - 27

m∠R = 129 degrees

Therefore,

m∠R = 1 / 2 (QPS)

129 = 1 / 2 × x

x = 129 ×2

x = 258 degrees

Hence,

mQPS = 258 degrees

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Answer:

The given expression 3x2 -11(2y+1)+4 is a polynomial expression with two variables, x and y. It can be simplified by distributing the -11 to the terms inside the parentheses and combining like terms:

3x2 -11(2y+1)+4 = 3x2 -22y -11 +4

= 3x2 -22y -7

Therefore, the statement "the expression is a polynomial with two variables" is true, while the statements "the expression is in simplest form" and "the expression has a constant term of 11" are false.

Step-by-step explanation:

The first student Z-score= 1.31, the second student= -0.82, the third student = 2.35, the fourth student = -0.37. None of these scores are unusual.

In statistics, a Z-score (also known as a standard score) is a measure of how many standard deviations a data point is from the mean of a population or sample. The Z-score is calculated by subtracting the mean from the data point and then dividing by the standard deviation.

A Z-score of 0 indicates that the data point is equal to the mean, while positive Z-scores indicate that the data point is above the mean and negative Z-scores indicate that the data point is below the mean.

Z-scores are often used in statistics to standardize data so that it can be compared across different populations or samples. Z-scores can also be used to identify outliers, which are data points that are unusually far from the mean and may skew the analysis of the data.

To find the z-score for each student, we use the formula:

z = (x - μ) / σ

where x is the student's score, μ is the mean test score, and σ is the standard deviation.

For the first student with a score of 1940:

z = (1940 - 1534) / 310

z = 1.31

For the second student with a score of 1280:

z = (1280 - 1534) / 310

z = -0.82

For the third student with a score of 2260:

z = (2260 - 1534) / 310

z = 2.35

For the fourth student with a score of 1420:

z = (1420 - 1534) / 310

z = -0.37

To determine if any of these values are unusual, we can use the rule of thumb that considers any z-score greater than 3 or less than -3 to be unusual.

None of the z-scores calculated above are greater than 3 or less than -3, so we can conclude that none of these scores are unusual.

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Answer:

7x+10

Step-by-step explanation:

2x(3x)+1x(x+10)

6x+1x+10

7x+10

The value of the inverse function of f(x), that is, f⁻¹(33) is 2.

A function takes in values, applies specific operations to them, and produces an output. The inverse function acts, agrees with the outcome, and returns to the initial function. In general, switching the coordinates x and y is how an inverse is calculated. Although not strictly a function, this freshly constructed inverse is a relation.

To guarantee that the original function's inverse will likewise be a function, the original function must be a one-to-one function. Only when every second element matches the first value is a function deemed to be a one-to-one function.

Given that,

f(x)=4x³+1

Also, f⁻¹(33).

Thus, using the input and output corresponding values we have:

f(x) = 4x³ + 1 = 33

x³ = 8

Taking the cube root of both sides, we get:

x = 2

Hence, the value of the inverse function of f(x), that is, f⁻¹(33) is 2.

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The calculated value of probability, P(X≥10) is option b) 0.1666

Probability refers to possibility. The subject of this mathematical discipline is the occurrence of random events. An event is occur by a chance.

Since the given distribution is uniform, the percentage of data set lying between any two consecutive values of X is given by:

[tex]= \frac{100\ \% }{Maximum \ possible\ X - minimum\ possible\ X}[/tex]

[tex]= \frac{100\ \%}{(12\ -\ 0)}[/tex]

[tex]= \frac{100\ \%}{12}[/tex]

[tex]= 8.33\ \%[/tex]

The probability of X having a value that is greater than or equal to 10 i.e.

P(X≥10) is given by:

[tex]= (12- 10) * Percentage \ of data set \ lying\ between \ any two \ values \ of\ x[/tex]

[tex]=2*8.33\%[/tex]

[tex]=16.66\%[/tex]

[tex]=0.1666[/tex]

The calculated value of probability, P(X≥10) is 0.1666.

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Parallelogram, In Euclidean geometry, a parallelogram is a simple quadrilateral with two(2) pairs of parallel sides.

A parallelogram is a geometric object with sides that are parallel to one another in two(2) dimensions. It is a form of a polygon with four(4) sides (sometimes known as a quadrilateral) in which each parallel pair of sides have the same length.

A Quadrilateral is a closed shape and a type of polygon that has four sides, four(4) vertices and four angles. It is formed by joining(connecting) four non-collinear points. The sum of the interior angles of quadrilaterals(4 sides) is always equal to 360 degrees.

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The area of triangle PYR, obtained using the formula for the area of a triangle; A = (1/2)·a·b·sin(C), is about 87.85 square units

The area of a triangle is the product of half multiplied by the base length and the height of the triangle. The height of the triangle is the length of a side of the triangle multiplied by the sine of the included angle between two sides of the triangle.

The dimensions in the triangle indicates that we get;

Length of the side PQ = 36

Length of the side PR = 22

PQ = QM + MP (Segment addition postulate)

MQ ≅ MP (Definition of bisected segment PQ)

MQ = MP (Definition of congruent segments)

PQ = MP + MP (Substitution property)

PQ = 2 × MP

PQ = 36

36 = 2 × MP

2 × MP = 36

MP = 36/2 = 18

MP = 18

Length of the segment, MY = 8

The Pythagorean Theorem indicates that we get;

PY = √(18² + 8²) = 2·√(97)

∠MPY = arctan(8/18) ≈ 23.92°

Therefore; ∠YPR ≈ 23.92°

The area, A, of the triangle PYR can be obtained from the formula;

A = (1/2)·a·b·sin(C)

The area of the triangle PYR is therefore;

A ≈ (1/2) ×  (2·√(97)) × 22 × sin(23.92°) ≈ 87.85

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An exponential function is a function in which the independent variable x appears as an exponent.

a is the y-intercept.

b is the rate of change.

The function is exponential growth when b > 1.

The function is exponential growth when b < 1.

y-intercept is the initial value of the function when x is equal to 0.

Asymptote is the line that the function approaches but does not cross.

In Mathematics, an exponential function can be modeled by using this mathematical expression:

f(x) = ab^x

Where:

In Mathematics, an asymptote can be defined as a line which the graph of a given function approaches but would never touch or cross.

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Answer:    i am pretty sure it is 33

Step-by-step explanation:

The value of x from the calculation is given as 10

We have ∠TUV = ∠STU + ∠UST

(7x+6) = (3x+14) + (2x+12)

open the bracket

7x+6 = 3x+14 + 2x+12

take the like terms

7x + 6 = 3x + 2x + 14 + 12

7x + 6 = 5x + 26

7x - 5x = 26 - 6

2x = 20

Divide through the equation that we have by 2 to get the value of x

2x / 2 = 20 / 2

x = 10

Hence the value of x is given as 10

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Answer:

hole at (- 1, - 4 )

Step-by-step explanation:

If f(x) has terms that cancel , on the numerator/ denominator then a discontinuity  corresponding to that factor is removable and there is a hole.

f(x) = [tex]\frac{x^2-2x-3}{x^2+3x+2}[/tex] ← factor numerator and denominator

    = [tex]\frac{(x-3)(x+1)}{(x+2)(x+1)}[/tex] ← cancel (x + 1) on numerator/ denominator

   = [tex]\frac{x-3}{x+2}[/tex]

this indicates that x + 1 = 0 ( or x = - 1) is a removable discontinuity and the graph has a hole in it.

substitute x = - 1 into the remaining f(x) for y- coordinate of hole

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

coordinates of hole are (- 1, - 4 )

Answer:

Yes, this function has one hole

This occurs at [tex](-1, -4)[/tex]

Step-by-step explanation:

Definition of hole
A hole on a rational function represents the fact that the function approaches the point, but is not actually defined on that precise x value.

Given function
[tex]f(x) = \dfrac{x^2-2x\:-3}{x^2+\:3x\:+\:2}\\[/tex]

Factor the numerator:
[tex]x^2 - 2x - 3 = (x+1)(x - 3)[/tex]

Factor the denominator:
[tex]x^2+\:3x\:+\:2 = (x + 1)(x + 2)[/tex]

Hence

[tex]\begin{aligned}f(x) & = \dfrac{x^2-2x\:-3}{x^2+\:3x\:+\:2}\\\\& = \dfrac{ (x+1)(x - 3)}{(x + 1)(x + 2)}\\\\ &= \dfrac{x - 3}{x+2} \end{aligned}[/tex]

The common factor is   [tex](x + 1)[/tex]

Set this = 0 to find the hole:
[tex]x + 1 = 0 = > x = -1[/tex]

So at[tex]x = -1[/tex] there is a hole.

Substitute this value of x in the factored function to get the y value for the hole

We have
[tex]f(x) = \dfrac{x - 3}{x+2}[/tex]

[tex]\begin{aligned}f(2) &= \dfrac{-1 - 3}{-1 + 2}\\& = \dfrac{-4}{1}\\&= -4\end{aligned}[/tex]

So the hole occurs at   [tex](-1, -4)[/tex]

The number of mangoes she picked is 2.

Number is the concept of a mathematical object used to count, measure, and label. It is an abstract concept, though most familiar in the form of numerals such as "3" or "10". Numbers allow us to compare and manipulate amounts, sequences, shapes, and patterns. They also allow us to create models and make predictions. As a result, they are essential tools in a variety of disciplines, from mathematics and physics to engineering and economics.

Let x be the number of mangoes in each group on Monday. The expression representing the number of mangoes in each group is x + 4/2 = x + 2.

b) Let y be the number of mangoes in each group on Tuesday. The expression representing the number of mangoes in each group is 2y + 2/3 = 2y + 2/3.

c) Let z be the number of mangoes left after subtracting the number of mangoes in each group on Tuesday from the number of mangoes in each group on Monday. The equation representing this information is x + 2 - (2y + 2/3) = z.

d) Substituting the expression for x and y in the equation, we get: (2y + 2/3) + 2 - (2y + 2/3) = z.

Solving the equation, we get z = 2.

Hence, the number of mangoes she picked is 2.

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The amount of money Leticia has after x time periods can be calculated using the formula:

f(x) = P(1 + r)^x

where P is the principal (initial amount invested), r is the annual interest rate (expressed as a decimal), and x is the number of time periods.

Substituting the given values, we get:

f(x) = 200(1 + 0.05)^x

Simplifying this expression, we get:

f(x) = 200(1.05)^x

Therefore, the exponential function that best represents this situation is:

f(x) = 200(1.05)^x

Brian will receive £[tex]1.75[/tex] from Sue buying [tex]500[/tex] grams of berries.

The metric measure for a tiny amount of mass and weight is the gram (g). It weighs the same as one milliliter (or one cubic centimeter) of water. 1,000 grams make to one kilogram (kg). One liter liquid water weighs exactly one kilogram (kg).

Using a scale is the most accurate technique to measure in grams. Kitchen spoons and cups, for example, offer an approximate approximation. Moreover, always have a conversion tool or chart on available so you can estimate grams without a scale.

Brian paid a total of £[tex]15.75[/tex] for [tex]4\frac{1}{2}[/tex] kg of strawberries. To find out the price per gram, we can divide the total cost by the total weight:

Price per gram [tex]=[/tex]Total cost / Total weight

Total weight of strawberries [tex]= 4.5 kg = 4500 grams[/tex]

Price per gram [tex]=[/tex] £[tex]15.75 / 4500[/tex] grams

Price per gram [tex]=[/tex] £[tex]0.0035[/tex] per gram

So, Brian paid £[tex]0.0035[/tex] for every gram of strawberries he picked.

Cost of [tex]500[/tex] grams [tex]=[/tex] [tex]500 grams *[/tex] £[tex]0.0035/gram[/tex]

Cost of [tex]500[/tex] grams [tex]=[/tex] £[tex]1.75[/tex]

Therefore, Sue should pay Brian £[tex]1.75[/tex] for [tex]500[/tex] grams of strawberries.

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Answer:-7

Step-by-step explanation:

|-5 - 4| = -9

|1 + -3|= -2

|-9 - -2| = -7

Answer:

Step-by-step explanation:

[tex]|a-b|-|c+d|=|-5-4|-|1+(-3)|=|-9|-|-2|=9-2=7[/tex]

Absolute value is just the distance from 0 and therefore is always positive.

Answer:-225

Step-by-step explanation:The arithmetic series is given by:

a1 = -1, d=-2, n = 15

where a1 is the first term, d is the common difference, and n is the number of terms.

To evaluate this series, we can use the formula for the sum of an arithmetic series:

Sn = n/2 * [2a1 + (n-1)d]

where Sn is the sum of the first n terms.

Substituting the given values, we get:

S15 = 15/2 * [2(-1) + (15-1)(-2)]

= 15/2 * [-2 - 28]

= 15/2 * (-30)

= -225

Therefore, the sum of the first 15 terms of the arithmetic series with a1 = -1, d=-2 is -225.

Step-by-step explanation:

Let's call the original length of the skirt "l". After cutting 7 inches off the bottom, the remaining length of the skirt is l-7. Then, when Christi adds a 5-inch ruffle, the final length of the skirt becomes:

(l-7) + 5

Simplifying this expression, we get:

l - 2

Therefore, the expression that best represents the final length of the skirt is:

O2 - s

The final function is f(x) =  ( [tex]b^3-4b^2[/tex]+[tex]63b[/tex]-487b)(b+7)+3021.

What is function?

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

Here the given function f(x)=[tex]b^4 + 3b^3-35b^2-46b+28[/tex] and d(x)=b+7.

Now using long division then,

=>           [tex]b^3-4b^2[/tex]+[tex]63b[/tex]-487b

    [tex]\\b+7)\overline{b^4 + 3b^3-35b^2-46b+28 }\\\\[/tex]

    (-)      [tex]{b^4+7b^3}[/tex]

                  [tex]\overline{-4b^3+35b^2+46b-28}[/tex]

    (-)           [tex]-4b^3-28b^2[/tex]

                             [tex]\overline{63b^2-46b+28}[/tex]

     (-)                     [tex]63b^2+441b[/tex]

                                     [tex]\overline{-487b-28}[/tex]

     (-)                             [tex]\underline {-487b-3049}[/tex]

                                                3021

Quotient q(x) =   [tex]b^3-4b^2[/tex]+[tex]63b[/tex]-487b

Remainder r(x) = 3021

Hence the final function is f(x) =  ( [tex]b^3-4b^2[/tex]+[tex]63b[/tex]-487b)(b+7)+3021.

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Answer:

Reflection only

The mirror line must pass through one of the sides of the triangle.

============================================================

Explanation:

The term "invariant" means "does not vary" i.e. "does not change".

An invariant point is fixed in place.

Here are the list of geometric transformations

Some geometry textbooks will mention "glide reflection", but that's really just a combination of translation and reflection. That means we can ignore it.

Let's go through each item to see if we can get two vertices to stay glued in their spot.  

In short we have eliminated: translations, rotations, and dilations. The only thing that works is reflections

Answer:  Large box weight 40 pounds, and small box weight 30 pounds.

Step-by-step explanation:

Let L be the weight of a large box, and S be the weight of a small box.

L + S = 70

50L + 60S = 3800

We change the first equation into the form of L, and we get:

L = 70 - S

And we plugin L = 70 - S into the second equation.

  50(70 - S) + 60S = 3800

3500 - 50S + 60S = 3800

          3500 + 10S = 3800

         -3500             -3500

                       10S = 300

                          S = 30

 L + S = 70

L + 30 = 70

  - 30     - 30

       L = 40

An insurance company's anticipated total profit from a single of these policies is $0.08.

Describe insurance. Insurance is a tool for risk management. You acquire security against unforeseen financial losses when you purchase insurance. If something terrible occurs to you, the insurance company compensates you or someone else of your choosing.

The expected net gain can be calculated as:

E(X) = P(Claim) × Net gain (Claim) + P(No claim) × Net gain (No claim)

P(Claim) = 1/500

P(No claim) = 1 - P(Claim) = 499/500

Net gain (Claim) = -$4960 + $5000 = $40

Net gain (No claim) = $40 - $40 = $0

Therefore,

E(X) = (1/500) × $40 + (499/500) × $0

E(X) = $0.08

So the expected net gain for the insurance company from one of these plans is $0.08.

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Answer:

30 dollars

Step-by-step explanation:

Khan Academy

Answer:

a. X ~ N(6.1, 1.2^2)

b. The median seedless watermelon weight is 6.1 kg.

c. The Z-score for a seedless watermelon weighing 7 kg is 0.75.

d. The probability that a randomly selected watermelon will weigh more than 5.1 kg is 0.7967.

e. The probability that a randomly selected seedless watermelon will weigh between 5.3 and 6 kg is 0.2454.

f. The 85th percentile for the weight of seedless watermelons is 7.2437 kg.

Step-by-step explanation:

a. X ~ N(6.1, 1.2^2)

b. The median of a normal distribution is equal to the mean, so the median seedless watermelon weight is 6.1 kg.

c. The Z-score for a seedless watermelon weighing 7 kg can be calculated as:

Z = (7 - 6.1) / 1.2 = 0.75

Therefore, the Z-score is 0.75.

d. To find the probability that a randomly selected watermelon will weigh more than 5.1 kg, we need to standardize the value using the formula:

Z = (X - μ) / σ

where X is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.

Z = (5.1 - 6.1) / 1.2 = -0.8333

Using a standard normal distribution table or a calculator, we can find the probability that Z is greater than -0.8333 to be 0.7967.

Therefore, the probability that a randomly selected watermelon will weigh more than 5.1 kg is 0.7967.

e. To find the probability that a randomly selected seedless watermelon will weigh between 5.3 and 6 kg, we need to standardize the values and find the area under the normal curve between the two Z-scores. The Z-scores for 5.3 kg and 6 kg are:

Z1 = (5.3 - 6.1) / 1.2 = -0.6667

Z2 = (6 - 6.1) / 1.2 = -0.0833

Using a standard normal distribution table or a calculator, we can find the probability that Z is between -0.6667 and -0.0833 to be 0.2454.

Therefore, the probability that a randomly selected seedless watermelon will weigh between 5.3 and 6 kg is 0.2454.

f. The 85th percentile for the weight of seedless watermelons can be found by finding the Z-score that corresponds to the 85th percentile of a standard normal distribution. Using a standard normal distribution table or a calculator, we can find the Z-score to be 1.0364.

To find the corresponding weight, we can use the formula:

Z = (X - μ) / σ

1.0364 = (X - 6.1) / 1.2

X = 7.2437

Therefore, the 85th percentile for the weight of seedless watermelons is 7.2437 kg.

Hope this helps! I'm sorry if it doesn't. If you need more help, ask me! :]

Answer:

a) N(6.1,1.2)
b) 6.1
c) 0.75
d) 0.7977   (or 79.77%)
e) 0.2143    (or 21.43%)
f) 7.3435 kg (using excel)    or      7.348 kg (using normal tables)

Step-by-step explanation:

a) normal distribution just need to be define we his mean and his standard deviation. You just need a sample higher than 30 or to be specified the population is normally distributed
X≈N(μ,σ)= N(6.1,1.2)

b) The median and mean are not necessarily the same, for a normal distribution, which is a symmetric distribution, those values are just the same


c) The equation for Z score is given by:

[tex]z=\frac{x-u}{\sigma}[/tex]

Replacing values:

[tex]Z=\frac{7-6.1}{1.2} =0.75[/tex]

d) first find the Zvalue for 5.1

[tex]Z=\frac{5.1-6.1}{1.2} =-0.83[/tex]

Now, find the probability from  Normal  Distribution table
P(Z>0.83) = 0.7977

e)

first find the Zvalue fo5.3 and 6

[tex]Z=\frac{5.3-6.1}{1.2} =-0.67[/tex]


[tex]Z=\frac{6-6.1}{1.2} =-0.08[/tex]

Now find probabilities from  Normal Distribution  table . Notice you need to subtract   P(Z>0.08)-P(Z>0.67) if you use a positive table.

P(-0.67<z<-0.08) = 0.2143


f) Find the Z score from a   Normal  Distribution table that give you an area of 0.8500 (or the closest value), im using excel for this one since the answer from tables have only TWO decimals and can be problematic if you need more decimal places.

this gives a Z of 1.036433389

now use the Zscore equation but solve for X

[tex]z\sigma+u=x[/tex]

x = 1.036433389(1.2)+6.1 = 7.3437   (Using excel)

If i use a table, the closes is 1.04 (0.8508 which is not exactly 0.85)

x = 1.04(1.2)+6.1= 7.348 (using table)

so be carefully with the input if the website needs the decimals places or not.

When a variable (or set of letters) in an algebraic statement is substituted, its numerical value is used instead. The expression's total value can then be calculated.

Using the given functions, we have:

[tex]f(x) = x^2 + 3[/tex]

[tex]g(x) =[/tex] [tex]\sqrt{(x-4)}[/tex]

To find f(g(x)), we substitute g(x) for x in the expression for f(x):

[tex]f(g(x)) = (g(x))^2 + 3[/tex]

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

[tex]= (x-4) + 3[/tex]

[tex]= x - 1[/tex]

Therefore,[tex]f(g(x)) = x - 1.[/tex]

To find g(f(x)), we substitute f(x) for x in the expression for g(x):

[tex]g(f(x)) =[/tex] [tex]\sqrt{((f(x)) - 4)}[/tex]

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

= [tex]\sqrt{(x^2 - 1)}[/tex]

= |x|[tex]\sqrt{(x^2 - 1)}[/tex]

Therefore,  [tex]g(f(x)) =[/tex] |x|  [tex]\sqrt{(x^2 - 1)}[/tex]

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12) the common multiples of 3 and 5 up to 50 are: 15, 30, 45.

13) the common multiples of 4 and 6 up to 50 are: 12, 24, 36, 48.

14) the common multiples of 3 and 7 up to 50 are: 12, 42.

15) the equivalent fraction of 3/4 is 6/8

16) the equivalent fraction of 2/3 is 4/6

17) the equivalent fraction of 4/5 is 8/10

18) In its simplest form, 4/12 is 1/3

19) In its simplest form, 6/10 is 3/5

20) In its simplest form, 4/8 is 1/2.

12) To find the common multiples of 3 and 5 up to 50, we can simply list the multiples of 15 (which is the least common multiple of 3 and 5) that are less than or equal to 50.

Multiples of 15: 15, 30, 45

Therefore, the common multiples of 3 and 5 up to 50 are:

15, 30, 45

13)
To find the common multiples of 4 and 6 up to 50, we can again find the least common multiple of 4 and 6, which is 12, and list its multiples that are less than or equal to 50.

Multiples of 12: 12, 24, 36, 48

Therefore, the common multiples of 4 and 6 up to 50 are:

12, 24, 36, 48


14)
To find the common multiples of 3 and 7 up to 50, we can again find the least common multiple of 3 and 7, which is 21, and list its multiples that are less than or equal to 50.

Multiples of 21: 21, 42

Therefore, the common multiples of 3 and 7 up to 50 are:

21, 42

15)

To write an equivalent fraction of 3/4, we can multiply or divide both the numerator and denominator of 3/4 by the same non-zero number. This will give us a fraction that has the same value as 3/4 but with a different numerator and denominator.

For example, if we multiply both the numerator and denominator of 3/4 by 2, we get:

3/4 * 2/2 = 6/8

Therefore, an equivalent fraction of 3/4 is 6/8.

replicating this logic for 16 and 17, we have:

16) the equivalent fraction of 2/3 is 4/6 and

17) the equivalent fraction of 4/5 is 8/10


18) to get the simplest form of 4/12, we can divide the numerator and denominator by 4 to get
(4/4)/(12/4)

= 1/3

replicate this logic for 19 and 20, and we get:

19) In its simplest form, 6/10 is 3/5

20) In its simplest form, 4/8 is 1/2.

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The required possible dimensions of the​ rug are 11   by  [tex]\frac{r^2 - 8r - 33}{11}[/tex].

The area of the rectangular rug is given by the trinomial:

[tex]$r^2 - 8r - 33$[/tex]

To find the possible dimensions of the rug, we need to factor this trinomial. We can do this by finding two numbers that multiply to -33 and add up to -8. These numbers are -11 and 3:

[tex]$r^2 - 8r - 33 = (r - 11)(r + 3)$[/tex]

Therefore, the possible dimensions of the rug are:

[tex]$r - 11 = 0$[/tex] or [tex]$r + 3 = 0$[/tex]

[tex]$r = 11$[/tex]or $[tex]$r = -3$[/tex]

Since the dimensions of the rug cannot be negative, we can only take the positive value of r, which is r = 11.

Therefore, the possible dimensions of the rug are:

11   by  [tex]\frac{r^2 - 8r - 33}{11}[/tex]

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[tex]\begin{array}{llll} \textit{Logarithm Cancellation Rules} \\\\ log_a a^x = x\qquad \qquad \stackrel{ \textit{we'll be using this one} }{a^{log_a (x)}=x} \end{array} \\\\[-0.35em] ~\dotfill\\\\ x^2\log(x)=10x^2\implies \log(x)=\cfrac{10x^2}{x^2}\implies \log(x)=10 \\\\\\ \log_{10}(x)=10\implies 10^{\log_{10}(x)}=10^{10}\implies x=10^{10}\implies x=10000000000[/tex]