1. f(x) = x2 + x and g(x) = 2x + 1
a. Evaluate f(g(3))
b. Evaluate g(f(3))

Figure J is rotated 90° clockwise and then translated by 3 units toward the right.

A point, line, or mathematical figure can be converted in one of four ways, and each has an effect on the object's structure and/or position.

Rotation does not change the shape and size of the geometry. But changes the orientation of the geometry.

The translation does not change the shape and size of the geometry. But changes the location.

Figure J is rotated 90° clockwise and then translated by 3 units toward the right.

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The calculated value of k on the line is 9

From the question, we have the following parameters that can be used in our computation:

XZ is the perpendicular bisector of segment WY

This means that

WX = XY

substitute the known values in the above equation, so, we have the following representation

3k - 4 = 2k + 5

So, we have

3k - 2k = 4 + 5

Evaluate

k = 9

Hence, the value of k is 9

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Answer:

$0.001967/gallon

Step-by-step explanation:

John uses alot of water.... and it sure is cheap.

19.67/10000 =

Answer:

72, 504, 172

Step-by-step explanation:

Let the unknown number be denoted as "x"

The second number = 7x

And the third number = 100 + x

(x) + (7x) + (100 + x) = 748

x + 7x + x + 100 = 748

9x + 100 = 748

9x = 748- 100

9x = 648

x = 648/9

x = 72

The first number = 72

The second number = 7x = 7 × 72 = 504

The third number = 100 + x = 100 + 72 = 172

The smallest integer is 10, the middle integer is 11, and the largest integer is 12.

What is the consecutive integers in algebra?

In algebra, consecutive integers are a sequence of numbers that follow each other in order, where each number is one more than the previous number.

Let's call the smallest integer "x". Then, the middle integer is "x + 1" and the largest integer is "x + 2".

Now, using the given information, we can set up an equation:

x + 2(x + 1) = 56 - 3(x + 2)

Expanding the equation and simplifying:

3x + 2 = 56 - 3x - 6

6x = 62

x = 10

So, the three consecutive integers are 10, 11, and 12.

Hence, The smallest integer is 10, the middle integer is 11, and the largest integer is 12.

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Answer:

L = 22

W = 16

Step-by-step explanation:

2L + 2W = 76

L = W + 6

Substitute for L in the first equation

2 (W + 6) + 2W = 76

2W + 12 + 2W = 76

4W = 64

W = 16

Plug W in the second equation to find L

L = W + 6

L = 16 + 6

L = 22

Plug both answers in the first equation to check

First equation =2(22) + 2(16) = 76

Second equation = 22 = 16 + 6

Hope that helps!

The sketch of the front elevation and the side elevation of the prism are added as an attachment

From the question, we have the following parameters that can be used in our computation:

The prism

Using the figure as a guide, we understand that:

The front elevation is a rectangle of 2m by 0.5m

While the side elevation is a rectangle merged with a trapezoid

Next, we draw the elevations (see attachment)

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Answer:

4.735 x 10^9 i believe

Step-by-step explanation:

Answer:2.7

Step-by-step explanation:

Just multily the 2 products

The most appropriate choice for equation of line in slope intercept form will be given by-

2) y intercept of the line is 13 units

3) Equation of line  f(x)

x - 2y = -26

What is equation of line in slope intercept form?

The most general form of equation of line in slope intercept form is given by y = mx + c

Where m is the slope of the line and c is the y intercept of the line.

Slope of a line is the tangent of the angle that the line makes with the positive direction of x axis.

If \(\theta\)  is the angle that the line makes with the positive direction of x axis, then slope (m) is given by

m = \(tan\theta\)

The distance from the origin to the point where the line cuts the x axis is the x intercept of the line.

The distance from the origin to the point where the line cuts the y axis is the y intercept of the line

Here,

2) From the table, it can be seen when x = 0, y = 13

y intercept of the line is 13 units

3) The line passes through (-4, 11) and (0, 13)

Slope = \(\frac{13 - 11}{0-(-4)}\)

          = \(\frac{2}{4}\)

          = \(\frac{1}{2}\\\)

Equation of line f(x)

\(y - 11 = \frac{1}{2}(x - (-4))\)

- \(2y - 22 = x+4\\x - 2y = -22 -4\\x - 2y = -26\)

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Answer:

a) false

b) true

c) true

Step-by-step explanation:

The y-intercept of a function is the point when x = 0

⇒ f(0) = 2|0 + 2| + 3

          = 2|2| + 3

          = 4 + 3

          = 7

From inspection of the graph, the y-intercept of g(x) is (0, 7)

Therefore, the y-intercepts of f(x) and g(x) are the same.

"f(x) has a greater y-intercept" → false

From inspection of the graph, g(1) = 6

⇒ f(-1) = 2|-1 + 2| + 3

          = 2|1| + 3

          = 2 + 3

          = 5

Therefore,  g(1) > f(-1) → true

The average rate of change of function f(x) over the interval a ≤ x ≤ b is given by:

\(\dfrac{f(b)-f(a)}{b-a}\)

From inspection of the graph:

g(0) = 7

g(3) = 4

\(\textsf{average rate of change of g(x)} \implies \dfrac{4-7}{3-0}=-1\)

f(0) = 7  (from part (a))

f(3) = 2|3 + 2| + 3

     = 2|5| + 3

     = 10 + 3

     = 13

\(\textsf{average rate of change of f(x)} \implies \dfrac{13-7}{3-0}=2\)

Function g(x) decreases over [0, 3] at the rate of -1

Function f(x) increases over [0, 3] at the rate of 2

Therefore, "f(x) increases at a faster rate than g(x) over interval [0, 3]" is true

**Please see the attached graph for reference**

The interval [0, 3] is shaded in grey

f(x) is shown in red

The added 2 points are f(-1) and g(1)

Answer:

120

Step-by-step explanation:

Corresponding angles of similar figures are congruent.

The measure for w is 120 degree.

We have ABCD ~ FECG

If two quadrilaterals are similar, it means that their corresponding angles are congruent, and their corresponding sides are proportional.

Then, we can write as

AB/ FE = CB/ EC = CD/ CD = AD/ FG

and, <A= <F,          

<B = <E,

<C= <C,

<D= <G

So, the measure for w is same as angle G.

Thus, the measure for w is 120 degree.

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Answer:

B).

Step-by-step explanation:

Add 8 to half the difference of 6 and 2 then subtract 1.

Answer:

I apologize but I haven't worked for an answer. but my mind is, what is the point of this question? everyone is familiar with the "when am I ever going to use this in life and why do I need to learn this?" but this is a whole new level of w t f.

Answer:

The inequality that represents this situation is 15x + 10y ≥ 1100, since Ezra needs to earn at least $1100 to buy the laptop.

To check if (60, 20) is a solution to this inequality, we need to substitute x = 60 and y = 20 into the inequality and see if it is true:

15(60) + 10(20) ≥ 1100

900 + 200 ≥ 1100

1100 ≥ 1100

Since 1100 ≥ 1100 is true, this means that (60, 20) is a solution to the inequality. This means that if Ezra works 60 hours mowing lawns and 20 hours walking dogs, he will earn enough money to buy the laptop.

The point (60, 20) represents the combination of hours worked for mowing lawns (x) and walking dogs (y) that will result in Ezra earning enough money to buy the laptop. The x and y values are the inputs or variables, while the inequality 15x + 10y ≥ 1100 is the constraint that must be satisfied. The solution (60, 20) is the output or result of the system of inequalities, and represents the combination of hours worked that will satisfy the constraint.

35 is a reasonable prediction of the number of times he will select 2 yellow socks?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 indicates that an event is impossible, and 1 indicates that an event is certain to occur. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

In probability theory, an event is a set of outcomes or a subset of a sample space. In simpler terms, an event is anything that can happen, or any possible outcome of an experiment or observation. An event can be a single outcome, or it can consist of multiple outcomes.

In the given question,

The theoretical probability of drawing two yellow socks with replacement from a bag containing equal numbers of red, yellow, green, blue, and purple socks is:

P(drawing two yellow socks) = P(yellow) * P(yellow) = (1/5) * (1/5) = 1/25

So, the probability of drawing two yellow socks from the bag in any given trial is 1/25.

To predict the number of times the child will select two yellow socks in 175 trials, we can use the formula for the expected value of a discrete random variable:

E(X) = n * p

where E(X) is the expected number of times the event occurs, n is the number of trials, and p is the probability of the event occurring in a single trial.

In this case, n = 175 and p = 1/25. So,

E(X) = 175 * (1/25) = 7

Therefore, a reasonable prediction of the number of times the child will select two yellow socks in 175 trials is 7. Since this prediction is not one of the answer choices, the closest option is 35, which is more than five times the expected value. However, this is within the range of possible outcomes due to the random nature of the experiment.

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Based on the given data, we can Reasonably infer that most knitters use some kind of specialty yarn and that there is a higher preference for boucle yarn compared to other specialty yarns among the surveyed knitters.

Based on the given data, we can make the following inferences:

b. Most knitters use some kind of specialty yarn:

Since the total number of knitters surveyed is 100, and each knitter was asked about their preference for specialty yarn, we can infer that most knitters (if not all) use some kind of specialty yarn. This is because all the surveyed knitters were specifically asked about their preference for specialty yarn, indicating that specialty yarn is commonly used among knitters.

c. More knitters like boucle yarn than other specialty yarns:

According to the data, 40 knitters preferred boucle yarn, which is more than the number of knitters who preferred any other specialty yarn. Therefore, we can infer that among the surveyed knitters, there is a higher preference for boucle yarn compared to other specialty yarns.

The other statements cannot be reasonably inferred from the given data:

a. The data does not provide information about the number of knitters using sequin yarns compared to fuzzy yarns. We only have information about the preference for boucle, mohair, and metallic yarns.

d. The data does not provide information about the preference of 500 knitters or the comparison of boucle yarn to other specialty yarns among a larger population.

e. The data does not provide information about the preference for metallic yarn compared to plain yarn among the surveyed knitters. The data only includes information about the preference for boucle, mohair, and metallic yarns.

In conclusion, based on the given data, we can reasonably infer that most knitters use some kind of specialty yarn and that there is a higher preference for boucle yarn compared to other specialty yarns among the surveyed knitters.

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Answer:

D is the correct answer

Step-by-step explanation:

A is not correct because it does not meet the fourth criteria, B is not correct because it doesnt meet the second criteria, C is not correct because it does not meet the third criteria, D IS CORRECT because it meets the fourth criteria.

The constant term in the binomial expansion of [-3x⁷+ (-2/x)]⁴⁸ is ⁴⁸C₆(3)⁶(-2)⁴² = 3.93446261119 × 10²²

Binomial expansion is the expansion of the trerms of a binomial of the form (a + b)ⁿ = ∑ⁿCₓaˣbⁿ⁻ˣ.

Since we want to find the constant term in the binomial expansion of [-3x⁷+ (-2/x)]⁴⁸, comparing [-3x⁷+ (-2/x)]⁴⁸ with the binomial expansion above, we see that

So, we have that

[-3x⁷+ (-2/x)]⁴⁸ =  ∑⁴⁸Cₓ(3x⁷)ˣ(-2/x)⁴⁸⁻ˣ

= ∑⁴⁸Cₓ(3)ˣx⁷ˣ(-2)⁴⁸⁻ˣ (1/x)⁴⁸⁻ˣ

= ∑⁴⁸Cₓ(3)ˣ(-2)⁴⁸⁻ˣx⁷ˣ(x)ˣ ⁻⁴⁸

= ∑⁴⁸Cₓ(3)ˣ(-2)⁴⁸⁻ˣx⁸ˣ⁻⁴⁸

So, the nth term is ⁴⁸Cₓ(3)ˣ(-2)⁴⁸⁻ˣx⁸ˣ⁻⁴⁸  (1)

For the constant term, we have that the power of x is zero.

So, 8x - 48 = 0

8x = 48

x = 48/8

x = 6

So, substituting x = 6 into the nth term, we have

⁴⁸Cₓ(3)ˣ(-2)⁴⁸⁻ˣx⁸ˣ⁻⁴⁸ = ⁴⁸C₆(3)⁶(-2)⁴⁸⁻⁶x⁸ˣ ⁶⁻⁴⁸

= ⁴⁸C₆(3)⁶(-2)⁴²x⁴⁸⁻⁴⁸

= ⁴⁸C₆(3)⁶(-2)⁴²x⁰

= ⁴⁸C₆(3)⁶(-2)⁴²

= 12271512 × 729 × 4.3980465111 × 10¹²

= 3.93446261119 × 10²²

The constant term is ⁴⁸C₆(3)⁶(-2)⁴² = 3.93446261119 × 10²²

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Answer:

y=3x+1

Step-by-step explanation:

Use the slope formula and slope-intercept form  y = m x + b  to find the equation.

Answer:

(-4,3),(2,-12)

slope = (-12 - 3) / (2 - (-4) = -15/6

Step-by-step explanation:

y = mx + b

slope(m) = -15/6

use either of ur points...(-4,3)...x = -4 and y = 3

now we sub into the formula and find b, the y int

3 = -15/6(-4) + b

3 = 10 + b

3 - 10 = b

-7 = b

so ur equation is : y = -15/6x - 7....but we need it in standard form

y = -15/6x - 7...add 15/6 to both sides

15/6x + y = -7...multiply both sides by 6

15x + 6y = -42....which reduces to 5x + 2y = - 14 <===

The equation passes through point (5,21) and (10,26).

Determine the equation of line passing through point (5,21) and (10,26).

So equation is y = x + 16.

Complete Question

The complete question is shown on the first uploaded image

Answer:

The  confidence interval is  \(64.86<\mu<67\)

Step-by-step explanation:

From the question we are given the following data

   The following heights are

66, 65, 67, 62, 62, 65, 61, 70, 66, 66, 71, 63, 69, 65, 71, 66, 66, 69, 68, 62, 65, 67, 65, 71, 65, 70, 62, 62, 63, 64, 67, 67      

 The  sample size is n  =32

  The confidence level is \(k = 95\)% = 0.95

The mean is evaluated as

          \(\= x = 66+ 65+ 67+ 62+ 62+ 65+ 61+ 70+ 66+ 66+ 71+63+ 69+ 65+ 71+ 66+ 66+ 69+ 68+ 62+ 65+ 67,+\\65+ 71+ 65+ 70+ 62+ 62+ 63+ 64+ 67+ 67 / 32\)

=>   \(\= x = \frac{2108}{32}\)

=>     \(\= x = 65.875\)

The standard deviation is evaluated as

           \(\sigma = \sqrt{ v}\)

Now  

   \(v = ( 66-65.875 )^2+(65-65.875)^2+( 67-65.875)^2+ (62-65.875)^2+ (62-65.875)^2+ (65-65.875)^2+( 61-65.875)^2+ (70-65.875)^2+ (66-65.875)^2+ (66-65.875)^2+ (71+63-65.875)^2+ (69-65.875)^2+ (65-65.875)^2+ (71-65.875)^2+( 66-65.875)^2+ (66-65.875)^2+ (69-65.875)^2+ (68-65.875)^2+ (62-65.875)^2+ (65-65.875)^2+ (67-65.875)^2,+\\(65-65.875)^2+ (71-65.875)^2+ (65-65.875)^2+ (70-65.875)^2+( 62-65.875)^2+( 62-65.875)^2+ (63-65.875)^2+ (64-65.875)^2+ (67-65.875)^2+ (67-65.875)^2 / 32\)

=>\(v= 8.567329\)

=>   \(\sigma = \sqrt{8.567329}\)

=>   \(\sigma = 2.927\)

The level of significance is evaluated as

        \(\alpha = 1 - 0.95\)

        \(\alpha = 0.05\)

The degree of freedom is  evaluated as

    \(Df = n- 1 \equiv Df = 32 -1 = 31\)

The critical values for the level of significance is obtained from the z -table as

      \(t_c = t_{\alpha/2 } , Df = t _{0.05/2}, 31 =\pm 1.96\)

The confidence interval is evaluated as

       \(\mu = \= x \pm t_c * \frac{\sigma }{\sqrt{n} }\)

substituting values

        \(\mu =65.875 \pm 1.96* \frac{2.927}{\sqrt{32} }\)

       \(\mu =65.875 \pm 1.01415\)

=>    \(64.86<\mu<67\)

Answer:

Kindly check explanation

Step-by-step explanation:

Acceptable = a

Fails = f

For machines X, Y, Z ;

Sample space, S = {aaa, aaf, aff, fff, faa, afa, ffa, faf}

Total elements in sample space = 8

B.)

Elements of the set:

ZF={circuit from Z fails},

XA ={circuit from X is acceptable}.

ZF = {aaf, aff, fff, faf}

XA = {aaa, aaf, aff, afa}

C.)

c. Are ZF and XA mutually exclusive?

Check if ZF n XA = ∅

ZF n XA = {aaf, aff}

Hence, ZF and XA are not mutually exclusive

d. Are ZF and XA collectively exhaustive?

To be collectively exhaustive :

ZF u XA = Sample space

ZF u XA = {aaa, aaf, aff, afa, faf, fff}

{aaa, aaf, aff, afa, faf, fff} ≠ sample space

Hence, ZF and ZA are not collectively exhaustive.

e. What are the elements of the sets

C={more than one circuit acceptable},

C = {aaa, aaf, faa, afa}

D={at least two circuits fail}.

D = {ffa, faf, aff, fff}

f. Are C and D mutually exclusive?

Check if C n D = ∅

C n D = ∅

Hence, A and D are mutually exclusive

g. Are C and D collectively exhaustive?

To be collectively exhaustive :

C u D = Sample space

C u D = {aaa, aaf, aff, afa, faf, fff}

{aaa, aaf, faa, ffa, aff, afa, faf, fff} = sample space

Hence, C and D are collectively exhaustive.

1/2x (4x-6)+3(x^2-1)

Solve:

Apply distributive property:

1/2x (4x) + 1/2x (-6) + 3 (x^2) + 3 (-1)

2x^2 - 3x + 3x^2 - 3

Combine like terms

2x^2+3x^2 -3x -3

5x^2 -3x -3

The monthly Payment for the loan is approximately $271.24.

To calculate the monthly payment for the loan, we can use the formula for calculating the monthly payment on an intermediate-term loan. The formula is:

Monthly Payment = (Loan Amount * Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate)^(-Number of Months))

Given:

Loan Amount = $50,000

Interest Rate = 6.5% per year

Number of Years = 5

First, we need to convert the annual interest rate to a monthly interest rate. Since there are 12 months in a year, the monthly interest rate is:

Monthly Interest Rate = (6.5% / 100) / 12 = 0.00541667

Plugging in the values into the formula, we get:

Monthly Payment = ($50,000 * 0.00541667) / (1 - (1 + 0.00541667)^(-5 * 12))

              = $271.24

Therefore, the monthly payment for the loan is approximately $271.24.

The owners of the restaurant want to use the loan to renovate the kitchen and expand the dining room, which they expect will generate additional monthly revenue between $2,000 and $5,000.

If we assume a conservative estimate of $2,000 per month in additional revenue, the loan payment of $271.24 represents approximately 13.56% of the additional revenue. This means that the owners would still have a significant portion of the additional revenue available for other expenses or profit.

On the other hand, if we consider the higher estimate of $5,000 per month in additional revenue, the loan payment would represent only about 5.42% of the additional revenue, leaving a substantial amount of money for other purposes.

Considering these calculations, it appears that taking this loan is a smart business decision. The monthly payment is manageable and leaves a significant portion of the additional revenue for the restaurant's operation and potential profit. However, it is essential for the owners to ensure that the projected additional revenue is realistic and sustainable to cover the loan payment and other business expenses.

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Area of circle of diameter with 13cm is 132.74

What is area of circle?

Given :

Diameter of circle, d = 13cm

Radius of a circle, r = d/2

=13/2

=6.5

Area of circle = πr²

= 3.142 * 6.5 * 6.5

= 132.74

Hence, the Area of circle of diameter with 13cm is 132.74.

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Answer:

Step-by-step explanation: