Match each given function with the description of its graph.
y-intercept at (0,2)
graph approaches 0 as x increases
y-interceptat (0,2)
graph approaches negative infinity as x increases
y-intercept at (0,1)
graph approaches positive infinity as x increases
y-intercept at (0,2)
graph approaches positive infinity as x increases
y-intercept at (0,1)
graph approaches 0 as x increases
y = 35
y =
y =

Answer:

\(\frac{2}{3} x + \frac{1}{3} - \frac{1}{3} x = \frac{1}{3} ( x - 1)\\\\ \\\frac{4}{3}x - \frac{4}{3} - \frac{2}{3} = \frac{4}{3} ( x - 1)\\\\( \frac{1}{3} x + \frac{2}{3} ) + ( -\frac{2}{3}x - \frac{4}{3}) = -\frac{1}{3} ( x + 2)\\\\(\frac{2}{3}x + \frac{1}{3} ) + (\frac{2}{3} x - \frac{2}{3} ) = \frac{1}{3} ( 4x - 1 )\)

Step-by-step explanation:

Simplify each equation by combining like terms, and then factoring

\(\frac{2}{3} x + \frac{1}{3} - \frac{1}{3} x\)

= \(\frac{1}{3} x + \frac{1}{3}\)

= \(\frac{1}{3}\) ( x + 1)

\(\frac{4}{3}x - \frac{4}{3} - \frac{2}{3}\)

= \(\frac{4}{3}x - \frac{6}{3}\)

= \(\frac{4}{3}\) ( x - 1 )

\(\ ( \frac{1}{3} x + \frac{2}{3} ) + ( -\frac{2}{3}x - \frac{4}{3})\)

=\(-\frac{1}{3}x - \frac{2}{3}\)

=\(- \frac{1}{3} x\) ( x + 2)

\(\ (\frac{2}{3}x + \frac{1}{3} ) + (\frac{2}{3} x - \frac{2}{3} )\)

= \(\ \frac{4}{3}x - \frac{1}{3}\)

= \(\(\frac{1}{3}\) ( 4x - 1)

Answer:

Step-by-step explanation:

\(1)\frac{2}{3}x + \frac{1}{3}-\frac{1}{3}x = \frac{2}{3}x-\frac{1}{3}x +\frac{1}{3}\\\\\\=\frac{(2-1)}{3}x+\frac{1}{3}\\\\\\=\frac{1}{3}x+\frac{1}{3}\\\\\\=\frac{1}{3}(x+1)\)

\(2) \frac{4}{3}x-\frac{2}{3}-\frac{2}{3}=\frac{4}{3}x + \frac{(-2-2)}{3}\\\\\\\)

                    \(=\frac{4}{3}x -\frac{4}{3}\\\\\\=\frac{4}{3}(x-1)\)

\(3) (\frac{1}{3}x+\frac{2}{3})+(\frac{-2}{3}x-\frac{4}{3})=\frac{1}{3}x-\frac{2}{3}x+\frac{2}{3}-\frac{4}{3}\)

                                     \(= \frac{-1}{3}x-\frac{2}{3}\\\\\\=\frac{-1}{3}(x+2)\\\)

4) \((\frac{2}{3}x +\frac{1}{3})+(\frac{2}{3}x-\frac{2}{3})=\frac{2}{3}x+\frac{2}{3}x+\frac{1}{3}-\frac{2}{3}\)

                                    \(=\frac{4}{3}x-\frac{1}{3}\\\\\\=\frac{1}{3}(4x-1)\)

Answer:

a = 28 / √2

Step-by-step explanation:

Isosceles triangle therefor a = b

use pythagorean theorem

a² + a² = c²

2a² = 28²

a² = 28² / 2

Take the square root of both sides

a = 28 / √2

Answer:

a = \(\frac{28}{\sqrt{2} }\)

Step-by-step explanation:

Using the sine ratio in the right triangle and the exact value

sin45° = \(\frac{1}{\sqrt{2} }\) , then

sin45° = \(\frac{opposite}{hypotenuse}\) = \(\frac{a}{28}\) = \(\frac{1}{\sqrt{2} }\) ( cross- multiply )

a × \(\sqrt{2}\) = 28 ( divide both sides by \(\sqrt{2}\) )

a = \(\frac{28}{\sqrt{2} }\)

A line segment that measures 92 millimeters in length can be drawn using scale.

Given that,

A line segment has to be drawn which has a measure of length 92 millimeters.

We know that,

10 millimeters = 1 centimeter

1 millimeter = 1/10 centimeters

92 millimeters = 92/10 = 9.2 centimeters

So it is enough to draw a line segment of length 9.2 centimeters.

In the scale, between a centimeter, there are 9 small lines which indicates the millimeters.

In between 9 and 10, there are 9 lines which indicates, 9.1, 9.2, 9.3, ....., 9.9 and after that is 10 cm.

So draw a line segment starting from 0 to 9.2.

Hence the line segment is drawn with scale.

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Answer:

a) 9676.1053731 rad/min

b) 1540 rpm

Step-by-step explanation:

A truck with 30-inch diameter wheels is traveling at 70 mi/h. Find the angular speed of the wheels in radians per minute. ___ radians per minute How many revolutions per minute do the wheels make? (Round your answer to three decimal places.) ____ revolutions per minute

truck with 30-in.-diameter wheels is traveling at 70 mi/h.

---------

1 mile per hour = 88 feet per minute

70 mi/hr = 6160ft/minute

----------

Each revolution = pi*d feet = 4pi feet

(b) How many revolutions per minute do the wheels make?

rev/min

-------------

Each revolution = pi*d feet = 4pi feet

rpm = 6160 ft/min / 4pi feet = 990/pi rpm

=~ 1540 rpm

---------------

(a) Find the angular speed of the wheels in rad/min.

rad/min

-------------

rad/min = rpm*2*pi

= 1540 rpm × 2× π

= 9676.1053731 rad/min

Answer:

An arithmetic sequence with a common difference of −2 is 20,18,16,14,12..

Step-by-step explanation:

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is same. Here, the common difference is -2, which means that each term in the sequence is obtained by subtracting 2 from the previous term.

To find the arithmetic sequence with a common difference of -2, you can start with an first term and then subtract 2 successively to find the subsequent terms.

Let the initial term is 20. Subtracting 2 from 20, we get 18. Subtracting 2 from 18, we get 16. Continuing this pattern, we subtract 2 from each subsequent term to generate the sequence. The arithmetic sequence with a common difference of -2 starting from 20 is

20,18,16,14,12

In this sequence, each term is obtained by subtracting 2 from the previous term, resulting in a common difference of -2.

After calculating the value of CB, we get a result of 3.45 cm. This means that the segment CB is 3.45 cm, given that the angle α is 25° and the segment AB is 8 cm.

CB = 8 cm * tan(25°)

≈ 3.45 cm

To find the segment CB, we need to use the tangent of the angle α (25°). We also need to know the length of the segment AB, which is 8 cm. To solve for CB, we use the equation CB = 8 cm * tan(25°). After calculating the value of CB, we get a result of 3.45 cm. This means that the segment CB is 3.45 cm, given that the angle α is 25° and the segment AB is 8 cm.

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An irrational number is a real number which cannot be represented as a simple fraction.

The given statement is the definition of an irrational number.

The probability is P(k) = (q^(k-1)) * p for k>=1, and P(0) = q. The expected number of times the loop will be executed is 1/p.The expected number of times that the repeat loop will be executed is 1/p.

To find the probability that the loop will be executed k times, we can consider the probability of the event that B is false k-1 times followed by B being true. This probability is q^(k-1) * p.

The event of the loop not being executed at all corresponds to B being true in the first trial, which has a probability of q. Therefore, the probability that the loop will be executed k times is P(k) = (q^(k-1)) * p for k>=1, and P(0) = q.

The expected number of times the loop will be executed is the sum of the probabilities of executing the loop k times, weighted by k, i.e., E = Sum(kP(k)) for k>=1, and E = 0 if P(0) = q.

By using the expression for P(k), we can simplify this to E = Sum(kq^(k-1)*p) for k>=1, and E = 0 if P(0) = q. By applying the formula for the sum of a geometric series, we get E = 1/p.

For the "repeat S until B" loop, the expected number of times that the loop will be executed is the expected number of trials in a Bernoulli process until the first success, where the success probability is p. By using the formula for the expected value of a geometric distribution, we get E = 1/p.

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Answer:

50 years

Step-by-step explanation:

a century is 100 years so half of that would be 50

The lower of cost or market (LCM) method states that when valuing a company's inventory, it is recorded on the balance sheet at either the historical cost or the market value. Historical cost refers to the cost at which the inventory was purchased.

The lower of cost or market rule states that a business must record the cost of inventory at whichever cost is lower – the original cost or its current market price.

The value of a good can shift over time. LCM holds significance, because if the price at which the inventory can be sold falls below the net realizable value of the item, thus triggering a loss for the company, then the lower of cost or market method can be employed to record the loss.

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Answer:

\(\textsf{A.} \quad (2+x)^x\left[\dfrac{x}{2+x}+\ln(2+x)\right]\)

Step-by-step explanation:

Replace f(x) with y in the given function:

\(y=(x+2)^x\)

Take natural logs of both sides of the equation:

\(\ln y=\ln (x+2)^x\)

\(\textsf{Apply the log power law to the right side of the equation:} \quad \ln a^n=n \ln a\)

\(\ln y=x\ln (x+2)\)

Differentiate using implicit differentiation.

Place d/dx in front of each term of the equation:

\(\dfrac{\text{d}}{\text{d}x}\ln y=\dfrac{\text{d}}{\text{d}x}x\ln (x+2)\)

First, use the chain rule to differentiate terms in y only.

In practice, this means differentiate with respect to y, and place dy/dx at the end:

\(\dfrac{1}{y}\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}}{\text{d}x}x\ln (x+2)\)

Now use the product rule to differentiate the terms in x (the right side of the equation).

\(\boxed{\begin{minipage}{5.5 cm}\underline{Product Rule for Differentiation}\\\\If $y=uv$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}\)

\(\textsf{Let}\; u=x \implies \dfrac{\text{d}u}{\text{d}x}=1\)

\(\textsf{Let}\; v=\ln(x+2) \implies \dfrac{\text{d}v}{\text{d}x}=\dfrac{1}{x+2}\)

Therefore:

\(\begin{aligned}\dfrac{1}{y}\dfrac{\text{d}y}{\text{d}x}&=x\cdot \dfrac{1}{x+2}+\ln(x+2) \cdot 1\\\\\dfrac{1}{y}\dfrac{\text{d}y}{\text{d}x}&= \dfrac{x}{x+2}+\ln(x+2)\end{aligned}\)

Multiply both sides of the equation by y:

\(\dfrac{\text{d}y}{\text{d}x}&=y\left( \dfrac{x}{x+2}+\ln(x+2)\right)\)

Substitute back in the expression for y:

\(\dfrac{\text{d}y}{\text{d}x}&=(x+2)^x\left( \dfrac{x}{x+2}+\ln(x+2)\right)\)

Therefore, the differentiated function is:

\(f'(x)=(x+2)^x\left[\dfrac{x}{x+2}+\ln(x+2)\right]\)

\(f'(x)=(2+x)^x\left[\dfrac{x}{2+x}+\ln(2+x)\right]\)

Answer:

The answer is n = 13

Step-by-step explanation:

1) Divide both sides by 3.

\( \frac{3}{24} = n - 5 \)

2) Simplify 24/3 to 8.

\(8 = n - 5\)

3) Add 5 to both sides.

\(8 + 5 = n\)

4) Simplify 8 + 5 to 13.

\(13 = n\)

5) Switch sides.

\(n = 13\)

Therefor, the answer for this equation is n = 13.

Check the picture below.

Michelle biked approximately 37.68 miles in the shape of a circle.

A circle's diameter and radius are connected because the diameter is twice as long as the radius. Mathematically, this connection may be written as d = 2r, where d stands for the diameter and r for the radius. On the other hand, we can alternatively state that the radius is equal to half the diameter's length: r = d/2

The circumference of a circle can be found using the formula:

C = πd

Given that,

Michelle biked around the trail 4 times.

The circumference is:

C = π x 3 = 9.42 miles

Also,

4 x 9.42 = 37.68 miles

Hence, Michelle biked approximately 37.68 miles in the shape of a circle.

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Answer:

4

Step-by-step explanation:

50 is 12\(\frac{1}{2}\)% of 400.

Given is the expression as -

50 is 12\(\frac{1}{2}\)% of {x}.

We can write the expression as -

50 = (25/2)/100 {x}

50 = (25/2) x (1/100) {x}

50 = {x}/8

{x} = 400

Therefore, 50 is 12\(\frac{1}{2}\)% of 400.

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Answer:About sixteen minutes.

Step-by-step explanation: After you find out that Ten minutes is six degrees you divide ten by six and get 1.666. Then you multiply that by the time remaining and get 16 minutes rounded. or seventeen if you are specific.

Answer:

\(x = \frac{3}{4} + \frac{ \sqrt{105} }{4} \: \: , \: \: \: x = \frac{3}{4} - \frac{ \sqrt{105} }{4} \\ \)

Step-by-step explanation:

2x² - 3x - 12 = 0

Using the quadratic formula which is

\(x = \frac{ - b \pm\sqrt{ {b}^{2} - 4ac} }{2a} \\ \)

From the question

a = 2 , b = - 3 , c = - 12

So we have

\(x = \frac{ - - 3\pm \sqrt{ ({ - 3})^{2} - 4(2)( - 12)} }{2(2)} \\ = \frac{3\pm \sqrt{9 + 96} }{4} \\ = \frac{3\pm \sqrt{105} }{4} \: \: \: \: \: \: \)

Separate the solutions

We have the final answer as

\(x = \frac{3}{4} + \frac{ \sqrt{105} }{4} \: \: , \: \: \: x = \frac{3}{4} - \frac{ \sqrt{105} }{4} \\ \)

Hope this helps you

The expected value of a $2 ticket is $0.63

To calculate the expected value of a $2 ticket, we need to multiply the probability of winning by the value of the prize, then sum those products:

The Probability of winning the grand prize = 1/2000

Product: (1/2000) x $500 = $0.25

The Probability of winning a secondary prize: 3/2000

Product: (3/2000) x $200 = $0.30

The Probability of winning a third level prize: 8/2000

Product: (8/2000) x $20 = $0.08

The sum of the products:

$0.25 + $0.30 + $0.08 = $0.63

So the expected value of a $2 ticket is $0.63, meaning that on average, a person can expect to win $0.63 for every $2 they spend on a ticket.

The charity is expected to make a profit, which is why the answer is negative.

That is -$2 + $0.63 = -$1.37

So the expected profit for the charity per ticket sold is -$1.37, which is approximately the same as the given answer of negative $1.358(rounding off)

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9514 1404 393

Answer:

  20

Step-by-step explanation:

The segment sum theorem tells you ...

  GH +HI = GI

  8 +12 = GI = 20

The football is approximately 96.4 feet from the base of the goal post.

The tangent function in trigonometry is used to determine the proportion between the lengths of the adjacent and opposite sides in a right triangle. Where theta is the angle of interest, the tangent function is defined as:

tan(theta) = opposing / adjacent.

When the lengths of one side and one acute angle are known, the tangent function is used to solve for the unknown lengths or angles in right triangles. In order to utilise the tangent function, we must first determine the angle of interest, name the triangle's adjacent and opposite sides in relation to that angle, and then calculate the ratio of those sides using the tangent function.

Given, the angle of elevation is 16 degrees 42'.

That is,

Angle of elevation = 16 degrees 42' = 16 + 42/60 = 16.7 degrees

Using tangent function we have:

tan(16.7) = 30/x

x = 30 / tan(16.7)

x = 96.4 feet

Hence, the football is approximately 96.4 feet from the base of the goal post.

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The slope-intercept equation of the line with a slope of 8 and a y-intercept of (0, 9) is y = 8x + 9.

To find the slope-intercept equation of a line, we use the form y = mx + b, where m represents the slope and b represents the y-intercept.

Given:

Slope (m) = 8

Y-intercept (0, 9)

We can substitute these values into the slope-intercept form:

y = mx + b

y = 8x + 9

Consequently, y = 8x + 9 represents the slope-intercept equation for the line with a slope of 8 and a y-intercept of (0, 9).

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The written description of x - 14 is given as the difference of variable x and integer 14 gives x -14.

Calculate the answer using a math operator is referred to as a mathematical operation.

Basic mathematical operations are addition, multiplication, subtraction and division.

Let x be a variable,

And 14 is an integer.

By using mathematical operations,

The difference of x and 14 can be given as,

x - 14.

The required expression x - 14 is the difference of variable x and 14.

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Answer:

Margin of Error decreases

Pvalue decreases

Power increases

Step-by-step explanation:

The margin of error decreases as sample size increases given the same level of confidence, hence, the interval gets narrower.

The margin of error = (Zcritical * σ/√n) ; where, n is the denominator, as the denomination increases, the obtained value will decrease, hence, the margin of error.

When we have a false H0, then we expect a statistical result which will reject H0, with all facts remaining unchanged , increase in sample size n, will lead to decrease in Pvalue, because a lower P value increases the significance of statistical test needed to reject H0.

The power of a fixed level test when the null hypothesis is true, higher power is required to reject the null , increasing n will increase the probability of rejecting H0, by increasing the power of a fixed level test.

The expansion of the function \((3x - 4)^4\) simplifies to \(81x^4 - 432x^3 + 864x^2 - 768x + 256.\)

To expand the function \(f(x) = (3x - 4)^4\), we can use the binomial theorem. According to the binomial theorem, for any real numbers a and b and a positive integer n, the expansion of \((a + b)^n\) can be written as:

\((a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^{(n-1)} b^1 + C(n, 2)a^{(n-2)} b^2 + ... + C(n, n-1)a^1 b^{(n-1)} + C(n, n)a^0 b^n\)

where C(n, k) represents the binomial coefficient, which is given by C(n, k) = n! / (k!(n-k)!).

Applying this formula to our function \(f(x) = (3x - 4)^4\), we have:

\(f(x) = C(4, 0)(3x)^4 (-4)^0 + C(4, 1)(3x)^3 (-4)^1 + C(4, 2)(3x)^2 (-4)^2 + C(4, 3)(3x)^1 (-4)^3 + C(4, 4)(3x)^0 (-4)^4\)

Simplifying each term, we get:

\(f(x) = 81x^4 + (-432x^3) + 864x^2 + (-768x) + 256\)

Therefore, the expanded form of the function \(f(x) = (3x - 4)^4\) is \(81x^4 - 432x^3 + 864x^2 - 768x + 256\).

Note that the coefficient of \(x^3\) is -432, the coefficient of \(x^2\) is 864, the coefficient of x is -768, and the constant term is 256.

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Note the complete question is

Hello!

x² - 5x + 6

= (x² - 2x) + (-3x + 6)

= x(x - 2) - 3(x - 2)

= (x - 2)(x - 3)

Hey there! :)

Answer:

A = 10 units².

Step-by-step explanation:

To solve this, we need to find the distance between the two points to derive the side-lengths of the square. Use the distance formula:

\(d = \sqrt{(x_2 - x_1)^2 + (y_2-y_1)^2}\)

Plug in points into the formula to find the distance:

\(d = \sqrt{(3 - 2)^2 + (4-1)^2}\)

Simplify:

\(d = \sqrt{(1)^2 + (3)^2}\)

\(d = \sqrt{(1) + (9)}\)

\(d = \sqrt{10}\)

Find the area of the square using the formula A = s² where s = √10:

A = (√10)²

A = 10 units².

Answer:

10

Step-by-step explanation:

We use the distance formula to find the distance between the two points, which is the side length of the square.. Therefore, the area of the square is 10.

Answer:

20

Step-by-step explanation:

12.50-8.00=3.00

The number of boxes of cookies that makes the cost equal is 6.

An equation is a mathematical statement containing two expressions on either sides which are connected with an equal to sign.

For the first store,

Cost for 5 boxes of cookies = $12.50

Cost for 1 box of cookie = 12.50 / 5 = $2.50

Cost for x box of cookies = 2.50x

For the second store,

Cost for 4 boxes of cookies = $8

Cost for 1 box of cookie = $8 / 4 = $2

Cost for x box of cookies = 2x

But you have to buy a gallon of milk for $3.00 to get that deal.

Total cost for x boxes of cookies = 2x + 3

Hence the cost model equations are :

C(x) = 2.50x and C(x) = 2x + 3

Cost will be equal when,

2.5x = 2x + 3

0.5x = 3

x = 6

Hence 6 boxes of cookies make both cost equal.

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Answer:

r=ln3/10

Step-by-step explanation:

The daily rate is given by 3 to the power x divided by 10.

We have given that a sample of bacteria is growing at a continuously compounding rate.

The sample triples in 10 days.

The daily rate of the bacteria growth is. exponential growth is in the form

y = abˣ

where y, x are variables.

a is the initial value

Sample triples, in 10 days. therefore the value of b=3.

Therefore we get,

\(y=3a^{\frac{x}{10}\)

\(y=a3^{\frac{x}{10}\)

Therefore the daily rate is given by \(3^{\frac{x}{10}\).

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Answer:

She will have paid $168 not including the original $75 and $243 with.

Using it's formula, the variance of the data-set is of 15 units squared.

For this problem, the mean is given by:

Mean = (3 + 4 + 5 + 7 + 10 + 12 + 15)/7 = 8.

Hence the variance is given as follows:

Variance = 1/8 x [(3-8)² + (4-8)² + (5-8)² + (7-8)² + (10-8)² + (12-8)² + (15-8)²] = 15.

Hence the variance of the data-set is of 15 units squared.

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