let g(x)=x2-1. what is the average rate of change of the function form x=3 to x=6

Answer:

  1a. increasing. It is increasing for all values of t.

  1b. no. The population asymptotically approaches 40,000.

  2. the sign of the slope of a reciprocal function is opposite the sign of the function's slope

Step-by-step explanation:

Given the population function P(t) = 20(4t+3)/(2t+5), you want to know if the population is increasing or decreasing, and if the population will reach 50,000. You want to have an explanation of why the reciprocal of an increasing linear function is decreasing.

The population function can be expanded to ...

  P(t) = 40 -140/(2t+5)

As we know from question 2, the second term is increasing, hence the population function is increasing. (The basic reciprocal function is decreasing, but it is subtracted here, so the overall effect is an increasing function.)

The magnitude of the second term of the above version of the population function starts at 28 for t=0 and decreases asymptotically to zero. Hence the population starts at 12000 and increases to an asymptote of 40,000. It will never reach 50,000.

The basic reciprocal function is f(x) = 1/x. It is decreasing everywhere it is defined. It is the reciprocal of a linear function with positive slope (an increasing function).

Translation or vertical or horizontal scaling of the basic function (using positive scale factors) does not change the sign of the slope, either of the original linear function or of its reciprocal. Hence the reciprocal of an increasing function is decreasing.

Looking at derivatives, if f'(x) is positive, then the derivative of g(x) = 1/f(x) is ...

  g'(x) = (-1/f(x)^2)f'(x)

That is, f(x)^2 is non-negative, and -f'(x) is negative, so the derivative g'(x) must be negative (wherever f(x)≠0). The reciprocal function is decreasing.

[tex](\stackrel{x_1}{36}~,~\stackrel{y_1}{-35})\hspace{10em} \stackrel{slope}{m} ~=~ - \cfrac{7}{12} \\\\\\ \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}{(-35)}=\stackrel{m}{- \cfrac{7}{12}}(x-\stackrel{x_1}{36}) \implies y +35 = - \cfrac{7}{12} ( x -36) \\\\\\ y+35=- \cfrac{7}{12}x+21\implies {\Large \begin{array}{llll} y=- \cfrac{7}{12}x-14 \end{array}}[/tex]

Answer: 50[tex]\sqrt{3\\[/tex] feet

Step-by-step explanation: Tan∅ = height of tower/ base

If sin(tetha) = √5/3, then the value of cos (tetha) =2/3

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.

Sin(tetha) = opp/hyp

cos(tetha) = adj/hyp

tan(tetha) = opp/adj

If sin(tetha) = √5/3, therefore the hypotenuse is 3 and the opposite is √5

Using Pythagoras theorem,

3² = √5)²+x²

x² = 9-5

x² = 4

x = 2

adjascent = 2

therefore the value of cos(tetha) = adj/hyp

= 2/3.

therefore the value of cos(tetha) = 2/3

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The range of the function is -8 < y < -3(option c)

What is range?

The set of all the outputs for a certain inputs of a function is known as the range of the function or after substituting the domain, the entire set of all values possible as outputs of the dependent variable.

Given points on the graph is given by:

(x, y) = (-5, -3)

(x, y) = (10, -8)

A diagonal curve connecting both points.

Removing the x values to determine the range

y = -3

y = -8

-8 is less than -3

So, the range can be represented as

-8 less than y less than -3

So, we have

Range: -8 < y < -3

Hence, the range of the given data is -8 < y < -3.

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x-intercept :(- 4, 0) . The given function is not a example of exponential decay.

We can determine the increase in pollution or the number of deaths using the exponential growth as well as decay equation, respectively. The exponential decay equation is used to calculate the radioactive elements' rate of decay.

There are two categories in which to classify exponential functions. Exponential growth as well as exponential decay are these. The formula for both of these functions is f = abˣ, but only varies in b's value.

The stated function:

f (x) = –3(–x – 3) + 3

x-intercept occurs when, f(x) = 0

0 = –3(–x – 3) + 3

–3(–x – 3) = - 3

-x - 3 = 1

x = -3 - 1

x = -4

x-intercept :(- 4, 0)

The graph for the function is plotted using the graphing tool.

Now, from the graph its is seen that the function forms a straight line in creasing order.

x → ∞, f (x) → 3 linearly.

Thus, the given function is not a example of exponential decay.

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Answer: 249.06

Step-by-step explanation: the area of the circle is 201.06

The area of the rectangle is 48 because 8*6 is 48. Add and you get 249.06

(m + n) has a magnitude of approximately 25.8.

We have,

Consider the two vectors as m and n.

To find (m + n), we need to add the two vectors.

To do this, we first need to determine the components of each vector.

Let's call the angle between vector m and the x-axis α, and the angle between vector n and the x-axis β.

Then we have:

|m| = 10, so the x-component of vector m is m_x = 10 cos α and the

y-component is m_y = 10 sin α.

|n| = 15, so the x-component of vector n is n_x = 15 cos β and the

y-component is n_y = 15 sin β.

We also know that the angle between vectors m and n is 75 degrees. Using the dot product, we can find the cosine of this angle:

m · n = |m| |n| cos 75

m · n = 10 * 15 * cos 75

m · n = 37.32

We also know that the dot product is equal to the sum of the products of the corresponding components:

m · n = (m_x) x (n_x) + (m_y) x (n_y)

37.32 = 10 cos α x 15 cos β + 10 sin α * 15 sin β

Now we have two equations with two unknowns:

m_x + n_x = (10 cos α) + (15 cos β)

m_y + n_y = (10 sin α) + (15 sin β)

We can solve for α and β using the equations we just derived:

m_x + n_x = (10 cos α) + (15 cos β)

10 cos α = (m_x + n_x - 15 cos β)

cos α = (m_x + n_x - 15 cos β) / 10

α = arccos[(m_x + n_x - 15 cos β) / 10]

m_y + n_y = (10 sin α) + (15 sin β)

10 sin α = (m_y + n_y - 15 sin β)

sin α = (m_y + n_y - 15 sin β) / 10

α = arcsin[(m_y + n_y - 15 sin β) / 10]

Now we can substitute these values into the equations for m_x + n_x and m_y + n_y to find (m + n):

m_x + n_x = 10 cos α + 15 cos β

m_y + n_y = 10 sin α + 15 sin β

We can use a graphical method to estimate the values of α and β. Start by drawing vector m with length 10 and angle α with respect to the x-axis. Then draw vector n with length 15 and angle β with respect to the x-axis, starting the tail of n at the head of m.

The parallelogram formed by the two vectors will have a diagonal that represents (m + n). Use a ruler to measure the length of this diagonal to the nearest hundredth.

We can use the law of cosines to find the magnitude of (m + n):

|(m + n)|² = |m|² + |n|² + 2|m||n| cos 75

|(m + n)|² = 10² + 15² + 2(10)(15) cos 75

|(m + n)|² = 665.19

|(m + n)| = 25.8 (to the nearest hundredth)

Therefore,

(m + n) has a magnitude of approximately 25.8.

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The age of the artifact is  13417 years from the calculations above.

Carbon-14 is useful for dating archaeological and geological materials, as well as for studying the carbon cycle in the environment as seen.

We know that;

0.693/t1/12 = 2.303/t ln(Ao/A)

t1/2 = half life

t = time taken

Ao = initial activity

A = activity at time A

Thus;

0.693/5730 = 2.303/t log (Ao/0.2Ao)

1.21 * 10^-4 = 2.303/t log 5

1.21 * 10^-4 = 1.61/t

t = 1.61/1.21 * 10^-4

t = 13417 years

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Answer:

about 149 ft

Step-by-step explanation:

cosine of 43 degrees is about 0.73 (can be found using calculator)

cosine is the adjacent side divided by the hypotenuse, so:

adjacent/hypotenuse=0.73

we know the adjacent is 109 so

109/hypotenuse=0.73

solve for hypotenuse

109/0.73=149.3

so the hypotenuse is about 149 ft

The volume of the rectangular prism is: 567 cubic inches

The volume of the triangular prism is: 183.6 cubic yards

The volume of the rectangular prism is:

Volume = length x width x height

Volume = 9 in x 7 in x 9 in

Volume = 567 cubic inches

It should be noted that to find the volume of a triangular prism, you need to multiply the area of the base by the height of the prism. In this case, the triangular prism has a base that is a triangle with dimensions of 9 yards by 3.4 yards, and a height of 12 yards. So the volume of the triangular prism is:

Volume = (base area) x height

Volume = (1/2 x base x height) x height

Volume = (1/2 x 9 yd x 3.4 yd) x 12 yd

Volume = 183.6 cubic yards

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Answer:cafe- 41

storage- 9

Step-by-step explanation:

Answer:

Since f(-1) = 0 and f(-2) = 0, we can set up two equations:

(-1)³ + p(-1)² - q(-1) - 4 = 0

(-2)³ + p(-2)² - q(-2) - 4 = 0

Simplifying each equation, we get:

-1 + p - q - 4 = 0

-8 + 4p - 2q - 4 = 0

Simplifying further, we get:

p - q = 5

2p - q = 6

Solving for p and q, we can add the two equations together to eliminate q:

p - q + 2p - q = 5 + 6

3p - 2q = 11

Then, we can substitute the value of q from the first equation into this equation to solve for p:

3p - 2(q + 5) = 11

3p - 2q - 10 = 11

3p - 2q = 21

3p - 2(5 + p) = 21

p - 10 = 7

p = 17

Substituting this value of p into either equation for q, we get:

q = p - 5 = 12

Therefore, the coefficients are p = 17 and q = 12. To factor the expression, we can use synthetic division or long division. Using synthetic division, we get:

-1 | 1 17 -12 -4

|__ -1 -16 28

1 1 12

This gives us the factorization:

f(x) = (x + 1)(x² + x + 12)

To find the solutions, we can use the quadratic formula on the quadratic factor:

x = (-1 ± sqrt(1 - 4(1)(12))) / 2(1)

x = (-1 ± sqrt(1 - 48)) / 2

x = (-1 ± sqrt(-47)) / 2

x1 = (-1 + i(sqrt(47))) / 2

x2 = (-1 - i(sqrt(47))) / 2

Therefore, the solutions are x1 = (-1 + i(sqrt(47))) / 2, x2 = (-1 - i(sqrt(47))) / 2, and x3 = -1.

Step-by-step explanation:

give me thanks for more! your welcome bud!

The additional cost that is incurred in dollars when production is increased from 4 units to 20 units is $516.80.

If ​C(4​)=203​,  C(20​) is given as $719.80

To determine the additional cost incurred when production is increased from 4 units to 20 units, we need to first obtain the difference between the cost of producing 20 units and the cost of producing 4 units.

The result is obtained by integration as follows:

C(x) = ∫ C'(x) dx = ∫ (0.09x^2 - 4x + 60) dx = 0.03x^3 - 2x^2 + 60x + C

Note that the constant is 'c'.

The additional cost when there is  an increment from 4 to 20 units will be:

C(20) - C(4)

= (0.03*20^3 - 2*20^2 + 60*20 + C) - (0.03*4^3 - 2*4^2 + 60*4 + C)

= $516.80

To solve the second part of the question, We are given these values:

C(4) = 203.

Also, from the expression we derived in part (a) for C(x), we can solve for the constant of integration C by substituting C(4) = 203 this way:

C(4) = 0.03*4^3 - 2*4^2 + 60*4 + C = 203

C = 83

To obtain C(20) we substitute C = 83 and x = 20 into the expression we derived in part (a) thus

C(20) = 0.03*20^3 - 2*20^2 + 60*20 + 83

= $719.80.

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Answer:

23 yards. Answer choice B is correct.

Step-by-step explanation:

Let's use the formula for the perimeter of a rectangle, which is:

P = 2L + 2W

where P is the perimeter, L is the length, and W is the width.

We know that the perimeter is 82 yards, and the length is 18 yards. Plugging these values into the formula, we get:

82 = 2(18) + 2W

Simplifying the right side of the equation, we get:

82 = 36 + 2W

Subtracting 36 from both sides, we get:

46 = 2W

Dividing both sides by 2, we get:

W = 23

Therefore, the width of the rectangle is 23 yards. Answer choice B is correct.

The upper bound of the 90% confidence interval is 49.044.

To calculate the upper bound of a 90% confidence interval, we first need to find the margin of error using the following formula:

Margin of Error = z * (σ / sqrt(n))

where:

z = the z-score for the level of confidence (90% confidence corresponds to a z-score of 1.645)

σ = the population standard deviation (unknown, so we use the sample standard deviation as an estimate)

n = the sample size (85 in this case)

Substituting the given values, we get:

Margin of Error = 1.645 * (13.5 / sqrt(85))

= 2.644

Next, we can find the upper bound of the confidence interval by adding the margin of error to the sample mean:

Upper bound = sample mean + margin of error

= 46.4 + 2.644

= 49.044

Therefore, the upper bound of the 90% confidence interval is 49.044.

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To solve this problem, we need to use the compound interest formula:

A = P(1 + r/n)^(nt)

where:

A = final amount

P = principal amount

r = annual interest rate (as a decimal)

n = number of times the interest is compounded per year

t = time in years

In this case, we have:

P = $700.00 (principal amount)

r = 4% = 0.04 (annual interest rate)

n = 4 (quarterly compounding)

t = 7 (time in years)

So, substituting the values into the formula:

A = $700.00(1 + 0.04/4)^(4*7)

A = $700.00(1.01)^28

A = $700.00(1.31976058416)

A = $923.83

Therefore, the balance after 7 years would be $923.83.

By answering the presented question, we may conclude that Therefore, cylinder the volume generated by rotating the region bounded by [tex]y = 8x - x^2, y = 12[/tex] about x = 2 is 128π/3 cubic units.

The cylinder, which is frequently a three-dimensional solid, is one of the most fundamental curved geometric shapes. In elementary geometry, it is referred to as a prism with a circle as its basis. A cylinder is also defined as an infinitely curved surface in several modern domains of geometry and topology. A "cylinder" is a three-dimensional object made up of curved surfaces with circular tops and bottoms. A cylinder is a three-dimensional solid figure with two identical circles as bases linked by a curving surface at the cylinder's height, which is defined by the distance between the bases from the centre. Cold beverage cans and toilet paper wicks are examples of cylinders.

We can see that the region is bounded by the parabola [tex]y = 8x - x^2[/tex] and the horizontal line y = 12. It is rotated about the vertical line x = 2.

[tex]h(x) = 12 - (8x - x^2)[/tex]

The radius of the cylindrical shell is the distance from the axis of rotation (x = 2) to the x-value of the slice, which is r(x) = x - 2.

The circumference of the cylindrical shell is 2πr(x), and the thickness is dx. Therefore, the volume of the cylindrical shell at x is:

dV = 2πr(x)h(x)dx

= 2π[tex](x - 2)(12 - (8x - x^2))dx[/tex]

To find the total volume, we integrate dV from x = 0 to x = 4 (the limits of the region):

V = ∫₀⁴ 2π[tex](x - 2)(12 - (8x - x^2))dx[/tex]

= ∫₀⁴ 2π[tex](-x^3 + 12x^2 - 40x + 48)dx[/tex]

= 2π[-¼[tex]x^4 + 4x^3 - 20x^2 + 48x[/tex]] from 0 to 4

= 2π(64/3)

= 128π/3

Therefore, the volume generated by rotating the region bounded by [tex]y = 8x - x^2, y = 12[/tex] about x = 2 is 128π/3 cubic units.

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We may infer after addressing the stated questiοn that  As a result, the prοbability that a prοduct chοsen at randοm wοuld endure between 154 and 180 days is arοund 0.3121.

Calculating the likelihοοd that an event will happen οr that a claim is true is the subject οf prοbability theοry, a branch οf mathematics. A prοbability is a number between 0 and 1, where 0 denοtes hοw prοbable an event is tο οccur and 1 denοtes certainty. An expressiοn οf chance οr likelihοοd expressed in numbers is called a prοbability.

Tο calculate the prοbability, use the usual nοrmal distributiοn fοrmula:

z = (x - μ) / σ

z = (166 - 146) / 19.5 = 1.03

z₁ = (154 - 146) / 19.5 = 0.41\sz₂ = (180 - 146) / 19.5 = 1.74

We can calculate the chance οf a z-scοre falling between 0.41 and 1.74 using a basic nοrmal distributiοn table. As a result, the likelihοοd that a prοduct chοsen at randοm wοuld endure between 154 and 180 days is arοund 0.3121.

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The algebraic expression for Desmond's height is x + 5.

If we let "x" represent Niki's height in inches, then Desmond's height can be represented by "x + 5", since Desmond is 5 inches taller than Niki.

So the algebraic expression for Desmond's height is x + 5.

Algebraic expressions is consist of variables, numbers, and mathematical operations (such as addition, subtraction, multiplication, and division).

They can contain one or more variables and can be evaluated for different values of the variables.

For example, the expression 3x + 5y - 2z is an algebraic expression that contains three variables (x, y, and z) and three coefficients (3, 5, and -2) that are multiplied by those variables. Algebraic expressions are commonly used in algebra to represent mathematical relationships and solve problems.

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Answer:  500 milliliters.

Step-by-step explanation:

There are 1000 milliliters in 1 liter.

To convert 0.5 liters to milliliters, you can multiply 0.5 by 1000.

0.5 liters * 1000 milliliters/liter = 500 milliliters

Therefore, 0.5 liters is equal to 500 milliliters.

Answer:

500

Step-by-step explanation:

I would give a step by step explanation step explanation but I don't want to waste you're time

Answer: A

Step-by-step explanation:

First, let's rewrite the given equation to have the constant term on the right side:

2x^2 + 12x = 32

2x^2 + 12x - 32 = 0

To use the completing the square method, we need to make the coefficient of the x^2 term equal to 1. We can do this by dividing the entire equation by 2:

x^2 + 6x - 16 = 0

Now, we will complete the square for the quadratic expression on the left side. To do this, we take half of the coefficient of the x term (6/2 = 3) and square it (3^2 = 9). Then, we add and subtract this value inside the parenthesis:

x^2 + 6x + 9 - 9 - 16 = 0

Now, the left side of the equation has a perfect square trinomial:

(x^2 + 6x + 9) - 25 = 0

The trinomial can be written as a square of a binomial:

(x + 3)^2 - 25 = 0

Now, let's move the constant term to the right side of the equation:

(x + 3)^2 = 25

The perfect square trinomial for the problem is (x + 3)^2 = 25

The vertices of the resulting image A'B'C'D'E' are A' = (2, -2), B' = (4, -2), C' = (6, -4), D' = (4, -6) and E' = (2, -6)

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

A = (2, 2), B = (2, 4), C = (4, 6), D = (6, 4) and E = (6, 2)

Rule: 90 degrees rotation

The rule of 90 degrees rotation is

(x, y) = (y, -x)

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

A' = (2, -2), B' = (4, -2), C' = (6, -4), D' = (4, -6) and E' = (2, -6)

Hence, the image = A' = (2, -2), B' = (4, -2), C' = (6, -4), D' = (4, -6) and E' = (2, -6)

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Answer:

The mean would be 45. But since that's not a option it might be C. The value of the mean would remain the same at about 46.

Step-by-step explanation:

Answer:

  (c)  The value of the mean would remain the same at about 46.

Step-by-step explanation:

You want to know how the mean of the given list would change if a value of 45 were added to the list.

The mean is the sum, divided by the number of items being summed. The calculator shows the mean of the given numbers is about 46 1/6.

This means the sum is (46 1/6)·12 = 554.

Adding 45 to the list would make the sum be 599, and the mean would change to ...

  599/13 ≈ 46.08

The value of the mean remains the same at about 46.

__

Additional comment

The added value is 46 1/6 -45 = 1 1/6 below the mean. Adding this value changes the mean by (-1 1/6)/13 = -7/78 ≈ -0.0897, so the nearest integer to the mean does not change.

The point of intersection of the two lines is (1/3, 4/3).
Option B is the correct answer.

We have,

To find the point of intersection of the two lines, we can solve the system of equations given by:

2x + y = 2 _____(1)

x + 2y = 3 ________(2)

We can solve this system by using the elimination method or substitution method.

Here, we will use the elimination method:

Multiplying equation (1) by 2, we get:

4x + 2y = 4  ________(3)

Subtracting equation (2) from equation (3), we get:

3x = 1

Solving for x, we get:

x = 1/3

Substituting this value of x in equation (1), we get:

2(1/3) + y = 2

Simplifying, we get:

y = 4/3

Thus,

The point of intersection of the two lines is (1/3, 4/3).

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The result of adding -4 (row 1) to row 3 is determined as (0  - 9   2)|12.

The result of the row multiplication in the matrice is calculated by applying the following method;

row 1 in the given matrices = [1  2  1] | -5

To multiply row by -4, we will multiply each entity by 4 as shown below;

= -4(1 2  1) | -5

= (-4  -8  - 4) |20

To add the result to 3;

(-4  -8  - 4)|20 + (4  - 1   6) | -8

= (0  - 9   2)|12

Thus, the result of the row multiplication is determined by multiplying each entry in row 1, by - 4.

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The length of the arc is 12π units.

The area of a sector is given by the formula:

Area of sector = θ/360 × πr²

Where θ is the angle subtended and r is the radius

Substituting:

16π = θ/360 × π × 8²

θ = 90° = 0.5π (in radian)

m∠WYX = 2π - m∠WX

m∠WYX = 2π - 0.5π

m∠WYX = 1.5π

Length of arc = θ/360 ×  2πr

Length of m∠WYX = (1.5π)/2π × 2πr

Length of m∠WYX = (1.5π)/2π × 2π × 8

Length of m∠WYX = 12π units.

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