Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Find the associated radius of convergence R. f(x) = 6(1 − x)−2

Answer:

  (4x -5)(16x^2 +20x +25)

Step-by-step explanation:

The factoring of the difference of cubes is something you might want to memorize, or keep handy:

  (a³ -b³) = (a -b)(a² +ab +b²)

Here, the minus sign in the middle tells you this is a difference. The power of x is a clue that this might be the difference of cubes. Your knowledge of cubes of small integers tells you ...

so you can recognize this as ...

  (4x)³ - 5³ . . . . . the difference of cubes.

___

Since you are familiar with the factorization above, you can easily write down the factoring of this expression using a=4x, b=5.

  (4x -5)(16x² +20x +25)

_____

For "completeness", here is the factorization of the sum of cubes:

  a³ +b³ = (a +b)(a² -ab +b²)

Note the linear factor (a +b) has the same sign as the sign between the cubes. The sign of the middle 2nd-degree term (ab) is opposite that.

Answer:

I think it is A. But i'm not 100% sure.

Step-by-step explanation:

Hope my answer has helped you!

Answer:

t=5.1 days

Step-by-step explanation:

We know that the formula is:

[tex]N = N_0e^{-kt}[/tex]

Where N is the amount of substance after a time t, [tex]N_0[/tex] is the initial amount of substance, k is the rate of decrease, t is the time in days.

[tex]k=0.1368\\N_0 =32[/tex]

We want to find the average life of the substance in days

The half-life of the substance is the time it takes for half the substance to disintegrate.

Then we equal N to 16 gr and solve the equation for t

[tex]16= 32e^{-0.1368t}[/tex]

[tex]0.5= e^{-0.1368t}[/tex]

[tex]ln(0.5)= ln(e^{-0.1368t})[/tex]

[tex]ln(0.5)= -0.1368t[/tex]

[tex]t= \frac{ln(0.5)}{-0.1368}\\\\t=5.1\ days[/tex]

Answer:

5.1

PLATO answer

Step-by-step explanation:

Answer:

  (4, 1)

Step-by-step explanation:

Since the first reflection is over the vertical line x=-3, the y-coordinate remains the same. The x-coordinate of A' will make the point (-3, 4) on the line of reflection be the midpoint between A and A':

  (-3, 4) = (A +A')/2

  2(-3, 4) -A = A' = (-6-(-7), 8 -4) = (1, 4)

The reflection over the line y=x simply interchanges the two coordinate values:

  A'' = (4, 1)

The point (-7,4) upon reflection over the lines x = -3 and y = x would be at point; (4,1).

According to the question;

For the first reflection;

However, the x-coordinate of P' will make the point (-3, 4) on the line of reflection be the midpoint between A and A':

For the second reflection;

Ultimately, the point P'' = (4, 1)

Read more;

https://brainly.com/question/20361910

Answer:

  3 1/3 bags

Step-by-step explanation:

5 × 2/3 = (5×2)/3 = 10/3 = 3 1/3

The total quantity of leaves gathered is equivalent to 3 1/3 full bags.

Answer:

  B.  -3 and 2/3

Step-by-step explanation:

One way to see the constants is to set the variables to zero. What's left is ...

  -3 + 2/3

These are the constants in the expression.

Answer:

c

Step-by-step explanation:

Answer:

20 years

Step-by-step explanation:

9=x

8=x+15

7=x+15+15

6=x+15+15+15

5=x+15+15+15+15

4=x+15+15+15+15+15

3=x+15+15+15+15+15+15

2=x+15+15+15+15+15+15+15

1=x+15+15+15+15+15+15+15+15

Xx6=6x

6x=x+15+15+15+15+15+15+15+15

6x=120

divide both sides by 6 to get X as 20

we had written the age of the youngest son as X so the youngest son is 20 years old.

Answer:

  42 square units

Step-by-step explanation:

The area of the rectangle is the product of its length and width:

  area = 6(x -8)

The area of the triangle is half the product of its base and height:

  area = (1/2)(x-3)·7

These two areas are equal, so we have ...

  6(x -8) = (1/2)(x -3)(7)

  6x -48 = 3.5x -10.5 . . . . eliminate parentheses

  2.5x = 37.5 . . . . . . . . . . . add 48 -3.5x

  x = 15 . . . . . . . . . . . . . . . divide by 2.5

The area of the rectangle is ...

  area = 6(15 -8) = 42 . . . . square units.

For this case we have the following expression:

[tex]\sqrt {29-12 \sqrt {5}}[/tex]

The expression can not be simplified, we can write its decimal form.

We have to:

[tex]12 \sqrt {5} = 26.83281572[/tex]

Then, replacing:

[tex]\sqrt {29-26.83281572} =\\\sqrt {2,16718428} =\\1.4721359584[/tex]

If we round up we have:

[tex]\sqrt {29-12 \sqrt {5}} = 1.47[/tex]

ANswer:

[tex]\sqrt {29-12 \sqrt {5}} = 1.47[/tex]

The answer is:

[tex]FirstCarSpeed=41mph\\SecondCarSpeed=55mph[/tex]

To calculate the speed of the cars, we need to write two equations in order to create a relation between the two speeds and be able to isolate one in function of the other.

So, let be the first car speed "x" and the second car speed "y", writing the equations we have:

For the first car:

[tex]x_{FirstCar}=x_o+v*t[/tex]

For the second car:

We know that the speed of the second car is the speed of the first car plus 14 mph, so:

[tex]x_{SecondCar}=x_o+(v+14mph)*t[/tex]

Now, we already know that both cars met after 2 hours and 45 minutes, meaning that positions will be the same at that moment, and the distance between A and B is 264 miles,  so, we can calculate the relative speed between them.

If the cars are moving towards each other the relative speed will be:

[tex]RelativeSpeed=FirstCarSpeed-(-SecondCarspeed)\\\\RelativeSpeed=x-(-x-14mph)=2x+14mph[/tex]

Then, since from the statement we know that the cars covered a combined distance which is equal to 264 miles of distance in 2 hours + 45 minutes, we have:

[tex]2hours+45minutes=120minutes+45minutes=165minutes\\\\\frac{165minutes*1hour}{60minutes}=2.75hours[/tex]

Writing the equation, we have:

[tex]264miles=(2x+14mph)*t\\\\264miles=(2x+14mph)*2.75hours\\\\2x+14mph=\frac{264miles}{2.75hours}\\\\2x=96mph-14mph\\\\x=\frac{82mph}{2}=41mph[/tex]

So, we have that the speed of the first car is equal to 41 mph.

Now, substituting the speed of the first car in the second equation, we have:

[tex]SecondCarSpeed=FirstCarSpeed+14mph\\\\SecondCarSpeed=41mph+14mph=55mph[/tex]

Hence, we have that:

[tex]FirstCarSpeed=41mph\\SecondCarSpeed=55mph[/tex]

Have a nice day!

Completing the square gives

[tex]y=8x-x^2=16-(x-4)^2[/tex]

and

[tex]16=16-(x-4)^2\implies(x-4)^2=0\implies x=4[/tex]

tells us the parabola intersect the line [tex]y=16[/tex] at one point, (4, 16).

Then the volume of the solid obtained by revolving shells about [tex]x=0[/tex] is

[tex]\displaystyle\pi\int_0^4x(16-(8x-x^2))\,\mathrm dx=\pi\int_0^4(x-4)^2\,\mathrm dx[/tex]

[tex]=\pi\dfrac{(x-4)^3}3\bigg|_{x=0}^{x=4}=\boxed{\dfrac{64\pi}3}[/tex]

Answer:

The weight of the yard waste = 1650 pounds

Step-by-step explanation:

* Lets revise the rule of the volume of the prism

- The rectangular prism has 6 rectangular faces

- Its base shaped a rectangle of dimensions length and width

- The volume of the rectangular prism = Area of its base × its height

∵ Area of the rectangle = length × width

∴ The volume of the prism = length × width × height

* Now lets solve the problem

∵ The length of the rectangular prism = 10 feet

∵ The width of it = 5.5 feet

∵ The height of it = 4 feet

∴ Its volume = 10 × 5.5 × 4 = 220 feet³

- The prism is completely full of yard waste

∴ The volume of the yard waste = the volume of the prism

∴ The volume of the yard waste = 220 feet³

- The average weighs of it is 7.5 pounds/cubic foot

∴ The weight of the yard waste = the average of weight × the volume

∴ The weight of the yard waste = 7.5 × 220 = 1650 pounds

Answer:

10(6y-9-2x)

Step-by-step explanation:

60y-90-20x

We can factor out 10 from each term

10(6y-9-2x)

Answer:

10(6y-9-2x)

Step-by-step explanation:

i dont know how to say it bit this is 100% true

h(t) = -16t² + 50t + 5

The maximum height is the y vertex of this parabola.

Vertex = (-b/2a, -Δ/4a)

The y vertex is -Δ/4a

So,

The maxium height is -Δ/4a

Δ = b² - 4.a.c

Δ = 50² - 4.(-16).5

Δ = 2500 + 320

Δ = 2820

H = -2820/4.(-16)

H = -2820/-64

H = 2820/64

H = 44.0625

So, the maxium height the ball will reach is 44.0625

The maximum height the ball will reach is 81.25 feet.

To find the maximum height the ball will reach, we can use the formula h(t) = -16t² + 50t + 5, where t represents time in seconds. The maximum height occurs at the vertex of the parabolic equation, which can be found using the formula t = -b/2a. In this case, a = -16 and b = 50. Plugging in these values, we get t = -50/(2*(-16)) = 1.5625 seconds.

Now, we can substitute this value of t back into the original equation to find the maximum height. h(1.5625) = -16(1.5625)² + 50(1.5625) + 5 = 81.25 feet. Therefore, the maximum height the ball will reach is 81.25 feet.

https://brainly.com/question/34973099

#SPJ3

Answer:

Step-by-step explanation:

The probability would be twice as large for eight coins I think

The probability distribution of flipping coins changes as the number of coins increases due to the law of large numbers. The probability of getting half heads is the highest when flipping four coins, whereas the probability distribution broadens with eight coins, making specific outcomes less likely.

When analyzing the probability of outcomes when flipping coins, we can compare the cases of flipping four coins to flipping eight coins. When flipping four coins, the most likely outcome is two heads, with a probability of 0.38. The probability of getting one head, which is one-fourth of the coins, has a probability of 0.25. However, these probabilities might change when flipping eight coins because as the number of coin flips increases, the distribution of outcomes tends to become more even due to the law of large numbers.

Tossing a fair coin should theoretically result in 50 percent heads over the long term, as demonstrated by Karl Pearson's experiment which approximated the theoretical probability after 24,000 coin tosses. When moving to eight coins, the probability distribution for getting different fractions of heads will widen, meaning that it's less likely to get a specific outcome (in terms of fractions of heads) because there are more possible combinations (microstates).

Overall, while the exact probabilities for eight coins are not provided, we can expect that the probability of getting exactly half heads will decrease since there are more total combinations possible, and the probability for one-fourth heads will also change but without specific numbers, we cannot determine the precise probabilities.

Final answer:

To establish that ΔABC is similar to ΔA'B'C' by the AA criterion, we need to verify that two pairs of angles are congruent. The sum of angles in each triangle should be equal to 180 degrees. However, the dilation scale factor does not affect angles, so both triangles are similar by AA criterion.

Explanation:

The question involves determining whether two triangles are similar by the AA (Angle-Angle) criterion. The original triangle ΔABC has angles ∠A = 4x + 7, ∠B = 2x + 3, and ∠C = 6x - 10. Similarly, the dilated triangle ΔA'B'C' has angles ∠A' = 5x - 8, ∠B' = 3x - 12, and ∠C' = 7x - 25. To confirm the similarity using the AA criterion, we need to prove that at least two angles of ΔABC are congruent to two angles of ΔA'B'C'.

However, upon examination of the provided angles, it's clear that there is an inconsistency. The angles of the original triangle ΔABC must add up to 180 degrees, as must the angles of the dilated triangle ΔA'B'C'. This gives us two equations to solve for x, 4x + 7 + 2x + 3 + 6x - 10 = 180 for ΔABC and 5x - 8 + 3x - 12 + 7x - 25 = 180 for ΔA'B'C'. Solving these would give us the values of x that we could use to find the angles. But the actual correspondence of angles and the fact that the sum of angles in both triangles must be equal indicates that the scale factor of dilation does not change the angles, which means ΔABC should be similar to ΔA'B'C' by AA criterion regardless of the values of x.

Final answer:

To find out how long it will take for an investment to grow with compound interest, we use the formula A = P(1 + r/n)^(nt). Substitute the given values into the formula and solve for t using logarithms to find the time needed for the investment to reach the desired amount.

Explanation:

To determine how long it will take for $1000 invested in an account earning 3% compounded monthly to grow to $1500, we use the formula for compound interest:

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

Where:

We want to solve for t when A is $1500, P is $1000, r is 0.03 (3%), and n is 12 (since interest is compounded monthly). Substituting these values into the formula we have:

Dividing both sides by $1000 and using algebra, we can solve for t:

[tex]1.5 = (1 + \frac{0.03}{12})^{12t}[/tex]

To solve for t, we take the natural logarithm of both sides:

[tex]ln(1.5) = ln((1 + \frac{0.03}{12})^{12t})[/tex]

[tex]ln(1.5) = 12t * ln(1 + \frac{0.03}{12})[/tex]

[tex]t = \frac{ln(1.5)}{12 * ln(1 + \frac{0.03}{12})}[/tex]

After calculating the above expression using a calculator, you will find the value of t, which is the time in years it will take for the investment to grow to $1500.

Answer:

Final answer is given by x=-4, y=3

Step-by-step explanation:

Question says to use the given graph to determine the number of solutions the system where given system is x = - 4,  y = - x - 1

Graph is missing but we can still find the solution.

plug first equation x=-4 into other equation y=-x-1

y=-x-1

y=-(-4)-1

y=4-1

y=3

Hence final answer is given by x=-4, y=3

or we can also write that as (-4,3)

Answer:

f(x) = 3 if x ≤ -2

     =  1 if x > -2 ⇒ attached figure

Step-by-step explanation:

* Lets explain how to answer the question

- For the part of the graph on the left side (2nd quadrant)

- There is a horizontal line start from x = -∞ and stop at x = -2

- The end of the line is black dot means x = -2 belongs to the function

- The horizontal line drawn at y = 3

∴ The equation of the horizontal line is y = 3

∴ The function represents this part of graph is y = 3 if x ≤ -2

- The other part of the graph is also horizontal line start from

  x = -2 to x = ∞

- The end of the line is white dot means x = -2 does not belong

 to the function

- The horizontal line drawn at y = 1

∴ The equation of the horizontal line is y = 1

∴ The function represents this part of graph is y = 1 if x > -2

* f(x) = 3 if x ≤ -2

       =  1 if x > -2

- The answer is attached

اقعى2اعلاع2ىخث

قرهغىع24عغع2ر

Answer:

That is correct

Step-by-step explanation:

If its reflecting over the y axis only the x is becoming opposite.  the x becomes -x, so (x, y) is reflected over the y-axis, the resulting point is (-x, y)

A reflection over the y-axis in mathematics results in changing the sign of the x-coordinate of a point, while the y-coordinate remains the same. This is a property associated with even functions, which are symmetric around the y-axis. The process is a transformation involving a change in direction.

In mathematics, a point's reflection over the y-axis is represented as (-x, y). When you reflect a point across the y-axis, the x-coordinate changes its sign while the y-coordinate stays the same. For example, if we start with point (2,3), its reflection would be at point (-2,3) across the y-axis.

Notably, this reflection is a property of even functions, which are symmetric around the y-axis. To visualize this, you can plot specific values for (x,y) data pairs and observe the symmetry.

The reflection process can be seen as a transformation involving a change in direction represented by unit vectors. In a plane, it’s customary that the positive direction on the x-axis is denoted by the unit vector i and the positive direction on the y-axis by the unit vector j. The reflection over the y-axis turns the x components in the opposite direction, hence -x.

https://brainly.com/question/30428419

#SPJ3

Answer:

[tex]\boxed{\text{4995 pages}}[/tex]

Step-by-step explanation:

The question is asking you to find the equation of a straight line that passes through two points

Let x = the number of pages

and y = the cost

Then the coordinates of the two points are (1543, 77.40) and (7361, 368.30).

(a) Calculate the slope of the line

[tex]\begin{array}{rcl}m & = & \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}\\\\ & = & \dfrac{368.30 - 77.40}{7361 - 1543}\\\\& = & \dfrac{290.90}{58.18}\\\\ & = & 0.05000\\\end{array}[/tex]

In other words, the cost is 5¢ per page.

(b) Calculate the y-intercept

[tex]\begin{array}{rcl}y & = & mx + b\\368.30 & = & 0.05 \times 7361 + b\\368.30 & = & 368.05 + b\\b & = & 0.25\\\end{array}[/tex]

(c) Write the equation for the line

y = 0.05x + 0.25

That is, the cost is 25¢ plus 5¢ per page

(d) Calculate the pages you can print for $250

[tex]\begin{array}{rcl}y & = & 0.05x + 0.25\\250 & = & 0.05x + 0.25\\249.75 & = & 0.05x\\x & = & 4995\\\end{array}\\\text{ You can print }\boxed{\textbf{4995 pages}}[/tex]

The figure below shows the graph of your equation, with slope 0.05 and y-intercept at (0,0.25).

It looks as if you could print 5000 pages, but you must pay that 25¢ (5 pages worth) up-front, so you can print only 4995.

Answer:

4995 pages

Step-by-step explanation:

The question is asking you to find the equation of a straight line that passes through two points

Let x = the number of pages

and y = the cost

Then the coordinates of the two points are (1543, 77.40) and (7361, 368.30).

(a) Calculate the slope of the line

In other words, the cost is 5¢ per page.

(b) Calculate the y-intercept

(c) Write the equation for the line

y = 0.05x + 0.25

That is, the cost is 25¢ plus 5¢ per page

(d) Calculate the pages you can print for $250

The figure below shows the graph of your equation, with slope 0.05 and y-intercept at (0,0.25).

It looks as if you could print 5000 pages, but you must pay that 25¢ (5 pages worth) up-front, so you can print only 4995.

x(x-7)=8

x^2-7x-8=0

(x-8)(x+1)=0

so x=8 (and -1 but width can’t be negative)

So your answer is C. 8

Answer:

-4y - 7x^3 + 2

Step-by-step explanation:

So first lets combine the y terms together, doing this we  get:

-4y + 1 - x^3 - 3x^3 + 1 - 3x^3

Now lets combine the terms with x^3 together

-4y + 1 - 7x^3 + 1

Lastly, lets add the constants

-4y - 7x^3 + 2

Answer:

[tex]A=9\pi\ square\ units[/tex]

Step-by-step explanation:

we know that

The base of the cylinder is a circle

The area of the circle is equal to

[tex]A=\pi r^{2}[/tex]

we have

[tex]r=6/2=3\ units[/tex] ----> the radius is half the diameter

substitute

[tex]A=\pi (3)^{2}[/tex]

[tex]A=9\pi\ units^{2}[/tex]

BBBBBBBBBBBBBBBBB

the answer is B

Answer:

  25,133 m^2

Step-by-step explanation:

The lateral area of a cone is found using the slant height (s) and the radius (r) in the formula ...

  A = πrs

So, we need to know the radius and the slant height.

The radius is half the diameter, so is (160 m)/2 = 80 m.

The slant height can be found using the Pythagorean theorem:

  s^2 = r^2 + (60 m)^2 = (80 m)^2 +(60 m)^2 = (6400 +3600) m^2

  s = √(10,000 m^2) = 100 m

Now, we can put these values into the formula to find the lateral area:

  A = π(80 m)(100 m) = 8000π m^2 ≈ 25,133 m^2

Answer:

2 miles

Step-by-step explanation:

Their combined speed (speed of closure) is 80 mph + 40 mph = 120 mph.

120 mi/h = 120 mi/(60 min) = 2 mi/min

The distance between the trains at the 1-minute mark is ...

(1 min) · (2 mi/min) = 2 mi

They were warned when two miles apart.

1. -x-5y=21

-5y=x+21

y= -1/5x-21/5

2.-2x-5y=25

-5y=2x+25

y= -2/5x-5

Answer:

I believe b is 5.

Step-by-step explanation:

When you place the points in the values, you get 5=3(0)+b. When you simplify it, you get 5=b. I really hope this is correct, I apologize if it isn't! I hope this helps! :)

I think it’s five too

First we must find out the slope of the points

The equation for slope is

[tex]\frac{y_{2} - y_{1}}{x_{2}-x_{1}}[/tex]

Let's plug in our points

[tex]\frac{3 - 3 }{5 -1 }  = \frac{0}{4}[/tex]

Our slope is 0

When lines are parallel, that means we have they have the same slope

So a line that is parallel to r would have a slope of 0!

[tex]3\tan3\theta-1=0[/tex]

[tex]3\tan3\theta=1[/tex]

[tex]\tan3\theta=\dfrac13[/tex]

Recall that the tangent function has a period of [tex]\pi[/tex] so that

[tex]3\theta=\tan^{-1}\dfrac13+k\pi[/tex]

for any integer [tex]k[/tex]. Then

[tex]\theta=\dfrac13\tan^{-1}\dfrac13+\dfrac{k\pi}3[/tex]

We get 6 solutions in the interval [0, 2π) for [tex]0\le k\le5[/tex],

[tex]\theta\approx0.107[/tex]

[tex]\theta\approx1.154[/tex]

[tex]\theta\approx2.202[/tex]

[tex]\theta\approx3.249[/tex]

[tex]\theta\approx4.296[/tex]

[tex]\theta\approx5.343[/tex]