Identify the area of the figure rounded to the nearest tenth.


Please give an Explanation :)

By the alternating series test, this series converges: for [tex]k\in\Bbb N[/tex],

[tex]\dfrac{k^5+1}{k^6+11}[/tex]

is a positive, decreasing sequence that converges to 0.

However,

[tex]\displaystyle \sum_{k=2}^\infty \left| (-1)^{k+1} \frac{k^5+1}{k^6+11} \right| = \sum_{k=2}^\infty \frac{k^5+1}{k^6+11} \approx \sum_{k=2}^\infty \frac1k[/tex]

is a divergent series by comparison to the harmonic series.

So the given series is conditionally convergent.

The given series is conditionally convergent. This can be obtained by using alternating series test first and then comparing the series to the harmonic series.

Initially we need to know what Absolute convergence and Conditional convergence,

If [tex]\sum|a_{n} |[/tex] → converges, and [tex]\sum a_{n}[/tex] → converges, then the series is Absolute convergence

If [tex]\sum|a_{n} |[/tex] → diverges, and [tex]\sum a_{n}[/tex] → converges, then the series is Conditional convergence

First use alternating series test,

[tex]\lim_{k \to \infty} \frac{k^{5} +1}{k^{6}+11 }[/tex] = [tex]\lim_{n \to \infty} \frac{5}{6k}[/tex] = 0,

The series is a positive, decreasing sequence that converges to 0.

Next by comparing the series to harmonic series,

[tex]\sum^{\infty} _{k=2}|(-1)^{k+1} \frac{k^{5} +1}{k^{6}+11 }|=\sum^{\infty} _{k=2}\frac{k^{5} +1}{k^{6}+11 }[/tex] ≈  [tex]\sum^{\infty} _{k=2}\frac{1}{k}[/tex] = 0

This implies that the series is divergent by comparison to the harmonic series.

First we got that the series is converging and then we got the series is divergent. Therefore the series is conditionally convergent.

[tex]\sum|a_{n} |[/tex] → diverges, and [tex]\sum a_{n}[/tex] → converges, then the series is Conditional convergence.

Hence the given series is conditionally convergent.

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Step-by-step explanation:

a ∈ (0; π/2) here means that our angle, a must lie between 0 and pi/2, exclusive.

So this mean our angle must be in between 0 and pi/2, but can not be neither 0 and pi/2.

Here we have

[tex] \sin( \alpha ) = \frac{3 \sqrt{11} }{10} [/tex]

We must find cos.

Using the Pythagorean theorem

[tex]( \sin( \alpha ) ) {}^{2} + ( \cos( \alpha ) ) {}^{2} = 1[/tex]

It is mostly notated as this,

[tex] \sin {}^{2} ( \alpha ) + \cos {}^{2} ( \alpha ) = 1[/tex]

But they mean the same thing, we know

[tex] \sin( \alpha ) = \frac{3 \sqrt{11} }{10} [/tex]

So we plug that in for sin a.

[tex]( \frac{3 \sqrt{11} }{10} ) {}^{2} + \cos {}^{2} ( \alpha ) = 1[/tex]

[tex] \frac{99}{100} + \cos {}^{2} ( \alpha ) = 1[/tex]

[tex] \cos {}^{2} ( \alpha ) = \frac{100}{100} - \frac{99}{100} [/tex]

[tex] \cos {}^{2} ( \alpha ) = \frac{1}{100} [/tex]

Since cos is Positve over the interval (0; π/2), we take the positive or principal square root.

[tex] \cos( \alpha ) = \frac{1}{10} [/tex]

2. We would get the same work for the second part, the only difference is that cosine is negative over the interval

(π/2, π)

So the answer for 2 is

[tex] \cos( \alpha ) = - \frac{1}{10} [/tex]

Disclaimer: Your work you did was correct, just remember for fractions like

[tex]1 - \frac{99}{100} [/tex]

Convert 1 into a fraction that has a denominator of 100.

[tex] \frac{100}{100} - \frac{99}{100} = \frac{1}{100} [/tex]

Answer:

Step-by-step explanation:

assuming that he has an annual interest of 5% then his initial deposit would be rounded to $2,238.65

The volume of the triangular prism as described is; 20 cubic foot.

A triangular pyramid refers to pyramid which has a triangular base

It follows from the formula for calculating the volume of a triangular prism that;

Volume = (1/3) × base area × height.

Consequently,

volume of the prism = (1/3) ×5 × 12

Volume = 20 cubic foot

Therefore, volume of the triangular prism as described is; 20 cubic foot.

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Answer:

C: 1 Quart

Step-by-step explanation:

2 pints are the equivalent of 1 quart

Answer:1 Quart

Step-by-step explanation: Because 2 pints equals=1 quart

The solutions using the relevant theorems are:

13. x = −3 or x = 4

14. x = 12

15. x = 12

16. x = 1/2 or x = 4

17. x = 5

18. x = −1 or x = 6

The midsegment of a triangle that joins two sides of a triangle divides the two sides proportionally based on the triangle midsegment theorem.

The angle bisector of a triangle, according to the angle bisector theorem divides the opposite angles sides in a proportional manner.

The theorem states that when three or more lines that are parallel is cut across by two transversals, they are divided by the transversal proportionally.

13. Apply the angle bisector theorem:

x/3 = (x + 4)/(x + 2)

Cross multiply

x(x + 2) = 3(x + 4)

x² + 2x = 3x + 12

x² + 2x - 3x - 12 = 0

x² - x - 12 = 0

Factorize

x = −3 or x = 4

14. Apply the angle bisector theorem:

16/x = 12/9

x(12) = (16)(9)

12x = 144

x = 12

15. Apply the angle bisector theorem:

(x - 4)/6 = x/9

9(x - 4) = 6x

9x - 36 = 6x

9x - 6x = 36

3x = 36

x = 12

16. Apply the triangle midsegment theorem:

3x/(x + 4) = (x - 1)/(x - 2)

Cross multiply

3x/(x + 4) = (x - 1)/(x - 2)

(x - 1)(x + 4) = 3x(x - 2)

x² + 3x - 4 = 3x² - 6x

x² + 3x - 4 - 3x² + 6x = 0

-2x² + 9x - 4 = 0

Factorize

x = 1/2 or x = 4

17. Apply the Parallel Lines and Proportionality Theorem:

(x + 3)/(2x + 2) = (2x + 2)/(4x - 2)

Cross multiply

4x² + 10x - 6 = 4x² + 8x + 4

4x² + 10x - 6 - 4x² - 8x - 4 = 0

2x - 10 = 0

2x = 10

x = 5

18. Apply the angle bisector theorem:

(2x + 3)/x = (x + 4)/(x - 2)

Cross multiply

(2x + 3)(x - 2) = x(x + 4)

2x² - x - 6 = x² + 4x

2x² - x - 6 - x² - 4x = 0

x² - 5x - 6 = 0

Factorize

x = −1 or x = 6

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Answer:

Step-by-step explanation:

To find:

Given:

[tex]\sf{7p+4=-p+9+2p+3p}[/tex]

Isolate the term of p, from one side of the equation.

Combine like terms.

[tex]\rightarrow \sf{7p+4=-p+2p+3p+9}[/tex]

Add the numbers from left to right.

-p+2p+3p=4p

7p+4=4p+9

Then, you subtract by 4 from both sides.

[tex]\sf{7p+4-4=4p+9-4}[/tex]

Solve.

7p=4p+5

Subtract by 4p from both sides.

7p-4p=4p+5-4p

Solve.

[tex]\boxed{\sf{3p=5}}[/tex]

Divide by 3 from both sides.

3p/3=5/3

Solve.

p=5/3

Divide is another option.

5/3=1.6666

So, the final answer is 3p=5.

I hope this helps, let me know if you have any questions.

Answer:

N = 2.1

M = 4.5

Step-by-step explanation:

<p is 25 degrees.  I know this because the exterior angle of p corresponds with 155.  Since this is a straight angle, angle p must be 25 (180-155).  Now I can use the cos and the sin of 25 to find M and N

sin 25 = N/5  multiply both sides by 5 to get

5 sin 25 = N  Use your calculator put in 5 sin 25

2.1 rounded to the nearest tenth.

cos 25 = M/5  Multiply both sides by

5 cos 25 = M Use your calculator

4.5

Answer:

a(1) = 2

a(n) = a(n-1)+4

Step-by-step explanation:

Using the given arithmetic sequence,

a1 is 2

a2 is 6

a3 is 10

a4 is 14

a5 is 18

and to get from a1 to a2 you have to +4, to get from a2 to a3 you have to +4 and so on.

In a reclusive formula you need to find two pieces of information:

1. The first term of the sequence

2. The pattern rule to get any term from the term that comes before it

Using the given sequence, the first term is 2 and the rule to get any term from its previous term is +4.

So, putting that information in the form of a recursive formula will read as the following:

an+1 = an+4

a1 = 2, n is greater or equal to 1 (n is an interger)

Which can be rearranged to

a(1) = 2

a(n) = a(n-1)+4

The range of the function y = -x² - 3x when the domain is {-5,0,2} is:

{-10, 0}.

The range of a function is the set that contains the output values for the given domain of the function.

In this problem, the domain is:

{-5,0,2}.

Hence the output values are:

Hence the range is:

{-10, 0}.

-10 repeats, but goes only once on the range.

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Answer:

C

Step-by-step explanation:

You are dividing BOTH sides by 3.

"The division property of equality states that when we divide both sides of an equation by the same non-zero number, the two sides remain equal."

Answer:144

Step-by-step explanation: a coin has 2 possibilities so 2 coins have 2x2 = 4 possibilities. A normal six-sided dice has 6x6 = 36 possibilities. 4 x 36 = 144 (this might be wrong)

Using the permutation formula, the number of different options for the top three finishers is given as follows:

[tex]P_{14,3} = \frac{14!}{11!} = 2184[/tex]

The order in which the players are chosen is important, as A, B, C is a different outcome than C,B,A for example, hence the permutation formula is used to solve this question.

The number of possible permutations of x elements from a set of n elements is given by:

[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]

In this problem, 3 students will be chosen from a set of 14, hence the number of ways is:

[tex]P_{14,3} = \frac{14!}{11!} = 2184[/tex]

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The function to represent the problem is f(x)=6x+24 and the range is 30[tex]\leq[/tex]y[tex]\leq[/tex]48.

If the terms of a sequence differ by a constant, we say the sequence is arithmetic. If the initial term ([tex]a_{0}[/tex]) of the sequence is a and the common difference is d, then we have,

[tex]a_{n}[/tex]=a+(n-1)d

Initial number of clients=30

Number increase per week= 6

So, we can make an arithmetic sequence fir six weeks

30,36,42,48

Here, first term, a=30

Common difference, d=6

Range is [30,48]

The explicit formula of an arithmetic sequence is

[tex]a_{n}[/tex]=a+(n-1)d

Put a=30, d=6, n=x

[tex]a_{x}[/tex]=30+(x-1)6

[tex]a_{x}[/tex]=30+6x-6

[tex]a_{x}[/tex]=6x+24

Therefore, The function to represent the problem is f(x)=6x+24 and the range is 30<=y<=48

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Answer:

4 degrees ( to nearest degree).

Step-by-step explanation:

The trig ratio you want is the tangent.

tan x = 10,000 / 160,000    where x is the angle of depression.

tan x = 0.0625

x = 3.576 degrees.

The price x which will maximize the company's revenue is 904

The price x which will maximize the company's revenue is 240

Let

Price of the good = x

Number of goods sold = 2712- 3x

The formula for revenue is given as;

Revenue = Price × Quantity

= x( 2712 - 3x)

Expand the bracket

2712x - 3x^2

Therefore, the price x which will maximize the company's revenue will be:

2712x - 3x^2 =0

Collect like terms

3x² = 2712x

3x =  2712

Divide both side by 3

3x/3 = 2712/3

x = 904

Hence, the price x which will maximize the company's revenue is 904

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Answer:

blank = inheritance

Step-by-step explanation:

Hope that helps

Answer:

  (a)  h^-1(x) = (x+6)/6

Step-by-step explanation:

The inverse relation can be found by solving for y: x = h(y).

  x = h(y) . . . . . the inverse relation

  x = 6y -6 . . . . . use the function definition

  x +6 = 6y . . . . add 6

  (x +6)/6 = y . . . . divide by 6

The inverse function is ...

  [tex]\boxed{h^{-1}(x)=\dfrac{x+6}{6}}[/tex]

Answer:

not linear since all first differences are not equal

Step-by-step explanation:

To find the first differences, subtract each y value from the next y value.

Use only y values.

here are the 4 first differences you need to calculate.

1 - 0 = 1

4 - 1 = 3

9 - 4 = 5

16 - 9 = 7

Now you place them in the appropriate column in the table.

x   y      first difference

0   0       1

1   1         3

2   4       5

3   9       7

4   16

If the first differences were all equal, the function would be a linear function.

In this case, the first differences are all different, 1, 3, 5, 7, so the function is not linear.

Answer:

x= 5/2

Step-by-step explanation:

Answer:

-4 =x

Step-by-step explanation:

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An equation which best represents the relationship between velocity and time is: B. v = 2.19x + 6.7.

A scatter plot can be defined as a type of graph which is used for the graphical representation of the values of two variables, with the resulting points showing any association (correlation) between the data set.

A linear function can be defined as a type of function whose equation is graphically represented by a straight line on the cartesian coordinate.

By critically observing the graph (see attachment) which models the data in the given table, we can infer and logically deduce that the linear function is given by:

v = 2.0852x + 6.3978

Therefore, an equation which best represents the relationship between velocity and time is:

v = 2.19x + 6.7.

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Step-by-step explanation:

Volume= lbh

= (12-2h)(12-2h)h

= 144h-48h²+4h³

The volume of the open-faced box is described by the polynomial equation:Volume = 4x³ - 48x² + 144x

To determine the volume of the open-faced box, we need to consider the dimensions of the box after cutting squares out of the corners and folding up the sides.

Let's assume that each side of the square cutout has a length of x inches.

When the squares are cut out and the sides are folded up, the resulting box will have dimensions:

Length = 12 - 2x inches

Width = 12 - 2x inches

Height = x inches

The volume of the box can be calculated by multiplying these dimensions:

Volume = Length * Width * Height

Volume = (12 - 2x) * (12 - 2x) * x

Volume = 4x³ - 48x² + 144x

Therefore, the volume of the open-faced box is described by the polynomial equation:

Volume = 4x³ - 48x² + 144x

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The function y = 2sin has a phase shift of pi/2 to the right (2x - pi).

To find the function which has a phase shift to the right:

The function has a phase shift of pi/2 to the right.

By definition, you have the phase shift is:

When you substitute the values from the function [tex]y=2sin(2x-\pi )[/tex], where  [tex]c=-\pi[/tex]  and [tex]b=2[/tex], you obtain:

Therefore, the function y = 2sin has a phase shift of pi/2 to the right (2x-pi).

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The complete question is given below:
Which function has a phase shift of pi/2 to the right?

The maximum area is achieved when the rectangle is a square of side 6 feet, with a domain, 0 < x < 12.

The perimeter available with Brayden is 24 feet.

The width of the rectangle is assumed to be x feet.

The length can be calculated using the formula:

2(length + width) = perimeter,

or, 2length + 2width = perimeter,

or, 2length = perimeter - 2width,

or, length = (1/2)(perimeter - 2 width).

Substituting the values, we get:

length = (1/2)(24 - 2x).

The area can be calculated using the formula:

Area = length*width.

Substituting the values, we get:

Area = (1/2)(24 - 2x)x = (1/2)x(24 - 2x).

Now, we need to maximize the area for the given perimeter.

For that, we differentiate the area function, with respect to its width x.

d(Area)/dx = (1/2)(24 - 2x) + (1/2)x(-2),

or, d(Area)/dx = 12 - x - x = 12 - 2x ... (i).

To check for the point of inflection, we equate this to zero, to get:

12 - 2x = 0,

or, 2x = 12,

or, x = 6.

To check whether this is maximum or  minimum, we differentiate (i) with respect to x to get:

d²(Area)/dx² = -2 which is less than 0, implying area is maximum at x = 6.

Thus, the maximum area is achieved when the width is 6 feet.

Length = (1/2)(24 - 2x) = (1/2)(24 - 2*6) = (1/2)12 = 6.

Thus, the maximum area is achieved when the length is 6 feet.

The area function is, area = (1/2)x(24 - 2x).

We know that the area is always greater than 0, thus, we can show that:

(1/2)x(24-2x) > 0,

or, x(24 - 2x) > 0,

or, x(12 - x) > 0, which is true when 0 < x < 12.

Thus, the domain of the area function is 0 < x < 12.

Thus, the maximum area is achieved when the rectangle is a square of side 6 feet, with a domain, 0 < x < 12.

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invest in each bond = $7000

An income statement, also known as a profit and loss account, is one of a company's financial statements and displays the company's receipts and outlays for a specific time period. It explains how revenues are converted into net income or net profit.

According to the given data:

Let n be the amount invested at 8%

the amount invested at 12% would be : 14000-n

so,

8n+12(14000-n) = 1400

8n + 168000 - 12n = 1400 * 100

4n + 168000 = 140000

n = 28000/4

   =  7000

Therefore,

At 12% amount invested

= 14000 - 7000

= $7000

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The equivalent expression of the radical expression ∛1080 is 6∛5

The radical expression is given as

∛1080

Express 1080 as 216 * 5

∛1080 = ∛(216 * 5)

Split the factors

∛1080 = ∛216 * ∛5

Evaluate the cube root of 216

∛1080 = 6 * ∛5

Evaluate the product

∛1080 = 6∛5

Hence, the equivalent expression of the radical expression ∛1080 is 6∛5

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Answer:

Part 1) Option B. y = 10x + 45

Part 2) The score is 95

Step-by-step explanation:

Linear equation that best describes the given model

Let

x ---> number of hours students spent studying

y ---> their score on the test

Looking at the line that was fit to the data to model the relationship

The slope is positive

The y-intercept is the point (0,45)

For x=1, y=55 ----> point (1,55)

Find the slope

The formula to calculate the slope between two points is equal to

substitute the points (0,45) and (1,55)

Find the equation of the line in slope intercept form

we have

substitute

Part 2) Estimate the score for a student that spent 5 hours studying.

For x=5 hours

substitute in the linear equation and solve for y

30 full cylindrical containers are required to completely fill the fish tank.

To find what is the least number of full cylindrical containers needed to completely fill the fish tank:

We know the volume of the cylinder is given by: [tex]V=\pi r^{2} h[/tex]

The volume of a cylinder:

[tex]V=\pi (\frac{6}{2} )^{2} (8)\\V=75\pi inches^{3}[/tex]

The volume of cube V = 24 × 24 × 12 = 6912 cubic inches

A number of full cylindrical containers are needed to completely fill the fish tank:

Therefore, 30 full cylindrical containers are required to completely fill the fish tank.

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