The volume of a sphere is 9,198.11 ft3. To the nearest foot​, what is the radius of the​ sphere? Use 3.14 for π.

5. Let [tex]x = \sin(\theta)[/tex]. Note that we want this variable change to be reversible, so we tacitly assume 0 ≤ θ ≤ π/2. Then

[tex]\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - x^2}[/tex]

and [tex]dx = \cos(\theta) \, d\theta[/tex]. So the integral transforms to

[tex]\displaystyle \int \frac{x^3}{\sqrt{1-x^2}} \, dx = \int \frac{\sin^3(\theta)}{\cos(\theta)} \cos(\theta) \, d\theta = \int \sin^3(\theta) \, d\theta[/tex]

Reduce the power by writing

[tex]\sin^3(\theta) = \sin(\theta) \sin^2(\theta) = \sin(\theta) (1 - \cos^2(\theta))[/tex]

Now let [tex]y = \cos(\theta)[/tex], so that [tex]dy = -\sin(\theta) \, d\theta[/tex]. Then

[tex]\displaystyle \int \sin(\theta) (1-\cos^2(\theta)) \, d\theta = - \int (1-y^2) \, dy = -y + \frac13 y^3 + C[/tex]

Replace the variable to get the antiderivative back in terms of x and we have

[tex]\displaystyle \int \frac{x^3}{\sqrt{1-x^2}} \, dx = -\cos(\theta) + \frac13 \cos^3(\theta) + C[/tex]

[tex]\displaystyle \int \frac{x^3}{\sqrt{1-x^2}} \, dx = -\sqrt{1-x^2} + \frac13 \left(\sqrt{1-x^2}\right)^3 + C[/tex]

[tex]\displaystyle \int \frac{x^3}{\sqrt{1-x^2}} \, dx = -\frac13 \sqrt{1-x^2} \left(3 - \left(\sqrt{1-x^2}\right)^2\right) + C[/tex]

[tex]\displaystyle \int \frac{x^3}{\sqrt{1-x^2}} \, dx = \boxed{-\frac13 \sqrt{1-x^2} (2+x^2) + C}[/tex]

6. Let [tex]x = 3\tan(\theta)[/tex] and [tex]dx=3\sec^2(\theta)\,d\theta[/tex]. It follows that

[tex]\cos(\theta) = \dfrac1{\sec(\theta)} = \dfrac1{\sqrt{1+\tan^2(\theta)}} = \dfrac3{\sqrt{9+x^2}}[/tex]

since, like in the previous integral, under this reversible variable change we assume -π/2 < θ < π/2. Over this interval, sec(θ) is positive.

Now,

[tex]\displaystyle \int \frac{x^3}{\sqrt{9+x^2}} \, dx = \int \frac{27\tan^3(\theta)}{\sqrt{9+9\tan^2(\theta)}} 3\sec^2(\theta) \, d\theta = 27 \int \frac{\tan^3(\theta) \sec^2(\theta)}{\sqrt{1+\tan^2(\theta)}} \, d\theta[/tex]

The denominator reduces to

[tex]\sqrt{1+\tan^2(\theta)} = \sqrt{\sec^2(\theta)} = |\sec(\theta)| = \sec(\theta)[/tex]

and so

[tex]\displaystyle 27 \int \tan^3(\theta) \sec(\theta) \, d\theta = 27 \int \frac{\sin^3(\theta)}{\cos^4(\theta)} \, d\theta[/tex]

Rewrite sin³(θ) just like before,

[tex]\displaystyle 27 \int \frac{\sin(\theta) (1-\cos^2(\theta))}{\cos^4(\theta)} \, d\theta[/tex]

and substitute [tex]y=\cos(\theta)[/tex] again to get

[tex]\displaystyle -27 \int \frac{1-y^2}{y^4} \, dy = 27 \int \left(\frac1{y^2} - \frac1{y^4}\right) \, dy = 27 \left(\frac1{3y^3} - \frac1y\right) + C[/tex]

Put everything back in terms of x :

[tex]\displaystyle \int \frac{x^3}{\sqrt{9+x^2}} \, dx = 9 \left(\frac1{\cos^3(\theta)} - \frac3{\cos(\theta)}\right) + C[/tex]

[tex]\displaystyle \int \frac{x^3}{\sqrt{9+x^2}} \, dx = 9 \left(\frac{\left(\sqrt{9+x^2}\right)^3}{27} - \sqrt{9+x^2}\right) + C[/tex]

[tex]\displaystyle \int \frac{x^3}{\sqrt{9+x^2}} \, dx = \boxed{\frac13 \sqrt{9+x^2} (x^2 - 18) + C}[/tex]

2(b). For some constants a, b, c, and d, we have

[tex]\dfrac1{x^2+x^4} = \dfrac1{x^2(1+x^2)} = \boxed{\dfrac ax + \dfrac b{x^2} + \dfrac{cx+d}{x^2+1}}[/tex]

3(a). For some constants a, b, and c,

[tex]\dfrac{x^2+4}{x^3-3x^2+2x} = \dfrac{x^2+4}{x(x-1)(x-2)} = \boxed{\dfrac ax + \dfrac b{x-1} + \dfrac c{x-2}}[/tex]

5(a). For some constants a-f,

[tex]\dfrac{x^5+1}{(x^2-x)(x^4+2x^2+1)} = \dfrac{x^5+1}{x(x-1)(x+1)(x^2+1)^2} \\\\ = \dfrac{x^4 - x^3 + x^2 - x + 1}{x(x-1)(x^2+1)^2} = \boxed{\dfrac ax + \dfrac b{x-1} + \dfrac{cx+d}{x^2+1} + \dfrac{ex+f}{(x^2+1)^2}}[/tex]

where we use the sum-of-5th-powers identity,

[tex]a^5 + b^5 = (a+b) (a^4-a^3b+a^2b^2-ab^3+b^4)[/tex]

Using the expected value of a discrete distribution, it is found that:

a) His expected value is of -$2.5.

b) The fair price of a ticket is of $0.5.

The expected value of a discrete distribution is given by the sum of each outcome multiplied by it's respective probability.

In this problem, the distribution for the net value of the ticket is:

Item a:

The expected value is given by:

[tex]E(X) = 497\frac{1}{1000} - 3\frac{999}{1000} = -2.5[/tex]

His expected value is of -$2.5.

Item b:

With an unknown ticket price, the distribution is:

The game is fair if E(X) = 0, hence:

[tex]\frac{500 - x}{1000} - \frac{999x}{1000} = 0[/tex]

0.5 - x = 0

x = 0.

The fair price of a ticket is of $0.5.

More can be learned about the expected value of a discrete distribution at https://brainly.com/question/24855677

Answer:

Option C.

Step-by-step explanation:

Perimeter = 2L + 2h

[tex]P=2(5x^{2}y) +2(3y^{3} )=10x^{2} y+6y^{3}[/tex]

Hope this helps

Answer:

(0,-2)

Step-by-step explanation:

one way to solve...

3x - y = 2 --> *4

-2x + 4y = -8

multiply the first equation by 4

12x - 4y = 8

-2x + 4y = -8

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

10x = 0

x = 0

y = -2

this method is called elimination

another way to solve...

3x - y = 2 simplify for y to get y = 3x - 2

-2x + 4y = -8 substitute y into this equation to solve for x

-2x + 4(3x - 2) = -8

-2x + 12x - 8 = -8

10x = 0

x = 0

y = -2

substitution

alsoo if there are options, just plug in the numbers to the correct variables

Answer:

3xy=2 2x4y=8

3x2=6x8=48x8

Answer:

$1.50

Step-by-step explanation:

There are 4 3-month periods in a year.

$45,000 per 3 months period × 4 = $180,000 cost

$180,000 ÷ 120,000 subscribers = $1.50

The banner ad costs about $1.50 per subscriber

Answer:

The first one should belong in the 54x+18 due to multiplication

The second answer choice has distributive property which has to belong in the 54x+18 category

The third answer choice fits into the 6 times 9x + 6 times one category due to multiplication like the first answer choice.

The last one should be in the 54x+18 category because of the distributive property

Step-by-step explanation:

The Objectives: 54x+18 and (6(9x) + 6

The first one should belong in the 54x+18 due to multiplication

The second answer choice has distributive property which has to belong in the 54x+18 category

The third answer choice fits into the 6 times 9x + 6 times one category due to multiplication like the first answer choice.

The last one should be in the 54x+18 category because of the distributive property

Answer: Angles 6 and 7

Step-by-step explanation:

These angles form a linear pair, and angles that form a linear pair are supplementary.

Answer:

b 34

Step-by-step explanation:

its 34 because if you do 46×2 because of the both sides you will get 92 then subtract 92 from 160 and you will get 68 then do 68÷2 to find out the second side and you will get 34

Answer:

About 5 bags of sand

Step-by-step explanation:

1. Create Equation

[(48x3.14)-(40x3.14)]/6

2. Solve

[(48x3.14)-(40x3.14)]/6

[150.72-125.6]/6

25.12/6

Now if you use a calculater you will get

4.1866666667

So lets just round the number to 5 instead of

4 beacuse if we round it to 4 it won't be enough

to cover the whole path.

so your answer is 5

Using the binomial distribution, it is found that Sam should have expected 5 people to say yes.

The formula is:

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

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

The parameters are:

The expected value is given by:

E(X) = np.

In this problem:

Hence, the expected value for the number of people that say yes is given by:

E(X) = np = 15 x 0.32 = 5.

More can be learned about the binomial distribution at https://brainly.com/question/24863377

Answer:

B

Step-by-step explanation:

-½(10+¼)

-½*10 + -½*¼

-5 + -⅛

-5-⅛

Don't forget a positive and negative makes a negative

Answer:

31.63495408

Step-by-step explanation:

Find area of trinagles and semicurucles and then add up

Answer:

Step-by-step explanation:

The figure comprises of two semicircles and a triangle.

The two semicircles add to one full circle with the diameter of 5 ft.

Area of circle

Area of triangle

Total area

Explanation:

Any point is of the form (x,y)

The first coordinate is x which tells us to go either left or right depending on whether x is negative or positive.

Example: (-2,3) means we go left 2 units

Another example: (5,7) means we go to the right 5 units.

The starting point is the origin where the x and y axis meet up.

Answer:

False

Step-by-step explanation:

equation: (x + 14)² + (y + 5)² = 149

Given:

=============

Formula's:

Find the radius:

[tex]\rightarrow \sf \sqrt{(-7-(-14))^2 + (5-(-5))^2}[/tex]

[tex]\sf \rightarrow \sqrt{\left(-7+14\right)^2+\left(5+5\right)^2}[/tex]

[tex]\sf \rightarrow \sqrt{149}[/tex]

Equation of circle:

Graph for clarification:

Answer:

[tex]\sf (x+14)^2+(y+5)^2=149[/tex]

Step-by-step explanation:

Standard equation of a circle:  [tex]\sf (x-a)^2+(y-b)^2=r^2[/tex]

(where (a, b) is the center and r is the radius of the circle)

Substitute the given center (-14, -5) into the equation:

[tex]\sf \implies (x-(-14))^2+(y-(-5))^2=r^2[/tex]

[tex]\sf \implies (x+14)^2+(y+5)^2=r^2[/tex]

Now substitute the point (-7, 5) into the equation to find r²:

[tex]\sf \implies ((-7)+14)^2+(5+5)^2=r^2[/tex]

[tex]\sf \implies (7)^2+(10)^2=r^2[/tex]

[tex]\sf \implies 149=r^2[/tex]

Final equation:

[tex]\sf (x+14)^2+(y+5)^2=149[/tex]

Let the equation be

[tex] \rightarrow y = mx + c[/tex]

[tex] \huge\rightarrow \: m = \frac{y' - y}{x' - x} \\ [/tex]

[tex] \huge\rightarrow \: m = \frac{4 - 0 }{0 \: - 1} = - 4 \\ [/tex]

hence ,our equation would be,

[tex]\huge \rightarrow 4x + y = 4[/tex]

Assuming  He wants to earn interest on this money rather than keep extra money he does not need. The choice that would be the best for his situation is  $250 to open a regular savings.

A regular savings account is a saving acount that enables you to save and earn interest based on the amount saved because it is an interest bearing account.

Although this account gives you interest but the interest rate is low. Since Tim has $1200 in his checking account  assuming he withdraw $100 twices, Tim will have $1,000 ($1,200-$200).

Now if he open a regular saving with $250 he will have $750 left ($1,000-$250) with which he can earn interest with.

Inconclusion the choice that would be the best for his situation is  $250 to open a regular savings.

Learn more about regular account here:https://brainly.com/question/25787382

In a parallelogram, opposite sides are of equal measure.

QR = TS

find x

1)

2)

Find value of y:

3)

QT:

4)

if x = 5

means the diagram is enlarged.

then y will be:

We need x and y

Opposite sides of a parallelogram are equal

And

Hence

#3

Segment QT is 4

so

Another way

So

scale factor:-

Find y

Answer:

  $6,160

Step-by-step explanation:

The total amount of dividend received is the product of the number of shares and the dividend per share:

  (2000 shares) × ($3.08/share) = $6,160.00 . . . dividend received

Answer:

may not be exact

1- surveying every fifth student who walks in

2-randomly selected people of both genders

3-asking 8 random students from each second period

4-asking 4 girls and 4 boys from each~~

5-randomly choosing a page and counting the words

Step-by-step explanation:

1-explanation; the first one is skewed two 7th graders, the third one is skewed to volleyball players, the fourth one is skewed to eight graders

The product of z₁ = 3 · (cos 14π/15 + i · sin 14π/15) and z₂ = 3 √3 · (cos 11π/15 + i · sin 11π/15) in rectangular form with fully simplified expressions is z₁ · z₂ = 7.794 - i · 13.5.

Let be two numbers of the form z = a + i · b, where i = √-1, the product of two of these numbers in rectangular form is described by the following formula:

z₁ · z₂ = (a + i · b) · (c + i · d) = (a · c - b · d) + i · (a · d + b · c)   (1)

If we know that a = 3 · cos 14π/15, b = 3 · sin 14π/15, c = 3√3 · cos 11π/15, d = 3√3 · sin 11π/15, then the result in rectangular form is:

z₁ · z₂ = 7.794 - i · 13.5

The product of z₁ = 3 · (cos 14π/15 + i · sin 14π/15) and z₂ = 3 √3 · (cos 11π/15 + i · sin 11π/15) in rectangular form with fully simplified expressions is z₁ · z₂ = 7.794 - i · 13.5. [tex]\blacksquare[/tex]

The statement presents typing mistakes and is poorly formatted, the correct form is introduced below:

Given the complex number z₁ = 3 · (cos 14π/15 + i · sin 14π/15) and z₂ = 3 √3 · (cos 11π/15 + i · sin 11π/15), express the result of z₁ · z₂ in rectangular form with fully simplified fractions and radicals.

To learn more on complex numbers, we kindly invite to check this verified question: https://brainly.com/question/10251853

Answer:

[tex]8.6\times 10^{-3}=0.0086[/tex]

Step-by-step explanation:

Answer:

1=32

2=44

3=56

4=70

Step-by-step explanation:

if need working please ask

Answer:

[tex]\sqrt{113}\\[/tex]

Step-by-step explanation:

[tex]distance= \sqrt{(5-[-3])^2+(2-[-5])^2}\\\\distance= \sqrt{(8)^2+(7)^2}\\\\distance= \sqrt{64+49}\\\\distance= \sqrt{113}\\[/tex]

Answer:

The first thing to do is translate the sentence into mathematical notation.

"no more than" means "less than or equal to" which is written as ≤

"seven less than a number" means we are subtracting 7 from a number. We don't know what the number is, so we can use a variable (like n) for the number. So "seven less than a number" becomes n - 7.

The whole thing becomes 12 ≤ n - 7.

To solve for n, we can add 7 to both sides.

19 ≤ n.

19 is less than or equal to n, which means the smallest (minimum) value of the number is 19.

We could also do it without an inequality:

If 12 is 7 less than a number, then the number is 19, because 19-7=12. If we chose a smaller number, like 18, then 7 less than 18 is smaller than 12, which is not allowed (12 is no more than 7 less than the number). So the smallest possible value of the number is 19.

Let the number be p.

Next, "seven times p" can be written like so:-

[tex]\pmb{7p}[/tex]

This expression is less than 18:-

[tex]\bigstar{\boxed{\pmb{7p < 18}}}[/tex]

Hope everything is clear; if you need any explanation/clarification, kindly let me know, and I will comment and/or edit my answer :)

Step-by-step explanation:

1) if the initial figure ABCD, then the figure A'B'C'D' is reduction of the initial one.

2) the scale factor is 0.5: the ratio of all the corresponded coordinates is 2:1 [D(8;4) - D'(4;2); C(4;0)-C'(2;0); B(0;-2)-B'(0;-1) and A(-4;4)-A'(-2;2)].

Mark the quadrilaterals ABcD and A'B'C'D'

The image is reduced in size hence it's reduction

Take 2 sides two calculate scale factor

Scale factor:-

Answer:

0.437

Step-by-step explanation:

Decimals are read the same as whole numbers, starting with the first digit, which is the largest, to the right. So, to find the larger decimal you need to look at the first digit. If this digit is the same then continue to the right until there is a difference in the number. In this case, the first digit is different. Therefore, whichever has the larger first digit must be the greater one. Since 4 > 3, 0.437 must be greater than 0.347.

Step-by-step explanation:

this is what i get , i hope this will help you