Find the volume of the solid whose base is the region bounded by the x-axis, the curve y=√x^2+1 , x=0 and x=3 and which has the property that each cross section perpendicular to the x-axis is an equilateral triangle.

Answers

Answer 1

The volume of the solid with a base bounded by the curve y=√(x^2+1), x=0 and x=3, and with equilateral triangle cross sections perpendicular to the x-axis, is 81√3/5 cubic units, obtained by integrating (1/4)(3/2)(x^2 + 1)√3(x^2 + 1) over [0,3].

To find the volume of the solid whose base is the region bounded by the x-axis and the curve y=√(x^2+1), and whose cross sections perpendicular to the x-axis are equilateral triangles, we can use the method of slicing.

Consider a vertical slice of the solid at an arbitrary value of x

This slice has width dx and thickness equal to the height of the equilateral triangle. Let h be the height of the equilateral triangle. Then the volume of this slice is:

dV = A(x) dx = (1/2)(h)(h√3/2) dx

where A(x) is the area of the cross section at x, which is an equilateral triangle with side length h.

To find h, we can use the Pythagorean theorem to relate h to y:

h^2 + (y/2)^2 = y^2

Simplifying, we get:

h^2 = (3/4)y^2

Taking the square root of both sides and substituting y = √(x^2 + 1), we get:

h = √(3/4)(x^2 + 1)

Substituting this expression for h into the formula for dV and integrating over the region of interest, we get:

V = ∫[0,3] (1/2)(h)(h√3/2) dx

 = ∫[0,3] (1/4)(3/2)(x^2 + 1)√3(x^2 + 1) dx

 = (1/8)(3√3)∫[0,3] (x^4 + 2x^2 + 1) dx

 = (1/8)(3√3)(243/5)

 = 81√3/5

Therefore, the volume of the solid is 81√3/5 cubic units.

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Related Questions

2-3 (8.5A) Hikaru is going to make x pies, he will put 3 pounds of berries in each one, which equation can be used to find b, the total number of pounds
of berries he will put in the pies
A
C b=x+3
5
8
b=x+3
35
b = x - 3
5
B
D
5
b = 3

2-3 (8.5A) Hikaru is going to make x pies, he will put 3 pounds of berries in each one, which equation

Answers

Answer:

D

Step-by-step explanation:

the answer is d because it's the 3 and 5/8ths per pie this it is multiplied by x.

very few tools are available to assist system analysts and end users in analyzing data.

Answers

False. very few tools are available to assist system analysts and end users in analyzing data.

There are many tools available to assist system analysts and end users in analyzing data. Some commonly used tools include spreadsheets (such as Microsoft Excel), data visualization software (such as Tableau or Power BI), statistical analysis software (such as R or SPSS), and business intelligence software (such as SAP or Oracle).

These tools allow analysts and end users to process, manipulate, and visualize large volumes of data, as well as to perform complex statistical analyses and modeling. Additionally, with the advent of big data technologies, such as Hadoop and Spark, analysts and end users can now analyze vast amounts of data in real-time, using distributed computing systems.

In summary, there are many tools available to assist system analysts and end users in analyzing data, and these tools continue to evolve and improve over time.

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coefficient of 3+4x-7

Answers

-4(1-x) I think that’s the correct answer

48 people are at a wedding reception. The tables are arranged so that the same number of people sit at each table. How many people sit at each table?
Give two possible answers
(I do know the answer but I just wanna test you guys)

Answers

Answer:

Step-by-step explanation:

Simply if there are 6 tables so 48 ÷ 6 = 8 people on each table

So it depends on the number of table

Complete the table to investigate dilations of
exponential functions.
3.2*
23x
-2
NAN
62
-1
2 / 를
0
a
b
с
1
d.
e
f
2
4
12
64
a =
b
с
e
f=
d =
DONE
Intro

Complete the table to investigate dilations ofexponential functions.3.2*23x-2NAN62-12 / 0ab1d.ef241264a

Answers

Answer:

a= 1

b= 3

c= 2

d= 2

e= 6

f= 8. These are the answers for the question. Love from Gauthmath

The standard exponential function is expressed as y =ab^x. The values of the variables are: b = c = 3, a = 1, d = 2,e = 6, f = 9

Functions and values

The standard exponential function is expressed as:

y =ab^x

wherex is the exponent

a is the base

According to the question, we are to fill the given table based on the given exponential functions.

If the exponential function is 2^x

a = 2^0 = 1

d = 2^1 = 2

If the exponential function is 3*2^x

b = 3* 2^0 = 3

e = 3 * 2^1 = 6

If the exponential function is 2^3x

c = 3^2(0) = 3^1 = 3

f = 3^2(1) = 3^2 = 9

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which coordinate plane represents the linear relationship 4x+5y=20

Answers

Answer:(0,4)

Step-by-step explanation:

4x+5y=20

4(0)+5y=20

5y=20

5x/5=20/5

y=4

4x+5(4)=20

4x+20=20

   -20   -20

4x=0

4x/4=0/4

x=0

4x+5y=20

4(0)+5(4)=20

0+20=20

20=20

I think this is right

I2. How many teachers are there in your school? How many male teachers and female

teachers are there?

I) Write the ratio of male teachers to the female teachers.

ii) Write the ratio of male teachers to the total number of teachers.

iii) Write the ratio of total of teachers to the female teachers.v​

Answers

There are 42 teachers 30 male 12 female. 30:12.  male teachers to the total number of teachers 42:30. total of teachers to the female teachers 42:12

                                     

Which technique(s) are appropriate to completely factor the
expression?
A)
Factor as a difference of squares.
B)
Factor out the greatest common factor.
C)
Factor out the greatest common factor; then factor
the difference of squares.
D)
Factor out the greatest common factor; then factor
the perfect square trinomial

Answers

Answer: C

Step-by-step explanation:

Here's a graph of a linear function. Write the
equation that describes that function.
Express it in slope-intercept form.

Here's a graph of a linear function. Write theequation that describes that function.Express it in slope-intercept

Answers

I took 2 points from the graph and found my slope by using the slope formula, then I found my y-int which is the where the y crosses and that is -3. Hope this helps good luck !!

Answer: y=1/2x-3
Here's a graph of a linear function. Write theequation that describes that function.Express it in slope-intercept


I need to find X and Y

Can someone pls help me

I need to find X and Y Can someone pls help me

Answers

y=35 x=110 however im a bit confused on x but im sure on y

Please help me I'm stuck. I will give 30 points for this one. Given triangle ABC tilde triangle PQR and your scale factor Complete the hotspots for these similar triangles and show work

Please help me I'm stuck. I will give 30 points for this one. Given triangle ABC tilde triangle PQR and

Answers

The value for the hotspots of the similar triangles ∆ABC and ∆PWR are:

(1). angle B = 68°

(2). PQ = 5cm

(3). BC = 19.5cm

(4). area of ∆PQR = 30cm²

What are similar triangles

Similar triangles are two triangles that have the same shape, but not necessarily the same size. This means that corresponding angles of the two triangles are equal, and corresponding sides are in proportion.

(1). angle B = 180 - (22 + 90) {sum of interior angles of a triangle}

angle B = 68°

Given that the triangle ∆ABC is similar to the triangle ∆PQR.

(2). PQ/7.5cm = 12cm/18cm

PQ = (12cm × 7.5cm)/18cm {cross multiplication}

PQ = 5cm

(3). 13cm/BC = 12cm/18cm

BC = (13cm × 18cm)/12cm {cross multiplication}

BC = 19.5cm

(4). area of ∆PQR = 1/2 × 12cm × 5cm

area of ∆PQR = 6cm × 5cm

area of ∆PQR = 30cm²

Therefore, the value for the hotspots of the similar triangles ∆ABC and ∆PWR are:

(1). angle B = 68°

(2). PQ = 5cm

(3). BC = 19.5cm

(4). area of ∆PQR = 30cm²

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The ratio of students to adults on a field trip is 48 to 6. Which table correctly represents this ratio?

The ratio of students to adults on a field trip is 48 to 6. Which table correctly represents this ratio?

Answers

Table B correctly represents the ratio of students to adult

What is a ratio?

Ratio is an ordered pair of numbers x and y, which can written in form x/ y where y is not 0.

For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is 8 :6 Similarly, the ratio of lemons to oranges is 6:8

ratio 48:6= 48/6

= 8

this means that 8 student to 1 adult

in table B all ratio of student to adult is 8:1

therefore only table 2 obeys this ratio

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Helpful conversions: 1 acre =43,560 square feet 1 hectare =10,000 square meters 1lb=454 grams 1 kg=2.2lbs=1000grams 1. Scouting for cover crop establishment in the field requires conversion of your massbased seeding rate (e.g., lbs/acre) to a population-based establishment rate (e.g., plants/f? ). Let's practice doing that. Convert the following seeding rates: a. Cereal rye - 60 ibs/acre = H seeds/square foot b. Crimson clover −15 kg/ha= H seeds/square meter 2. Convert the following population-based seeding rates to mass-based seeding rates. a. Oats −30 plants/square foot = Ibs seedlacre b. Hairy vetch −20 plants/square meter = kg seedsha 3. Your farm manager wants you to drill crimson clover seed, but the seed hasn't arrived yet and you don't know if it's coated or not. If the seed is not coated, the seeding rate is 20 blacre. Will you need to increase or decrease your seeding rate if the seed is coated?

Answers

Coating the seed typically improves seed quality and germination rates, so the same seeding rate can be maintained even with coated seed.

To convert mass-based seeding rates to population-based establishment rates, we need to consider the area conversion factors provided:

a. Given a seeding rate of 60 lbs/acre for cereal rye, we can convert it to seeds per square foot by using the conversion factor of 1 acre = 43,560 square feet.

b. Given a seeding rate of 15 kg/ha for crimson clover, we can convert it to seeds per square meter using the conversion factor of 1 hectare = 10,000 square meters.

To convert population-based seeding rates to mass-based seeding rates, we can use the given area conversion factors:

a. Given a seeding rate of 30 plants/square foot for oats, we can convert it to pounds of seed per acre.

b. Given a seeding rate of 20 plants/square meter for hairy vetch, we can convert it to kilograms of seed per hectare.

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simplify 2√18a²x5
please :)

Answers

Step-by-step explanation:

2×18a ×5

36a×5

180a for simplification

quadrilateral WXYZ with vertices W (-1,-1), X (4,1), Y (4,5) and Z (1,7) over the x-axis

Answers

The new coordinates of the image include the following:

W' (-1, 1)

X' (4, -1)

Y' (4, -5)

Z' (1, -7)

What is a reflection over the x-axis?

In Mathematics and Geometry, a reflection over or across the x-axis is represented by this transformation rule (x, y) → (x, -y). This ultimately implies that, a reflection over or across the x-axis would maintain the same x-coordinate while the sign of the y-coordinate changes from positive to negative or negative to positive.

Next, we would apply a reflection over or across the x-axis to quadrilateral WXYZ;

(x, y)                →      (x, -y)

W (-1, -1)         →      (-1, -(-1)) = W' (-1, 1)

X (4, 1)           →      (4, -(1)) = X' (4, -1)

Y (4, 5)         →      (4, -(5)) = Y' (4, -5)

Z (1, 7)         →      (1, -(7)) = Z' (1, -7)

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Complete Question:

Quadrilateral WXYZ with vertices W (-1,-1), X (4,1), Y (4,5) and Z (1,7) reflected over the x-axis. What are the new coordinates of the image?

Need help ASAP! Directions in picture ;)

Need help ASAP! Directions in picture ;)

Answers

h= 92 and h and f equal the same cuz they’re congruent (i think is the right term) so 5x+4=92 so x would be 17.6

the day length in manila, philippines, varies over time in a periodic way that can be modeled by a trigonometric function. assume the length of the year (which is the period of change) is exactly 365365365 days long. the shortest day of the year is december 212121, and it's 675.85675.85675, point, 85 minutes long. manila's longest day is 779.60779.60779, point, 60 minutes long. note that december 212121 is 111111 days before january 111. find the formula of the trigonometric function that models the length lll of the day ttt days after january 111. define the function using radians. \qquad l(t)

Answers

The length of the day in Manila, Philippines, varies periodically and can be modeled by a trigonometric function. The formula is l(t) = (779.607 - 675.856) xcos((2π/365) x (t - 111111)) + (779.607 + 675.856) / 2.

To model the variation in the length of the day in Manila, a trigonometric function is used, considering the period of change as exactly 365365365 days. The shortest day occurs on December 21, with a length of 675.85675.85675, point, 85 minutes, while the longest day occurs on June 21, with a length of 779.60779.60779, point, 60 minutes.

To find the formula for the trigonometric function, we need to consider the general form of a cosine function, which includes an amplitude, angular frequency, and phase shift. Here, the amplitude is (779.607 - 675.856), the angular frequency is (2π/365), and the phase shift is (t - 111111), where t represents the number of days after January 1.

Therefore, the formula for the length of the day can be expressed as l(t) = (779.607 - 675.856) xcos((2π/365) x (t - 111111)) + (779.607 + 675.856) / 2. This equation represents the variation in the day length throughout the year in Manila, with the cosine function capturing the periodic nature of the changes.

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a traffic light is red for 25 seconds, yellow for 5 seconds, and green for 55 seconds. what is the probability that when you reach the light, a) the light is green b) the light is yellow c) the light is not red d) the light is not green

Answers

The probability of light being green ,yellow, not red ,and not green are 11/17, 1/17, 12/17 and 6/17 respectively.

Traffic light is red for 25 sec

Traffic light is yellow for 5 sec

Traffic light is green for 55 sec

Total of 85 seconds

The probability of the light being yellow when you reach it, all other things being equal, is 5 / 85 = 1/17

The probability of the light being green when you reach it, all other things being equal, is 55/ 85 = 11/17

The probability of the light being not red when you reach it, all other things being equal, is 5+55/ 85 = 60/85 = 12/17

The probability of the light being not green when you reach it, all other things being equal, is 5+25/ 85 = 30/85 = 6/17

As a result, the odds of a light being green, yellow, not red, or not green are respectively 11/17, 1/17, 12/17, and 6/17.

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Choose the correct formula for combination.
A . n!/r!(n-r)!
B. n!/(n-r)!

Answers

Answer:

B

Step-by-step explanation:

The number of combinations of n objects taken r at a time is determined by the following formula: C(n,r)=n!/(n−r)!

Answer: The answer is A n!/r!(n-r)!

search it up

jamie is a senior in college contemplating the next steps after graduation (graduate school or getting a job). one factor that influences jamie's decision is the annual salary of a college graduate versus the annual salary of an individual with a master's degree. jaime decides to look up the average salary of college graduates in that profession. which information would be more useful to jaime in making the decision between pursuing graduate studies or applying for a job, the mean annual salary or the median annual salary? please provide your answer in a few sentences.

Answers

The median annual salary would be more useful to Jaime in making the decision between pursuing graduate studies or applying for a job.

The median annual salary represents the middle value in a salary distribution, which indicates the typical or central salary of college graduates in that profession. It is less affected by extreme values or outliers in the data. Therefore, the median provides a more reliable measure of the average salary that Jaime can expect as a college graduate. On the other hand, the mean annual salary can be influenced by a few individuals with exceptionally high or low salaries, which may not accurately reflect the typical salary range. Hence, the median salary would be a better indicator for Jaime to make an informed decision.

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Evaluate ab - 0.50 when a = 1 and b = 5

Answers

Answer: 4.5

Step-by-step explanation:

To evaluate ab - 0.50, we want to plug in the given values for a and b and then solve.

(1)(5)-0.5                       [multiply]

5-0.5                             [subtract]

4.5

After plugging in the values of a and b, we found that the solution is 4.5.

PLZZ HELPP I HAVE TO GO SOMEWHERE RN ASAP! (:
Which statement is true about the theory of plate tectonics and the theory of continental drift?

A. The theory of plate tectonics explains how continental movements could occur.

B. The theory of plate tectonics tells exactly where the continents were before Pangaea divided.

C. The theory of plate tectonics shows that continental movements could not have happened.

D. The theory of plate tectonics shows that Pangaea was impossible, but the continents did move.

Answers

The correct answer is ...

D
the answer is gonna be

D

Select the protractor measuring an angle that is 55°.

A. Figure A
B. Figure B
C. Figure C
D. Figure D

Select the protractor measuring an angle that is 55.A. Figure AB. Figure BC. Figure CD. Figure D

Answers

Answer:

A. Figure A

Step-by-step explanation:

The protractor on the top left has an arrow halfway inbetween 50° and 60, making the angle 55°.

Therefore, the answer is Figure A.

The figure above shows the graph of f', the derivative of a function f. The domain of f is the set of all real numbers x such that -3 < x < 5. a. For what values of x does f have a relative maxlmum? Why? b. On what intervals is the graph of concave upward? Use ' to justify your answer. C. Suppose that f(1) = 0. In the xy-plane provided, draw a sketch that shows the general shape of the graph of the function f on the open interval 0 < x < 2. у X O 2

Answers

a) For value of x=-2 given function have a relative maximum.

b) The graph is concave upward in the interval of (-1,1) and (3,5)

c)

What is a function?

A function in mathematics from a set X to a set Y allocates precisely one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively.

What is Differentiation?

Finding a function's derivative, or rate of change, is the process of differentiation in mathematics. The practical approach of differentiation may be performed utilizing just algebraic operations, three fundamental derivatives, four principles of operation, and an understanding of how to manipulate functions, in contrast to the theory's abstract character.

solution of above questions are:

a)Function 'f' has relative maximum at x=-2 because f'(x)=0 and f' is   changing from +ve to -ve

b) (-1,1 ) and (3,5) are intervals of graph that is concave upward  because f' is increasing and f'' is +ve.

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The figure above shows the graph of f', the derivative of a function f. The domain of f is the set of

brainliest if right!!!!

brainliest if right!!!!

Answers

Answer:

B

Step-by-step explanation:

-4a + 139 = 4a + 123

a = 2

8 + 123 =  131

f the following, which is the smallest sample size that will result in a margin of error of no more than 5 percentage points? responses 73 73 97 97 271 271 385 385 1,537 1,537 skip to navigation

Answers

The smallest sample size that will result in a margin of error of no more than 5% for a 95% confidence interval is given as follows:

385.

What is a confidence interval of proportions?

A confidence interval of proportions has the bounds given by the rule presented as follows:

\(\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}\)

In which the variables used to calculated these bounds are listed as follows:

\(\pi\) is the sample proportion, which is also the estimate of the parameter.z is the critical value.n is the sample size.

The margin of error is modeled as follows:

\(M = z\sqrt{\frac{\pi(1-\pi)}{n}}\)

The confidence level is of 95%, hence the critical value z is the value of Z that has a p-value of \(\frac{1+0.95}{2} = 0.975\), so the critical value is z = 1.96.

We have no estimate, hence we consider that:

\(\pi = 0.5\)

The minimum sample size is obtained as n when M = 0.05, hence:

\(M = z\sqrt{\frac{\pi(1-\pi)}{n}}\)

\(0.05 = 1.96\sqrt{\frac{0.5(0.5)}{n}}\)

\(0.05\sqrt{n} = 1.96 \times 0.5\)

\(\sqrt{n} = 1.96 \times 10\) (0.5/0.05 = 10).

\((\sqrt{n})^2 = (1.96 \times 10)^2\)

n = 384.16

Hence rounded to 385, as a sample size of 384 would have a margin of error slightly above 0.05.

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Precalculus to wa
Please help Ik the language

Answers

Answer:

In mathematics education, precalculus or College algebra is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus. Schools often distinguish between algebra and trigonometry as two separate parts of the coursework.

Step-by-step explanation:

Dude you can search this stuff up

What is the range of the function y = x??
Oyo
oxio
all real numbers

Answers

Answer:

All Real Numbers

General Formulas and Concepts:

Algebra I

Range is the set of y-values that are outputted by function f(x)

Step-by-step explanation:

If we graph the function, we see that the line stretches out from negative infinity to infinity for both x-values and y-values.

If that is the case, then that means that there is no restrictions on the y-values.

Therefore, our range is (-∞, ∞) or All Real Numbers.

What is the range of the function y = x??Oyooxioall real numbers

Can anyone help me please

Can anyone help me please

Answers

Answer: 12pi in squared

12pi is the answer!!

Jose invests money in an account paying a simple interest of 9% per year. If no money will be added or removed from the investment, what should he multiply his current balance by to find his total balance in a year in one step?

Answers

Jose invests money in an account paying a simple interest of 9% per year, he should multiply his current balance by 9% to find his total balance.

Simple Interest is a simple way for computing interest on a loan or principle amount. The notion of simple interest is used in many sectors, including banking, finance, and vehicles. The monthly interest is deducted first, followed by the principal amount, when you make a loan payment.

We have rate of interest as 9%

So, SI = PNR/100

SI = P x 1 x 9/100

SI = P x 9%.

Therefore, current balance should be multiplied by 9% .

Simple Interest (S.I.) is a way of determining the amount of interest for a given principle amount. You put that money to use for the reason you borrowed it in the first place. The money is then returned after you receive your following month's pocket money from your parents. This is how borrowing and lending operate in the real world.

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