f(x)=x^2-2x+3; f(x)=-2x+28

F(x)=x^2-2x+3; F(x)=-2x+28

Answers

Answer 1

Answer:

(-5, 38) and

(5,18)

Step-by-step explanation:

[tex]x^2-2x+3=-2x+28\\<=> x^2-2x+3+2x=28\\<=> x^2 = 28-3=25\\<=> x^2-25=0\\<=> x^2-5^2 =0\\<=> (x-5)(x+5)=0\\<=> x = 5 \ or \ x=-5[/tex]

so the solutions are

(-5, 38) and

(5,18)


Related Questions

Find the exact solution of 3x^2+7=28

Answers

[tex]\text{Solve:}\\\\3x^2+7=28\\\\\text{Subtract 7 from both sides}\\\\3x^2=21\\\\\text{Divide both sides by 3}\\\\x^2=7\\\\\text{Square root both sides}\\\\\sqrt{x^2}=\sqrt7\\\\x=\pm\sqrt7\\\\\boxed{x=\sqrt7\,\,or\,\,x=-\sqrt7}[/tex]

A hiker starts at an elevation of 65 feet and descends 30 feet to the base camp . What is the elevation of the base camps ?

Answers

Answer:

the elevation of base camp is 35 ft

Step-by-step explanation:

Starting at 65 feet elevation, and the descending 30 feet to reach base camp, that means that base camp is at: 65 ft - 30 ft = 35 ft elevation

Answer:

35 feet

Step-by-step explanation:

65 feet- 30 feet= 35 feet is the elevation of the base

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Unit 2 Tutorials
Question 20
Mark this question
For the arithmetic sequence beginning with the terms (-2,0,2,4,6,8...), What is the sum of the first
18 terms?
0 238
0 340
o 304
0 270
Sove and continue

Answers

Answer:

270

Step-by-step explanation:

For any arithmetic sequence

nth term is given by

nth term = a + (n-1)d

where a is first term,

d is common difference

d is given by nth term - (n-1)th term

sum of n terms given by

sum = n/2(2a + (n-1)d)

________________________________________________

Given arithmetic sequence

-2,0,2,4,6,8...

first term a = -2

lets take third term as nth term and second term as (n-1)th term to find common difference d.

d = 2 - 0 = 2

using a = -2 , d = 2, n = 18

thus, sum of first 18 terms = n/2(2a + (n-1)d)

                                           =18/2( 2*(-2) + (18-1) 2)

                                           =9 ( -4 + 34)

                                           =9 ( 30) = 270

Thus, sum of first 18 terms is 270.

I need help
On these two

Answers

Answer:

10.

A. 10240

6.

B. 2^18 = 262144

Step-by-step explanation:

Help needed please!!!!!!!!
Olivia recorded the prices of 10 paperback books and 10 hard cover books. Her data is shown.
Paperback: $6.99, $7.49, $12.99, $9.99, $5.99, $8.99, $9.99, $10.00, $3.99, $4.99
Mean: 8.14
Hard cover: $9.99, $12.99, $34.99, $16.99, $15.00, $19.99, $9.99, $10.99, $18.99, $24.99
Mean: 17.49
Which statement is true given the data?

Answers

Answer:

C

Step-by-step explanation:

b) A man purchased 5 dozen of eggs at Rs 5 each. 10 eggs were broken and he
sold the remaining at Rs 5.70 each. Find
(ii) Profit or loss percent.
(i) his total profit or loss.​

Answers

Answer:

Dear User,

Answer to your query is provided below

(i) Total Loss = Rs.15

(ii) Loss percent = 5%

Step-by-step explanation:

Eggs purchased = 5x12 = 60

Total Cost = 60x5 = Rs 300

Eggs Broken = 10

Eggs Broken cost = 10x5= Rs. 50

Eggs sold = 60-10 = 50

Egg Sale cost = 50x5.70 = Rs 285

(i) Total Loss = C.p. - S.p. = 300 - 285 = 15

(ii) Loss Percent = (Loss/CP)x100 = (15/300)x100 = 5%

If 3 boxes of apples weigh 105 pounds, how much would 2 boxes of apples weigh?​

Answers

Answer:

70 pounds

Step-by-step explanation:

3 boxes= 105 pounds

2boxes= x pounds

Cross Multiply

3*x=105 *2

3x=210

3x/3=210/3

x=70 pounds

Answer:

70

Step-by-step explanation:

105/3=35

35x2=70

So 70 is the answer

Fifteen different Thursday evening programs reported that a commercial cost an average of $169,000 with a standard deviation of $81,000. What is the 95% confidence interval for the true mean

Answers

Answer:

The 95% confidence interval for the true mean is between $0 and $342,729

Step-by-step explanation:

We have the standard deviation for the sample, so we use the t-distribution to solve this question.

The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So

df = 15 - 1 = 14

95% confidence interval

Now, we have to find a value of T, which is found looking at the t table, with 14 degrees of freedom(y-axis) and a confidence level of [tex]1 - \frac{1 - 0.95}{2} = 0.975[/tex]. So we have T = 2.1448

The margin of error is:

M = T*s = 2.1448*81000 = 173,729.

In which s is the standard deviation of the sample.

The lower end of the interval is the sample mean subtracted by M. So it is 169,000 - 173,729 = -4,... = $0(cannot be negative)

The upper end of the interval is the sample mean added to M. So it is 169,000 + 173,729 = $342,729

The 95% confidence interval for the true mean is between $0 and $342,729

There is a bag with only red marbles and blue marbles. The probability of randomly choosing a red marble is 2 9 . There are 45 marbles in total in the bag and each is equally likely to be chosen. Work out how many red marbles there must be

Answers

Answer:

10 red marbles

Step-by-step explanation:

Total= 45 marbles

Probability of red= 2/9

Number of red= 45*2/9= 10

What is 27 ÷ 4 rounded to the nearest tenth?​

Answers

Answer:

6.8

Step-by-step explanation:

27 / 4 = 6.75, which rounded to the nearest tenth, is 6.8.

In the circle above, P is the center,​What is the value, in degrees, ​of θ?

Answers

Answer:

45°

Step-by-step explanation:

There is a propiety that says "The measure of the inscribed angle is half that of the arc that the two sides cut out of the circle."

So the central angle is 90, the inscribed angle will be 90/2=45°

Quadrilateral BCDE is a kite. What is BF?
B
20
С
12
E
F
D

Answers

Answer:

32

Step-by-step explanation:

if u do pythagoras, sq root of 20^2-12^2=16

16x2=32

If f(x) = 4–1 and g(x) = 8x, which expression is equivalent to (g-1)(3)?
O 8-3-(4 + 3)
08-3-(4-32
813)-4432
O 6(3) 4-32

Answers

Answer:

Option (3)

Step-by-step explanation:

Given functions are f(x) = 4 - x² and g(x) = 6x

We gave to find the expression for (g - f)(3).

(g - f)(x) = g(x) - f(x)

            = 6x - (4 - x²)

            = 6x - 4 + x²

By substituting x = 3 in this expression,

(g - f)(x) = 6(3) - 4 + (3)²

Therefore, option (3) will be the answer.

Fill in the following for a possible study with one independent variable (IV) with two conditions/treatments and a dependent variable (DV) that is measured on a continuous scale (interval or ratio): • Independent variable = ______________ • Condition A = ______________ • Condition B = ______________ • Dependent variable = _______________ • How do you know this DV is measured on a continuous scale? • How would you word the null hypothesis for your sample study? • How would you word the alternative hypothesis for your sample study? • What alpha level would you set to test your hypothesis? Why?

Answers

Answer:

Step-by-step explanation:

A possible study is to compare the prices of items in a two different online auction platform: the Dutch auction and the first-priced sealed auction.

Independent variable = the two types of auction

• Condition A = Dutch auction

• Condition B = First-price sealed auction

The Dependent variable in my case study is the prices for each pair of identical items I place in each auction using a known pair sample. The depends variable is measured in the continuous scale because prices are in numbers and these numbers vary continuously, it is not fixed.

The null hypothesis for my study would be: there is no difference in the prices of identical items in the two different auction.

The alternative hypothesis for my study would be: there is a difference in the prices of identical items in the two different auction.

I would set it to the 0.05 level of significance because this is the standard level of significance normally set in a study although this varies.

In triangle FGH, F = 830 inches, g = 460 inches and h=500 inches. Find the measure of angle H
to the nearest degree.​

Answers

9514 1404 393

Answer:

  32°

Step-by-step explanation:

The law of cosines can be used for this:

  h^2 = f^2 +g^2 -2fg·cos(H)

  cos(H) = (f^2 +g^2 -h^2)/(2fg)

  cos(H) = (650,500/763,600)

  H = arccos(6505/7636) ≈ 31.5826°

Angle H is about 32°.

A polygon will be dilated on a coordinate grid to create a smaller polygon. The polygon is dilated using the origin as the center of dilation. Which rule could represent this dilation

Answers

Answer:

(x, y) → (4/5 x, 4/5 y)

Question:

The answer choices to determine the rule that represent the dilation were not given. Let's consider the following question:

A polygon will be dilated on a coordinate grid to create a smaller polygon. The polygon is dilated using the origin as the center of dilation. Which rule could represent this dilation?

A) (x, y) → (0.5 − x, 0.5 − y)

B) (x, y) → (x − 7, y − 7)

C) (x, y) → ( 5/4 x, 5/4 y)

D) (x, y) → (4/5 x, 4/5 y)

Step-by-step explanation:

To determine the rule that could represent the dilation, we would multiply each coordinate by a dilation factor (a constant) to create a dilation. Since the dilation would be used to create a smaller polygon, the constant multiplied with the coordinates of x and y would be less than 1.

Let's check the options out.

In option (A), the coordinates is subtracted from the constant (0.5).

In option (B), the constant (7) is subtracted from the coordinates.

In option (C), the coordinates are multiplied by constant (5/4).

But 5/4 = 1.25. This is greater than 1.

In option (D), the coordinates are multiplied by constant (4/5).

4/5 = 0.8

The constant multiplied with the coordinates of x and y is less than 1 in option (D) = (x, y) → (4/5 x, 4/5 y)

4/5 = 0.8

0.8 is less than 1

Please answer this correctly

Answers

Answer:

514 square meters

Step-by-step explanation:

Consider the length of p;

[tex]11 * p * 3 = 528,\\33 * p = 528,\\p = 528 / 33 = 16 meters[/tex]

11, 3, and p act as the length, width, and height of this rectangular prism. We can apply the volume formula length * width * height, and thus made 11 * p * 3 equivalent to the volume 528. Now let us determine the surface area;

[tex]Area of Side 1 = 16 * 11 = 176 square meters,\\Area of Side 2 = 3 * 16 = 48 square meters,\\Area of Side 3 = 11 * 3 = 33 square meters,\\\\Surface Area = 2 * ( 176 ) + 2 * ( 48 ) + 2 * ( 33 ) = 352 + 96 + 66 = 514 square meters[/tex]

Hope that helps!

Answer:

514 square meters

Step-by-step explanation:

Since the volume of a rectangular prism is the product of the width, length, and height, 11*3*p=528. Therefore, p=528/(11*3)=16. Now, you can find the surface area. The surface of a rectangular prism is made up of 3 pairs of rectangles. One pair has dimensions of 11 by 3, one pair has dimensions of 16 by 3, and the last pair has dimensions of 16 by 11. The surface area of this figure is therefore:

[tex]2(11\cdot 3)+2(16\cdot 3) + 2(16\cdot 11)=66+96+352=514 m^2[/tex]

Hope this helps!

Printed circuit cards are placed in a functional test after being populated with semiconductor chips. A lot contains 140 cards. A sample of 20 cards are selected from the lot without replacement for functional testing. (a) If 20 cards are defective, what is the probability that at least one defective card appears in the sample

Answers

Answer:

The probability that at least one defective card appears in the sample

P(D) = 0.9644 or 96.44%

Step-by-step explanation:

Given;

Total number of cards t = 140

Number of defective cards = 20

Number of non defective cards x = 140-20 = 120

The probability that at least one defective card = 1 - The probability that none none is defective

P(D) = 1 - P(N) ........1

For 20 selections; r = 20

-- 20 cards are selected from the lot without replacement for functional testing

The probability that none none is defective is;

P(N) = (xPr)/(tPr)

P(N) = (120P20)/(140P20)

P(N) = (120!/(120-20)!)/(140!/(140-20)!)

P(N) = (120!/100!)/(140!/120!) = 0.035618370821

P(N) = 0.0356

The probability that at least one defective card appears in the sample is;

P(D) = 1 - P(N) = 1 - 0.0356 = 0.9644

P(D) = 0.9644 or 96.44%

Note: xPr = x permutation r

The sum of three numbers is 10. Two times the second number minus the first number is equal to 12. The first number minus the second number plus twice the third number equals 7. Find the numbers. Listed in order from smallest to largest, the numbers are , , and .

Answers

Answer:

[tex]x =\frac{14}{3} , y = \frac{25}{3} and z = -3[/tex]

The numbers are    [tex]-3 ,\frac{14}{3} , \frac{25}{3}[/tex]

Step-by-step explanation:

Step(i):-

Given sum of the three numbers is 10

Let x , y , z be the three numbers is 10

x +y + z = 10  ...(i)

Given two times the second number minus the first number is equal to 12

2 × y - x = 12 ...(ii)

Given the first number minus the second number plus twice the third number equals 7

x + y + 2 z = 7 ...(iii)

Step(ii):-

Solving (i) and (iii) equations

                       x + y +   z     =    10  ...(i)

                       x + y + 2 z   =     7 ..   (iii)

                     -      -     -         -              

                     0    0    -z      =   3              

Now we know that    z = -3 ...(a)

from (ii)  equation

           2 × y - x = 12 ...(ii)

               x = 2 y -12  ...(b)

Step(iii):-

substitute equations (a) and (b) in equation (i)

                x+y+z =10

           2 y - 12 + y -3 =10

              3 y -15 =10

              3 y = 10 +15

              3 y =25

               [tex]y = \frac{25}{3}[/tex]

Substitute   [tex]y = \frac{25}{3}[/tex]  and   z = -3 in equation(i) we will get

        x+y+z =10

       [tex]x + \frac{25}{3} -3 = 10[/tex]

       [tex]x +\frac{25-9}{3} = 10[/tex]

      [tex]x +\frac{16}{3} = 10[/tex]

      [tex]x = 10 - \frac{16}{3}[/tex]

     [tex]x = \frac{30 -16}{3} = \frac{14}{3}[/tex]

Final answer :-

[tex]x =\frac{14}{3} , y = \frac{25}{3} and z = -3[/tex]

The numbers are  [tex]-3 ,\frac{14}{3} , \frac{25}{3}[/tex]

       

Answer:

-2, 5, 7 on Edge.

Step-by-step explanation:

I got the Answer right.

Chris Evans drives 300 miles per week in his Honda Civic that gets 22 miles per gallon of gas. He
is considering buying a new fuel-efficient car for $20,000 (after trade-in of your Honda Civic)
that gets 50 miles per gallon. Insurance prerniums for the new car and old care are $900 and
$500 per year respectively. If he decides to keep his car, he will need to spend $1200 on repairs
per year. Assume gas costs $3.50 per gallon over a 5-year period,


a, what is the cost of the old car?



b. what is the cost of the new car?

Answers

Answer:

old car $20,909new car: $29,960

Step-by-step explanation:

At 300 miles per week, Chris drives 300×52 = 15,600 miles per year. His gas cost can be figured as ...

  (5 years)×(miles per year)÷(miles/gallon)×($ per gallon) = $273,000/(miles per gallon)

__

a) old car cost = repair cost + gas cost + insurance cost

  = 5($1200) + $273,000/22 + 5($500) ≈ $20,909 . . . over 5 years

__

b) new car cost = purchase cost + gas cost + insurance cost

  = $20,000 + $273,000/50 +5($900) = $29,960 . . . over 5 years

Donte simplified the expression below. 4(1+3i) - (8-5i)
4 + 3i - 8 + 5i
-4 + 8i
What mistake did donte make?

Answers

Answer:

Donde didn't multiply 4(1+3i)

Answer: it’s A he did not apply distributive property yo

Step-by-step explanation:

John leaves school to go home.his bus drives 6 kilometers north and then goes 7 kilometers west.how far is John's house from the school?

Answers

Answer:

John is 9.21 km form the school.

Step-by-step explanation:

John leaves school to go home. His bus drives 6 kilometres north and then goes 7 kilometres west. It is required to find John's distance from the school. It is equal to the shortest path covered or its displacement. So,

[tex]d=\sqrt{6^2+7^2} \\\\d=9.21\ km[/tex]

So, John is 9.21 km form the school.

Solve the inequality and graph the solution set. Write the answer in interval notation. Write your answer in exact simplified form

0> 20x+2>-32

what is the solution?

Answers

Answer:

The solution is [tex]\:\left(-\frac{17}{10},\:-\frac{1}{10}\right)[/tex].

Step-by-step explanation:

An inequality is a mathematical relationship between two expressions and is represented using one of the following:

≤, "less than or equal to"<, "less than">, "greater than" ≥, "greater than or equal to"

To find the solution of the inequality [tex]0>\:20x+2>\:-32[/tex] you must:

[tex]\mathrm{If}\:a>u>b\:\mathrm{then}\:a>u\quad \mathrm{and}\quad \:u>b\\\\0>20x+2\quad \mathrm{and}\quad \:20x+2>-32[/tex]

First, solve [tex]0>20x+2[/tex]

[tex]\mathrm{Switch\:sides}\\\\20x+2<0\\\\\mathrm{Subtract\:}2\mathrm{\:from\:both\:sides}\\\\20x+2-2<0-2\\\\\mathrm{Simplify}\\\\20x<-2\\\\\mathrm{Divide\:both\:sides\:by\:}20\\\\\frac{20x}{20}<\frac{-2}{20}\\\\\mathrm{Simplify}\\\\x<-\frac{1}{10}[/tex]

Next, solve [tex]20x+2>-32[/tex]

[tex]20x+2-2>-32-2\\\\20x>-34\\\\\frac{20x}{20}>\frac{-34}{20}\\\\x>-\frac{17}{10}[/tex]

Finally, combine the intervals

[tex]x<-\frac{1}{10}\quad \mathrm{and}\quad \:x>-\frac{17}{10}\\\\-\frac{17}{10}<x<-\frac{1}{10}[/tex]

The interval notation is [tex]\:\left(-\frac{17}{10},\:-\frac{1}{10}\right)[/tex] and the graph is:

Find the term that must be added to the equation x2−2x=3 to make it into a perfect square. A. 1 B. 4 C. -3 D. 2

Answers

Answer:

1

Step-by-step explanation:

x^2−2x=3

Take the coefficient of x

-2

Divide by 2

-2/2 =-1

Square it

(-1)^2 = 1

Add this to each side

Here It Is !!
More Otw

Answers

Answer:

3

Step-by-step explanation:

0 pairs mean when two "boxes" add together to make 0. For the x's we only have one because x + (-x) = x - x = 0. For the other ones we have two (the + means 1 and the - means -1) because 1 + (-1) = 1 - 1 = 0. Therefore the answer is 1 + 2 = 3.

determine whether the forces in the pair are pulling at right angles to each other for the values. a-3.4 and b=2.6, which are legs of a right triangle, find c, the hypotenuse, to the nearest tenth

Answers

Answer:

4.3 units

Step-by-step explanation:

In this question we use the Pythagorean Theorem which is shown below:

Data are given in the question

Right angle

a = 3.4

b = 2.6

These two are legs of the right triangle

Based on the above information

As we know that

Pythagorean Theorem is

[tex]a^2 + b^2 = c^2[/tex]

So,

[tex]= (3.4)^2 + (2.6)^2[/tex]

= 11.56 + 6.76

= 18.32

That means

[tex]c^2 = 18.56[/tex]

So, the c = 4.3 units

Problem 10: A tank initially contains a solution of 10 pounds of salt in 60 gallons of water. Water with 1/2 pound of salt per gallon is added to the tank at 6 gal/min, and the resulting solution leaves at the same rate. Find the quantity Q(t) of salt in the tank at time t > 0.

Answers

Answer:

The quantity of salt at time t is [tex]m_{salt} = (60)\cdot (30 - 29.833\cdot e^{-\frac{t}{10} })[/tex], where t is measured in minutes.

Step-by-step explanation:

The law of mass conservation for control volume indicates that:

[tex]\dot m_{in} - \dot m_{out} = \left(\frac{dm}{dt} \right)_{CV}[/tex]

Where mass flow is the product of salt concentration and water volume flow.

The model of the tank according to the statement is:

[tex](0.5\,\frac{pd}{gal} )\cdot \left(6\,\frac{gal}{min} \right) - c\cdot \left(6\,\frac{gal}{min} \right) = V\cdot \frac{dc}{dt}[/tex]

Where:

[tex]c[/tex] - The salt concentration in the tank, as well at the exit of the tank, measured in [tex]\frac{pd}{gal}[/tex].

[tex]\frac{dc}{dt}[/tex] - Concentration rate of change in the tank, measured in [tex]\frac{pd}{min}[/tex].

[tex]V[/tex] - Volume of the tank, measured in gallons.

The following first-order linear non-homogeneous differential equation is found:

[tex]V \cdot \frac{dc}{dt} + 6\cdot c = 3[/tex]

[tex]60\cdot \frac{dc}{dt} + 6\cdot c = 3[/tex]

[tex]\frac{dc}{dt} + \frac{1}{10}\cdot c = 3[/tex]

This equation is solved as follows:

[tex]e^{\frac{t}{10} }\cdot \left(\frac{dc}{dt} +\frac{1}{10} \cdot c \right) = 3 \cdot e^{\frac{t}{10} }[/tex]

[tex]\frac{d}{dt}\left(e^{\frac{t}{10}}\cdot c\right) = 3\cdot e^{\frac{t}{10} }[/tex]

[tex]e^{\frac{t}{10} }\cdot c = 3 \cdot \int {e^{\frac{t}{10} }} \, dt[/tex]

[tex]e^{\frac{t}{10} }\cdot c = 30\cdot e^{\frac{t}{10} } + C[/tex]

[tex]c = 30 + C\cdot e^{-\frac{t}{10} }[/tex]

The initial concentration in the tank is:

[tex]c_{o} = \frac{10\,pd}{60\,gal}[/tex]

[tex]c_{o} = 0.167\,\frac{pd}{gal}[/tex]

Now, the integration constant is:

[tex]0.167 = 30 + C[/tex]

[tex]C = -29.833[/tex]

The solution of the differential equation is:

[tex]c(t) = 30 - 29.833\cdot e^{-\frac{t}{10} }[/tex]

Now, the quantity of salt at time t is:

[tex]m_{salt} = V_{tank}\cdot c(t)[/tex]

[tex]m_{salt} = (60)\cdot (30 - 29.833\cdot e^{-\frac{t}{10} })[/tex]

Where t is measured in minutes.

Analyze the function for domain, range, continuity, symmetry, boundedness, extrema, and asymptotes. f(x)=-2cot x

Answers

Answer:

(See explanation below for further details)

Step-by-step explanation:

The domain of the function is:

[tex]x \in \mathbb{R} - \{ \pm \pi \cdot i \}[/tex] for [tex]i \in \mathbb{N}_{O}[/tex]

The range of the function is:

[tex]f(x) \in \{-\infty, +\infty \}[/tex]

There are no absolute extrema and such function is not bounded.

Function is symmetric, whose period is π.

Lastly, the set of asymptotes is:

[tex]x = \pm \pi \cdot i[/tex], for [tex]i \in \mathbb{N}_{O}[/tex]

Answer:

Step-by-step explanation:

edge

A school needs 1,860 pencils for its students. The pencils are sold in boxes of 12. How many boxes does the school need to order?

Answers

Answer:

Step-by-step explanation:

155

The number of boxes required by the school to order is 155.

What is the unitary method?

The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value.  If an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.

We have been given that the school needs 1,860 pencils for its students. Also, the pencils are sold in boxes of 12.

We need to find the school needs to requires boxes to order.

Total number of pencil = 1,860

Number of boxes = 12

Therefore, boxes needed = 1,860 / 12

= 155

Hence, the number of boxes required by the school to order is 155.

To learn more about the unitary method, please visit the link given below;

https://brainly.com/question/23423168

#SPJ5

Write an integral for the area of the surface generated by revolving the curve y equals cosine (2 x )about the​ x-axis on negative StartFraction pi Over 5 EndFraction less than or equals x less than or equals StartFraction pi Over 5 EndFraction .

Answers

Answer:

The integral is

∫ˣ²ₓ₁ 2π cos 2x √[1 + 4 sin² 2x] dx

x₁ = (-π/5)

x₂ = (π/5)

And the area of the surface generated by revolving = 9.71 square units

Step-by-step explanation:

When a function y = f(x) is revolved about the x-axis, the formula for the area of the surface generated is given by

A = 2π ∫ˣ²ₓ₁ f(x) √[1 + (f'(x))²] dx

A = 2π ∫ˣ²ₓ₁ y √[1 + y'²] dx

For this question,

y = cos 2x

x₁ = (-π/5)

x₂ = (π/5)

y' = -2 sin 2x

1 + y'² = 1 + (-2 sin 2x)² = (1 + 4 sin² 2x)

So, the Area of the surface of revolution is

A = 2π ∫ˣ²ₓ₁ y √[1 + y'²] dx

= ∫ˣ²ₓ₁ 2πy √[1 + y'²] dx

Substituting these variables

A = ∫ˣ²ₓ₁ 2π cos 2x √[1 + 4 sin² 2x] dx

Let 2 sin 2x = t

4 cos 2x dx = dt

2 Cos 2x dx = (dt/2)

dx = (1/2cos 2x)(dt/2)

Since t = 2 sin 2x

when x = (-π/5), t = 2 sin (-2π/5) = -1.90

when x = (π/5), t = 2 sin (2π/5) = 1.90

A

= ∫¹•⁹⁰₋₁.₉₀ π (2 Cos 2x) √(1 + t²) (1/2cos 2x)(dt/2)

= ∫¹•⁹⁰₋₁.₉₀ (π/2) √(1 + t²) (dt)

= (π/2) ∫¹•⁹⁰₋₁.₉₀ √(1 + t²) (dt)

But note that

∫ √(a² + x²) dx

= (x/2) √(a² + x²) + (a²/2) In |x + √(a² + x²)| + c

where c is the constant of integration

So,

∫ √(1 + t²) dt

= (t/2) √(1 + t²) + (1/2) In |t + √(1 + t²)| + c

∫¹•⁹⁰₋₁.₉₀ √(1 + t²) (dt)

= [(t/2) √(1 + t²) + (1/2) In |t + √(1 + t²)|]¹•⁹⁰₋₁.₉₀

= [(1.90/2) √(1 + 1.90²)+ 0.5In |1.90+√(1 + 1.90²)|] - [(-1.9/2) √(1 + -1.9²) + (1/2) In |-1.9 + √(1 + -1.9²)|]

= [(0.95×2.147) + 0.5 In |1.90 + 2.147|] - [(-0.95×2.147) + 0.5 In |-1.90 + 2.147|]

= [2.04 + 0.5 In 4.047] - [-2.04 + 0.5 In 0.247]

= [2.04 + 0.70] - [-2.04 - 1.4]

= 2.74 - [-3.44]

= 2.74 + 3.44

= 6.18

Area = (π/2) ∫¹•⁹⁰₋₁.₉₀ √(1 + t²) (dt)

= (π/2) × 6.18

= 9.71 square units.

Hope this Helps!!!

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