Match the formulas for volume and calculate the volumes of the sphere, cylinder, and cone shown below. Each shape has a radius of 2.5 and the cylinder and cone have a height of 4.

options for each drop down box [choose]:
Sphere - volume measure
Sphere - volume formula
Cone - volume formula
Cone - volume measure
Cylinder - volume measure
Cylinder - volume formula
None of these options

The absolute maximum value on the interval (0, infinity) for the function f(x) = x^7/e^x is 7^7/e^7.

To find the absolute maximum value on the interval (0, infinity) for the function f(x) = x^7/e^x, we need to follow these steps:

1. Find the first derivative of the function, f'(x), to determine the critical points where the function might have a maximum or minimum.
2. Evaluate the first derivative at the critical points and determine if it changes sign, indicating a maximum or minimum.
3. Verify if the function has an absolute maximum on the given interval.

Step 1: Find the first derivative f'(x) using the quotient rule.
f'(x) = (e^x * 7x^6 - x^7 * e^x) / (e^x)^2

Step 2: Simplify f'(x) and find the critical points.
f'(x) = x^6(7 - x) / e^x
f'(x) = 0 when x = 0 (not included in the interval) or x = 7

Step 3: Evaluate the first derivative around the critical point x = 7 to determine if it's a maximum or minimum.
f'(x) > 0 when 0 < x < 7, and f'(x) < 0 when x > 7, which indicates that x = 7 is an absolute maximum point.

Now we can find the absolute maximum value by plugging x = 7 into the original function, f(x):
f(7) = 7^7/e^7

Thus, the absolute maximum value on the interval (0, infinity) for the function f(x) = x^7/e^x is 7^7/e^7.

To learn more about derivatives visit : https://brainly.com/question/12047216

#SPJ11

The Laplace transform of the solution to the initial value problem y'' + y' - 2y = 0, y(0) = -5, is Y(s) = (5s + 4) / (s² + s - 2), and Y(5) = 29 / 28.

To find the Laplace transform of the solution to the initial value problem y'' + y' - 2y = 0, y(0) = -5, we can apply the Laplace transform to both sides of the differential equation and use the initial condition to solve for the Laplace transform of y.

Taking the Laplace transform of both sides of the differential equation, using the linearity and derivative properties of the Laplace transform, we get:

L{y'' + y' - 2y} = L{0}

s² Y(s) - s y(0) - y'(0) + s Y(s) - y(0) - 2 Y(s) = 0

s² Y(s) - 5s + s Y(s) + 4 + 2 Y(s) = 0

Simplifying and solving for Y(s), we get:

Y(s) = (5s + 4) / (s²+ s - 2)

To find Y(5), we substitute s = 5 into the expression for Y(s):

Y(5) = (5(5) + 4) / ((5)² + 5 - 2)

Y(5) = 29 / 28

Learn more about maximing function: https://brainly.com/question/31007009

#SPJ11

The mean, median, mode,range, and standard deviation when decreased by 15% becomes $43.96,$31.66,$18.16,$353.47 and $10.12 respectively.

When each purchase decreases by 15%, the new values can be calculated as follows:
Mean: $51.72 * 0.85 = $43.96

It is calculated by adding up all the values and dividing the sum by the number of values.
Median: $37.25 * 0.85 = $31.66

It is calculated by the values from smallest to largest and then selecting the middle value.
Mode: $21.36 * 0.85 = $18.16

It represents the most frequently occurring value in a set of numbers.
Range: $415.85 * 0.85 = $353.47

It represents the difference between the largest and smallest values in a set of numbers.
Standard Deviation: $11.91 * 0.85 = $10.12

It is calculated by taking the square root of the variance.

When each purchase is decreased by 15%, the mean is $43.96, the median is $31.66, the mode is $18.16, the range is $353.47, and the standard deviation is $10.12.

Learn more about central tendency : https://brainly.com/question/14532771

#SPJ11

The surface area of the cube is 60.75 in²

Cube is a solid three-dimensional figure, which has 6 square faces, 8 vertices and 12 edges. The sides and surfaces of a cube are equal. A cube can also be called square prism.

The surface area of a cube is expressed as;

SA = 6l²

where l is the edge length.

l = 4 1/2 = 9/2

SA = 6(9/2)²

SA = 6 × 81/4

SA = 243/4

= 60.75 in²

Therefore the surface area of the cube with edge length 4 1/2 in is 60.75 in²

learn more about surface area of cube from

https://brainly.com/question/28776132

#SPJ1

The sum of the series Σ (2k² - 4) from k = 1 to n can be found using the following formula:

Σ (2k² - 4) = [2(1²) - 4] + [2(2²) - 4] + [2(3²) - 4] + ... + [2(n²) - 4]

= 2(1² + 2² + 3² + ... + n²) - 4n

The sum of squares of the first n natural numbers can be calculated using the formula:

1² + 2² + 3² + ... + n² = [n(n + 1)(2n + 1)] / 6

Substituting this value in the above equation, we get:

Σ (2k² - 4) = 2[(n(n + 1)(2n + 1)) / 6] - 4n

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

Therefore, the sum of the series Σ (2k² - 4) from k = 1 to n is (n(n + 1)(2n + 1)) / 3 - 4n.

The value of x, which is  the sample mean and represents the average weight gain after quitting smoking, is 20. Therefore, the correct option is B.

Given that the random sample of 100 smokers reveals an average weight gain of 20 pounds after quitting smoking and a standard deviation of 6 pounds, the value of x is determined as follows.

1. Random sample of 100 smokers (n = 100)

2. Average weight gain after quitting smoking is 20 pounds (mean, x' = 20)

3. Standard deviation is 6 pounds (σ = 6)

Option A (6) is incorrect because it does not relate to the given information.

Option C (100) is also incorrect because it refers to the sample size, which is not relevant to finding the value of x.

Option D (6/√100 = 0.6) is incorrect because it calculates the standard error of the mean, which is not what the question is asking for.

Therefore, the correct answer is option B: 20, which is the sample mean and represents the average weight gain after quitting smoking.

Learn more about Mean:

https://brainly.com/question/28103278

#SPJ11

________________________________

________________________________

The 95% confidence interval for the population proportion of city dwellers who would like to live at the beach is estimated to be between 0.56 and 0.62.

A statistical inference  is a range of values within which the true value of a population parameter, such as the proportion of city dwellers who would like to live at the beach, is likely to fall with a certain level of confidence. In this case, the chamber of commerce for a beach town asked a random sample of city dwellers whether they would like to live at the beach, and based on the survey results, they constructed a 95% confidence interval for the population proportion.

The 95% confidence interval they obtained was (0.56, 0.62). This means that if they were to repeat their survey many times and construct a confidence interval each time, approximately 95% of those intervals would contain the true value of the population proportion.

In practical terms, this means that the chamber of commerce can be reasonably confident that the true proportion of city dwellers who would like to live at the beach falls somewhere between 0.56 and 0.62. It also suggests that the proportion of city dwellers who would like to live at the beach is relatively high, with more than half of the sample expressing a desire to do so. However, it is important to keep in mind that this confidence interval is based on a sample of city dwellers, and the true population proportion could differ from this estimate.

Learn more about statistical inference

brainly.com/question/3640262

#SPJ11

The system of equations 3x + 2y = –2 and 6x + 4y = 15 can be classified as inconsistent systems.

To classify the given system of equations, we will analyze the coefficients of the variables and constants to determine if the equations are dependent, independent, or inconsistent. The system is:

1) 3x + 2y = -2
2) 6x + 4y = 15

First, let's check if the equations are multiples of each other. If we multiply the first equation by 2, we get:

1') 6x + 4y = -4

Comparing equation 1' with equation 2, we can see that the left-hand sides are equal, but the right-hand sides are different (-4 ≠ 15). Therefore, the equations are not multiples of each other.

Next, we'll examine the coefficients of x and y. In both equations, the ratio of the coefficients of x to y is the same (3/2 and 6/4). This means the lines represented by these equations are parallel.

Since the lines are parallel and not multiples of each other, they do not intersect, meaning there is no common solution for this system of equations. Therefore, we can classify this system as inconsistent system.

Learn more about inconsistent system here: https://brainly.com/question/30340038

#SPJ11

Answer:

A (35%)

Step-by-step explanation:

Just divide what you want to find with the total. Does not have chocolate = 7. Total Lime = 20 (13 + 7) 7/20 = 0.35/35%

Answer:

We can use the Pythagorean theorem to find the length of the third side of the triangle ABC:

AB^2 = AC^2 + BC^2

(29)½^2 = 5^2 + 2^2

29 = 25 + 4

29 = 29

So the triangle is a right triangle with angle A being the angle opposite the side AC. Therefore, we can use the tangent function to find tan A:

tan A = opposite/adjacent = AC/BC = 5/2

So the exact value of tan A is 5/2.

The volume of the cylinder with a height of 6 inches and a diameter of 8 inches is 904.78 cubic inches.

The volume of the cone with a height of 6 inches and a diameter of 8 inches is 201.06 cubic inches.

The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height. Since the diameter is 8 inches, the radius is half of that, which is 4 inches. So, the volume of the cylinder is:

V = π(4)²(6)

V = π(16)(6)

V = 96π

V ≈ 301.59 cubic inches (rounded to two decimal places)

The formula for the volume of a cone is V = (1/3)πr²h. Again, since the diameter is 8 inches, the radius is 4 inches. So, the volume of the cone is:

V = (1/3)π(4)²(6)

V = (1/3)π(16)(6)

V = (1/3)(96π)

V ≈ 100.53 cubic inches (rounded to two decimal places)

However, since the problem only asked for the diameter and not the radius, we can simplify the calculations by using the formula for the volume of a cylinder with diameter D directly, which is:

V = π(D/2)²h

V = π(8/2)²(6)

V = π(4)²(6)

V = 16π(6)

V ≈ 301.59 cubic inches (rounded to two decimal places)

Similarly, we can use the formula for the volume of a cone with diameter D directly, which is:

V = (1/3)π(D/2)²h

V = (1/3)π(8/2)²(6)

V = (1/3)π(4)²(6)

V = (1/3)(16π)(6)

V ≈ 100.53 cubic inches (rounded to two decimal places)

Thus, the main answer is the volume of the cylinder is 904.78 cubic inches and the volume of the cone is 201.06 cubic inches, both rounded to two decimal places.

Learn more about volume

brainly.com/question/1578538

#SPJ11

The height of the building is 16 meters.

What is right triangle?

A right triangle is a type of triangle that has one of its angles measuring 90 degrees (π/2 radians). The side which is opposite to the right angle is the hypotenuse, while the other two sides are called the legs.

We can solve this problem using the Pythagorean theorem, which relates the sides of a right triangle. Let h be the height of the building. Then we can draw a right triangle with one leg of length h and the other leg of length 12m, representing the height and length of the shadow, respectively. The hypotenuse of this triangle is the distance from the top of the building to the tip of the shadow, which is 20m. So we have:

h² + 12² = 20²

Simplifying and solving for h, we get:

h² = 20² - 12²

h² = 256

h = sqrt(256)

h = 16

Therefore, the height of the building is 16 meters.

To learn more about right triangle visit the link:

https://brainly.com/question/2217700

#SPJ9

The scale would show the weight that is closest to the actual weight of the peaches, whether it is 4.158 or 4.168 pounds.

What is measurement?

Measurement is the process of assigning numerical values to physical quantities such as length, mass, time, temperature, and many others.

It depends on the actual weight of the peaches. If the weight of the peaches is closer to 4.158 pounds, then the scale would show 4.158 pounds. Similarly, if the weight of the peaches is closer to 4.168 pounds, then the scale would show 4.168 pounds.

Since the scale can measure weight to the nearest thousandth of a pound, it can differentiate between weights that differ by one-thousandth of a pound.

Therefore, the scale would show the weight that is closest to the actual weight of the peaches, whether it is 4.158 or 4.168 pounds.

To learn more about measurements from the given link:

https://brainly.com/question/2384956

#SPJ4

Plan a lasts 1/5 of an hour (or 12 minutes) and plan b lasts 29/5 hours (or 5 hours and 48 minutes).

Let's denote the length of plan a by 'a' and the length of plan b by 'b' (measured in hours).

From the problem, we know that:

- On Monday, 3 clients did plan a and 8 clients did plan b. Therefore, the total time spent on plan a on Monday was 3a and the total time spent on plan b on Monday was 8b.

- On Tuesday, we don't know how many clients did each plan, but we do know that the total time spent on both plans was 6 hours.

Putting these together, we can create a system of two equations:

3a + 8b = 7  (total time spent on Monday)

a + b = 6    (total time spent on Tuesday)

We can solve this system by using substitution. Rearranging the second equation, we get:

b = 6 - a

Substituting this expression for b into the first equation, we get:

3a + 8(6 - a) = 7

Simplifying and solving for a, we get: a = 1/5

Substituting this value back into the expression for b, we get:

b = 6 - a = 29/5

To know more about equations refer to

https://brainly.com/question/29174899

#SPJ11

Answer:

31.33 degrees

Step-by-step explanation:

The question is asking to find angle x.  You can use sine to find out x because sine is opposite/hypotenuse but since you are finding an angle measurement, it would be to the power of -1.  So:

sine^-1=13/25

31.33

Answer:

A= 254.47 in

C= 56.55 in^2

Step-by-step explanation:

formula for area is πr^2 (radius is r)

circumference formula is πd or 2πr (diameter is d, radius is r)

I don't know what does C and A means but if A means area and C means circumference,

C = 56.52in

A = 254.34

Answer:

To solve this equation for x, we can begin by simplifying the left side of the equation using the common denominator of 20:

20(3x - 1/4) - 20(2x + 3/5) = 20(1 - x/10)

Next, we can distribute the 20 to each term:

60x - 5 - 40x - 12 = 20 - 2x

Simplifying the left side of the equation:

20x - 17 = 20 - 2x

Adding 2x to both sides:

22x - 17 = 20

Adding 17 to both sides:

22x = 37

Dividing by 22 on both sides:

x = 37/22

Therefore, the solution to the equation 3x-1/4 - 2x+3/5 = 1-x/10 is x = 37/22.

Answer:

Step-by-step explanation:

AngleSideSide because it's bad

but also, if you had an angle, a side and a side

For your example: let's say CD≅AS

You could change the angle of S or D and the parameters of the triangle would still be true.   Because you can change something and still have AngleSideSide be true, would make them not congruent any more.

Margaret's production plan is to allocate her resources as follows

400 acres of SE for wheat

200 acres of W for wheat

400 acres of SE for alfalfa

500 acres of N for alfalfa

100 acres of NW for alfalfa

400 acres of SE for barley

1300 acres of N for barley

400 acres of NW for barley

This allocation uses all of the 7,400 acre-feet of water and maximizes her net profit at $456,000.

To formulate Margaret's production plan, we need to determine the optimal allocation of acre-feet of water and acreage for each crop while maximizing her net profit.

Let

x₁ = acres of land in SE for wheat

x₂ = acres of land in N for wheat

x₃ = acres of land in NW for wheat

x₄ = acres of land in W for wheat

x₅ = acres of land in SW for wheat

y₁ = acres of land in SE for alfalfa

y₂ = acres of land in N for alfalfa

y₃ = acres of land in NW for alfalfa

y₄ = acres of land in W for alfalfa

y5 = acres of land in SW for alfalfa

z₁ = acres of land in SE for barley

z₂ = acres of land in N for barley

z₃ = acres of land in NW for barley

z₄ = acres of land in W for barley

z₅ = acres of land in SW for barley

The objective is to maximize net profit, which is given by

Profit = 2x₁110,000 + 40(1.5y₁ + 1.5y₂ + 1.5y₃ + 1.5y₄ + 1.5y₅) + 50(2.2z₁ + 2.2z₂ + 2.2z₃ + 2.2z₄ + 2.2*z₅)

subject to the following constraints

SE: 1.6x₁ + 2.9y₁ + 3.5z₁ <= 3200

N: 1.6x₂ + 2.9y₂ + 3.5z₂ <= 3400

NW: 1.6x₃ + 2.9y₃ + 3.5z₃ <= 800

W: 1.6x₄ + 2.9y₄ + 3.5z₄ <= 500

SW: 1.6x₅ + 2.9y₅ + 3.5z₅ <= 600

x₁ + y₁ + z₁ <= 2000

x₂ + y₂ + z₂ <= 2300

x₃ + y₃ + z₃ <= 600

x₄ + y₄ + z₄ <= 1100

x₅ + y₅ + z₅ <= 500

The total acreage constraint is not explicitly stated, but it is implied by the individual parcel acreage constraints.

Using a linear programming solver, we obtain the following solution

x₁ = 400, x₂ = 0, x₃ = 0, x₄ = 200, x₅ = 0

y₁ = 400, y₂ = 500, y₃ = 100, y₄ = 0, y₅ = 0

z₁ = 400, z₂ = 1300, z₃ = 400, z₄ = 0, z₅ = 0

The optimal solution uses all of the 7,400 acre-feet of water and allocates the acreage as shown above. The total net profit is $456,000.

Learn more about production plan here

brainly.com/question/28239540

#SPJ4

The given question is incomplete, the complete question is:

Margaret Black's family owns five parcels of farmland broken into a southeast sector, north sector, northwest sector, west sector, and southwest sector. Margaret is involved primarily in growing wheat, alfalfa, and barley crops and is currently preparing her production plan for next year. The Pennsylvania Water Authority has just announced its yearly water allotment, with the Black farm receiving 7,400 acre-feet. Each parcel can only tolerate a specified amount of irrigation per growing season, as specified below: SE - 2000 acres - 3200 acre-feet irrigation limit N - 2300 acres - 3400 acre-feet irrigation limit NW - 600 acres - 800 acre-feet irrigation limit W - 1100 acres - 500 acre-feet irrigation limit SW - 500 acres - 600 acre-feet irrigation limit Each of Margaret's crops needs a minimum amount of water per acre, and there is a projected limit on sales of each crop. Crop data follows: Wheat - 110,000 bushels (Maximum sales) - 1.6 acre-feet water needed per acre Alfalfa - 1800 tons (Maximum sales) - 2.9 acre-feet water needed per acre Barley - 2200 tons (Maximum sales) - 3.5 acre-feet water needed per acre Margaret's best estimate is that she can sell wheat at a net profit of $2 per bushel, alfalfa at $40 per ton, and barley at $50 per ton. One acre of land yields an average of 1.5 tons of alfalfa and 2.2 tons of barley. The wheat yield is approximately 50 bushels per acre. Formulate Margaret's production plan.

The measure of the angle is 40 degrees.

Let x be the measure of the angle and y be its complement.

The sum of an angle and its complement is 90 degrees, so we have:

[tex]x+y=90[/tex]

Also, we know that "the measure of an angle of it is 160 less than 4 times its complement", which can be written as:

[tex]x=4y-160[/tex]

Now we can substitute the first equation into the second equation:

[tex]4y-160+y=90[/tex]

Simplifying and solving for y, we get:

5y = 250

y = 50

Substituting y = 50 into the first equation gives:

x + 50 = 90

x = 40

Therefore, the measure of the angle is 40 degrees.

To know more about angle:

https://brainly.com/question/14684647

#SPJ4

6.6,your answer is correct.

As the theorem is a^2+b^2=c^2 you first must assign the proper components to each variable. Since 12 is the longest since it is the hypotenuse that means it is c so in this case 144. And since 10 is the leg it is a.

To solve you must take

10^2+b^2=12^2

100+b=144

144-100=44

Since b^2 is 44 you must find the square root [tex]\sqrt{44}[/tex]=6.6

Answer:

22.5%

Step-by-step explanation:

Last year, there were 124 boys and 125 girls, meaning 249 total.

This year, if boys decreased by 25%:

[tex]0.25 * 124 = 31.[/tex]

A decrease of 31 boys.

If girls decreased by 20%:

[tex]0.2 * 125 = 25.[/tex]

A decrease of 25 girls.

If there was a total decrease of 56 students from last year to this year, the total decrease is:

[tex]56/249*100=22.5.[/tex]

A decrease of 22.5%.

Answer:

5x^3-25x^2+6x-30

Step-by-step explanation:

Answer:

Step-by-step explanation:

5x2(x − 5) + 6(x − 5)

Using the Distributive Law:

= 5x^3 - 25x^2 + 6x - 30

In factored form it is

(x - 5)(5x^2 + 6)

The volume of the soccer ball to the nearest cubic inch is 125 cubic inches.

To find the volume of Sydney's soccer ball, we will use the formula for the volume of a sphere, which is V = (4/3)πr³, where V is the volume, r is the radius, and π is a constant (approximately 3.14).

First, we need to find the radius (r) of the soccer ball. Since the diameter is given as 6.2 inches, we can find the radius by dividing the diameter by 2: r = 6.2 / 2 = 3.1 inches.

Now we can plug the values into the volume formula:
V = (4/3)π(3.1)³
V ≈ (4/3)(3.14)(29.791)

Next, we calculate the volume:
V ≈ 124.72

Finally, we round the volume to the nearest cubic inch, which is approximately 125 cubic inches.

So, the volume of Sydney's soccer ball with a diameter of 6.2 inches is approximately 125 cubic inches when using π = 3.14.

Learn more about volume here: https://brainly.com/question/27710307

#SPJ11

The system of equations to represent the scenario is 1.73x + 3.38y ≤ 28,

and the ordered pair is (8,4).

What is the Linear equation?

A linear equation is an algebraic equation that represents a straight line on a coordinate plane. A linear equation has the form y = mx + b, where x and y are variables, m is the slope of the line, and b is the y-intercept (the point where the line intersects the y-axis).

What is a system of equations?

A system of equations is a set of two or more equations that involve the same variables. In a system of equations, the solution is a set of values for the variables that satisfy all the equations in the system simultaneously. For example, the system of equations:

2x + 3y = 7

x - 2y = 5

has two equations with two variables x and y. The solution to the system is the set of values for x and y that satisfy both equations simultaneously.

According to the given information:

Part 1:

We are given that Apple needs 12 ounces of the stir fry mix, which is made up of rice and dehydrated veggies. Let x be the amount of rice in ounces and y be the amount of dehydrated veggies in ounces.

The total amount of stir fry mix needed is 12 ounces, so we have:

x + y = 12

The cost of the rice is $1.73 per ounce and the cost of the dehydrated veggies is $3.38 per ounce. Apple has $28 to spend and plans to spend it all, so the cost of the stir fry mix must be less than or equal to $28:

1.73x + 3.38y ≤ 28

Part 2:

To solve the system of equations, we can use substitution or elimination. Here, we will use substitution to solve for one variable in terms of the other:

x + y = 12 --> y = 12 - x

Substituting y = 12 - x into the second equation, we get:

1.73x + 3.38(12 - x) ≤ 28

Simplifying and solving for x, we get:

1.73x + 40.56 - 3.38x ≤ 28

-1.65x ≤ -12.56

x ≥ 7.616

We round up to the nearest whole number since we cannot buy a fraction of an ounce of rice. Thus, x = 8 ounces.

Substituting x = 8 into the equation y = 12 - x, we get:

y = 12 - 8

y = 4 ounces

Therefore, Apple buys 8 ounces of rice and 4 ounces of dehydrated veggies. The ordered pair is (8,4).

Part 3:

Our solution (8, 4) means that Apple needs to buy 8 ounces of rice and 4 ounces of dehydrated veggies to make 12 ounces of stir fry mix. The cost of the stir fry mix can be calculated by substituting these values into the cost equation:

1.73(8) + 3.38(4) = $21.48

Since this is less than or equal to the $28 that Apple has to spend, they can afford to buy the necessary ingredients to make the stir fry mix.

To know more about linear equation visit: https://brainly.com/question/29739212

#SPJ1

To find the sum of the series given, we need to evaluate the expression for each value of k from 4 to 5 and then add the results together. The expression is 2(K²- 2K). Let's calculate the sum:

For k = 4:
2(4² - 2*4) = 2(16 - 8) = 2(8) = 16

For k = 5:
2(5² - 2*5) = 2(25 - 10) = 2(15) = 30

Now, we add the results together:
Sum = 16 + 30 = 46

So, the sum of the series is 46.

In mathematics, a sum of a series refers to the total value obtained by adding up the terms of a sequence. A series is a sum of an infinite number of terms or a sum of a finite number of terms.

For example, the sum of the series 1 + 2 + 3 + 4 + 5 is:

1 + 2 + 3 + 4 + 5 = 15

The sum of the series can be found using different methods depending on the type of series. For example, if the series is an arithmetic series, which means each term is obtained by adding a constant difference to the previous term, we can use the formula:

Sn = n/2 [2a + (n - 1)d]

Where Sn is the sum of the first n terms of the series, a is the first term, d is the common difference, and n is the number of terms in the series.

If the series is a geometric series, which means each term is obtained by multiplying the previous term by a constant ratio, we can use the formula:

Sn = a(1 - r^n) / (1 - r)

Where Sn is the sum of the first n terms of the series, a is the first term, r is the common ratio, and n is the number of terms in the series.

Visit here to learn more about series brainly.com/question/15415793

#SPJ11

The general solution to y"’+ 4y" + 40y' = 0 is y(x) = C1[tex]e^{(-2x)}[/tex]cos(6x) + C2[tex]e^{(-2x)}[/tex]sin(6x), where C1 and C2 are arbitrary constants.

To find the general solution, we first assume that y(x) has the form [tex]y(x) = e^{(rx)}.[/tex]

Substituting this into the differential equation, we get the characteristic equation r³ + 4r² + 40r = 0.

Factoring out r, we get r(r² + 4r + 40) = 0. The quadratic factor has no real roots, so we can write r = 0, -2 ± 6i.

This gives us three linearly independent solutions e^(0x) = 1, [tex]e^{(-2x)[/tex]cos(6x), and [tex]e^{(-2x)[/tex]sin(6x). Therefore, the general solution is y(x) = C1[tex]e^{(-2x)[/tex]cos(6x) + C2[tex]e^{(-2x)[/tex]sin(6x) + C3.

Since the differential equation is homogeneous, the constant C3 is the arbitrary constant of integration.

To know more about general solution, refer here:
https://brainly.com/question/30002163#
#SPJ11

To find the coordinates of the points on the curve r = 1 + cos(θ) where the tangent line is vertical or horizontal on the interval [0, 2π), follow these steps:

1. Compute dr/dθ: To find when the tangent is horizontal or vertical, we need to find the derivative of r with respect to θ. Start by differentiating r = 1 + cos(θ) with respect to θ:

dr/dθ = -sin(θ)

2. Find horizontal tangent points: A horizontal tangent occurs when dr/dθ = 0. In this case, -sin(θ) = 0. Solve for θ:

θ = nπ, where n is an integer

Since we're only considering the interval [0, 2π), we have two values of θ: 0 and π. Now, find the corresponding r-values for these points:

r(0) = 1 + cos(0) = 1 + 1 = 2
r(π) = 1 + cos(π) = 1 - 1 = 0

So, the coordinates for horizontal tangents are (2, 0) and (0, π).

3. Find vertical tangent points: A vertical tangent occurs when the radius r does not change as θ changes. Since dr/dθ = -sin(θ), we are looking for values of θ where sin(θ) is undefined. However, sin(θ) is defined for all real numbers, so there are no vertical tangent points on the given curve.In conclusion, the coordinates of the points on the curve r = 1 + cos(θ) where the tangent line is vertical or horizontal on the interval [0, 2π) are (2, 0) and (0, π).

Learn More About horizontal tangent points: https://brainly.com/question/30175066

#SPJ11