An integrating factor for the linear differential equation y' - y/x = x is Select the correct answer. a. x b. x² c. 1/x d. 1/x² e. e⁻ˣ

Answers

Answer 1

The correct answer is d. 1/x².

The integrating factor for the given equation is 1/x².

To solve a linear differential equation, we must first find an integrating factor so that the equation can be integrated. The integrating factor for the given equation is 1/x². To calculate it, we must multiply both sides of the equation by the integrating factor.

Let's start by rearranging the equation: y' - y/x = x

We multiply both sides of the equation by the integrating factor, 1/x²:

(1/x²) (y' - y/x) = (1/x²) x

We can simplify the left side of the equation by factoring out the y': y'/x² + (y/x³) = (1/x²) x

We can then solve for the integrating factor: 1/x² + (y/x³) = x/x²

We can then solve for the integrating factor: 1/x². Therefore, the integrating factor for the given equation is 1/x².

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Related Questions

If a translation maps H onto J, which of the following statements is true?
G
H
Ol=K
O
GH = 2 GJ
GK
HI 2 JK
K

Answers

If a translation maps H onto J, the option from the following statements which is true is:  ∠I ≅ ∠K (Option 1)

What is a translation in math?

A translation is a sort of transformation that involves sliding each point in a figure the same distance in the same direction. The four fundamental translations or transformations in geometry are:

Translation. Reflection.Rotation.Resizing or dilation.

With regard to the above answer, note that, because both triangles have the same vertex, angle HGI is equivalent to angle JGK. Angle H is parallel to angle J.

As a result, by converting angle I to angle K, their measurements will be equivalent. As a result, the angle I is congruent with angle K.

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Full Question:

If a translation maps H onto J, which of the following statements is true?

1. ∠I ≅ ∠K

2. GH = 2GJ

3. ∠G ≅ ∠K

4. HI = 2JK

A pail hold 220. 5 ounce (oz) of water when full. The bucket loe 0. 2 oz of water per econd. In how many econd will the bucket be 30% full? Round your anwer to the nearet econd

Answers

The bucket will be 30% full at 772 seconds

What is velocity?

It is a physical quantity that indicates the displacement of a mobile per unit of time, it is expressed in units of distance per time, for example (miles/h, km/h).

The formula and procedure we will use to solve this problem is:

v= x/t

Where:

x = volumet = timev = velocity

Information about the problem:

Total = 220.5 ozv = 0.2 oz/svolume = total *30%

Calculating how many ounces the bucket will be when it has 30% full:

Volume = 220.5 * 30%

Volume = 66.15oz

The bucket will lose:

x = 220.5oz - 66.15oz

x = 154.35 oz

The time that the bucket will have 66.15 oz (30%) is:

t = x/v

t = 154.35 oz / 0.2 oz/s

t = 771.75 s = 772 s

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Correctly written question:

A pail holds 220.5 ounces (oz.) of water when full. The bucket loses 0.2oz of water per second. In how many seconds will the bucket be 30% full? Round your answer to the nearest second.

1/7 of what is 1/35 thanks!

Answers

Answer:

1/5

Step-by-step explanation:

1/7 * x = 1/35    multiply both sides by 7

x = 7/35 = 1/5

The system of equations models the prices per share, y, for two stocks over time, x. Which represents the viable solutions to the system?
−y ≤ 3x + 4
−3x + 3y ≤ −9

A. All real number values of x and y
B. (5, 15)
C. All real number values, with y ≥ 0
D. (15, 5)

Answers

The option that describes the viable solution for the system is all real number values, with y ≥ 0 which is denoted as option C.

What is Inequality?

This is referred to as a relation which makes a non-equal comparison between two numbers or other mathematical expressions.

all real number values, with x >=0 and y>=0​

The following system of inequalities is given below:

-y ≤ 3x + 4

-3x - 3y ≤ 9 which can be rewritten as:

-3y ≤ 9 + 3x

-3y/3 ≤ 9/3 + 3x/3

-y ≤  3 + x

The system of inequalities is therefore:

-y ≤  3 + x

-y ≤ 3x + 4

Since x and y are both positive or 0, the system is true and we can conclude that the correct choice is option C which is all real number values, with y ≥ 0

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I need help with this please ​

Answers

Equations with exponents, coefficients, and variables are called polynomial equations.

What is meant by polynomial equation?

Polynomial equations are equations that contain variables, exponents, and coefficients. The greater exponent is referred to as the degree of the equation and it can have many exponents.

A polynomial is an expression in mathematics made up of coefficients and indeterminates, which are also known as variables. It can only be solved using the addition, subtraction, multiplication, and powers of positive-integer variables.

When a variable in an equation like the quadratic equation, cubic equation, etc. has only non-negative integer powers or only positive integer exponents, the function is said to be polynomial.

The absence of square roots of variables, fractional or negative powers on the variables, and the absence of variables in any fraction's denominator are specifically required for an expression to qualify as a polynomial term.

Therefore, the correct answer is option iii) -7 v² - 6v + 4.

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By rearranging the terms, the correct answer is option iii) -7 v² - 6v + 4. Equations with exponents, coefficients, and variables are called polynomial equations.

What is meant by the phrase "polynomial equation"?

Polynomial equations are those that have exponents, coefficients, and variables. The higher exponent of the equation, which may have multiple exponents, is known as the equation's degree.

A polynomial is an equation in mathematics made composed of coefficients and indeterminates, often known as variables. The only operations that may be used to solve it are addition, subtraction, multiplication, and powers of positive-integer variables.

If a variable in an equation, such as a quadratic equation or a cubic equation, only has positive integer exponents or non-negative integer powers, the variable is said to be a polynomial.

Expressions must expressly be free of variable square roots, fractional or negative powers on the variables, and variable presence in any fraction's denominator in order to be classified as polynomial terms.

Therefore, the correct answer is option iii) -7 v² - 6v + 4.

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can the graph of intersect a vertical asymptote? can it intersect a horizontal asymptote? illustrate by sketching graphs.

Answers

A graph cannot intersect its vertical asymptote. A graph can intersect is horizontal asymptote. A graph that intersect its vertical asymptote is not a valid function.

Asymptote is a line that touches the graph of a function only at infinity but it never touches its vertical asymptotes but it can touch its horizontal asymptote and that too at +∞ , -∞. The older definition of asymptotes implies it cannot intersect the graph but recent discoveries shows it is not the case. For example vertical asymptotes,

The graph of [tex]x^{2}[/tex]  / [tex]x^{2} +1[/tex] has x- axis as its horizontal asymptotes and it intersects at origin. Also [tex]\frac{sin(x)}{x}[/tex] is asymptotic to y=0 as x→∞

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A sunflower's height was measured at 42 inches. The second time the flower was measured its height was 70 inches. What was the percent change in the height of the sunflower between the two measurements? Increase or decrease? amount of change original amount = percent= Substitute and solve:​

Answers

Answer:

67%

Step-by-step explanation:

Percent change is the change divided by the original value.  The change for the flower is (70" - 42") = 28"

The original value was 42".

Percent change is (28"/42")*(100%) = 67%

Adela paid $54.60 for 4 tickets at the skating rink. Calculate the unit rate per ticket.

Answers

Answer: $13.65 per ticket

Step-by-step explanation:

54.6/4

two of the interior angles of a triangle are 45.5 degrees and 14.6 degrees. what is the third interior angle of the triangle?

Answers

The third interior angle of the triangle is 119.9 degrees

Triangle is a closed polygon consist of three sides, three angle or three segment.

Triangle has three angle which if we sum all of those three angle it will be 180 degrees.

Triangle can be divided into several type :

1.  Equilateral triangle which all of the angle is equal ( 180 degrees / 3 = 60 degrees each )

2. Isosceles triangle which have two equal angle, and the third angle is different

3. Scalene triangle which all of the angle is different

From the question, we have data :

1. First Angle is 45.5 degrees

2. Second Angle is 14.6 degrees

For the Third angle, we just simply subtract 180 degrees with the first and second angle. So it will be :

Third Angle = 180 degrees - ( 45.5 degrees + 14.6 degrees )

                    = 180 degrees - 60.1 degrees

                    = 119.9 degrees

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What’s the measurements

Answers

The basic units for length or distance measurements in the English system are the inch, foot, yard, and mile. Other units of length also include the rod, furlong, and chain. survey foot definition. In the English system, areas are typically given in square feet or square yards.

if the total number of hot lunches sold at school this week was 1,350 and the relative frequency on friday was 0.22, how many lunches were sold on friday? provide your answer and explain how you arrived at it.

Answers

The number of lunches were sold on friday this week with a relative frequency of 0.22 is 297.

The word "relative" is used to describe how something is being viewed in connection to or in comparison to another event. How frequently an event occurs can be determined by looking at its frequency. On the other hand, relative frequency is a means to compare the frequency of a specific event to all other occurrences.

Let's learn about the relative frequency formula,

Relative frequency = f/n

f= number of times data occurred in an observationn= total frequency

So 0.22 = f / 1350

so f= 1350 x 0.22

f = 297

So  number of lunches were sold on friday this week were 297.

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certain standardized math exams have a mean of 100 and a standarad deviation of 60. of a sample of 36 students who take this exam, what percent could you expect to score above 110

Answers

Using the Normal Z-distribution and the values from the Z-table, We can calculate that 12.5% can be expected to score between 70 and 90 according to the value of the mean and standard deviation provided.

A unique type of normal distribution with a mean of 0 and a standard deviation of 1 is known as the standard normal distribution or z-distribution. By transforming the values of any normal distribution into z scores, it is possible to standardize it.

mean =100

standard deviation =60

70≤ x ≤ 90

70 - 100 ≤ x - 100 ≤ 90 -100

-0.5 ≤  Z ≤ -0.167

P = 0.125(From Z-Table)

Required percentage  = 12.5%

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The correct question is:

certain standardized math exams have a mean of 100 and a standard deviation of 60. of a sample of 36 students who take this exam, what percent could you expect to score between 70 and 90?

.What is the probability an individual student likes between 5.5 and 8.5 songs from Eminem?​

Answers

It is not possible to determine the probability that an individual student likes between 5.5 and 8.5 songs from Eminem without any additional information. Probability is a measure of the likelihood of an event occurring, and it requires data or information about the event in question. In this case, without knowing the number of songs from Eminem the student has heard, the student's taste in music, or any other relevant information, it is impossible to determine the probability that the student likes between 5.5 and 8.5 songs from Eminem.

What values of x satisfy the following equations?
(a) P(-x ≤ T22 ≤ x)=0.98
(b) P(T13x)=0.85
(c) P(T26 (d) P(T₂≥x)=0.025

Answers

For (a), x must be the 98th percentile of T22. For (b), x must be the 85th percentile value  of T13. For (c), x must be the 26th percentile of T26. For (d), x must be the 97.5th percentile of T2.

For (a), the value of x must be the 98th percentile of T22, which is the value of x such that 98% of the data points of T22 are less than or equal to x. For (b), the value of x must be the 85th percentile of T13, which is the value of x such that 85% of the data points of T13 are less than or equal to x. For (c), the value of x must be the 26th percentile of T26, which is the value of x such that 26% of the data points of T26 are less than or equal to x. For (d), the value of x must be the 97.5th percentile of T2, which is the value of x such that 97.5% of the data points of T2 are less than or equal to x. In order to find these values of x, one must first calculate the percentiles of the given variables. This can be done by sorting the data points in ascending order and then calculating the percentile of each data point. After the corresponding percentile of each data point has been calculated, the desired values of x can be determined by finding the data point whose percentile matches the desired value.

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Select all the expressions that are equivalent to (12⁻₂)₈

Answers

Step-by-step explanation:

1/4096

0.000244140625

2^-12

1/(2^12)

All of the above expressions are equivalent to (12⁻₂)₈, which is a decimal representation of a number that is obtained by dividing 1 by 2 raised to the power of 12. The first and second expressions are decimal representation of the number, the third one is the representation of the number by using the base 2 logarithm, and the fourth one is the mathematical representation of the number by dividing 1 by 2 raised to the power of 12.

question 117 please !

Answers

The simplified form of the expression (3x + 8y)/(x² + 8xy + 16y²) × (x + 4y)/4

is  [(3x² + 20xy + 32y²)/(4x² + 32xy + 64y²)].

What is a polynomial?

A polynomial is an algebraic expression.

A polynomial of degree n in variable x can be written as,

a₀xⁿ + a₁xⁿ⁻¹ + a₂xⁿ⁻² +...+ aₙ.

Given, (3x + 8y)/(x² + 8xy + 16y²) × (x + 4y)/4.

= [(3x + 8y)×(x + 4y)/(x² + 8xy + 16y²)×4].

= [3x² + 8xy + 12xy + 32y²/4x² + 32xy + 64y²].

= [(3x² + 20xy + 32y²)/(4x² + 32xy + 64y²)].

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What is an angle of 180° called?

Answers

Answer:

straight angles

Step-by-step explanation:

Angles that are 180 degrees (θ = 180°) are known as straight angles.

What is called a straight angle?In Maths, a straight angle is an angle equal to 180 degrees. It is called straight because it appears as a straight line.A flat surface has an angle of 180 degrees. A plane inclined staircase. The angle between the minute hand and hour hand at 6:00.Angles that are 180 degrees (θ = 180°) are known as straight angles. Angles between 180 and 360 degrees (180°< θ < 360°) are called reflex angles.Squares and rectangles are a type of right angled shape that has exactly 4 right angles that measure a total of 360 degrees. Other shapes like trapezoids, pentagons, and hexagons can also have a right angle

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A bagel factory produced 647 bags of bagels. There were 14 bagels in each bag. How many bagels did the factory produce?

Answers

Answer:

9058 Bagels

Step-by-step explanation:

14x657 I hope this helped!

A quadrilateral with vertices P (1,-1), Q(3,-7), R(-9,-11) and S (-11,-5) is a parallelogram. Select the correct statement to justify the diagonals of a parallelogram bisecct each other

Answers

The distances from the midpoint M to the vertices are all the same and equivalent to 5·√2, therefore, the diagonals of the quadrilateral bisect each other.

What is a diagonal of a quadrilateral?

A diagonal is a line that joins the opposite corners of the quadrilateral.

The question is without the possible options for the response.

The vertices of the quadrilateral are;

P(1, -1), Q(3, -7), R(-9, -11), and S(-11, -5)

The midpoint of the side PR = ((-9 + 1)/2, (-11 + (-1))/2) = (-4, -6)

Midpoint of the diagonal SQ = ((3 + (-11))/2, (-7 - 5)/2) = (-4, -6)

Let M represent the point of intersection of the diagonals, we get;

PM = √((1 - (-4))² + (-1 - (-6))² = 5·√2

MR = √((-9 - (-4))² + (-11 - (-6))² = 5·√2

The length of PM = The length of MR, therefore, the diagonal SQ bisects PM

Similarly, we get;

SM = √((-9 - (-4))² + (-11 - (-6))²) = 5·√2

MQ = √((3 - (-4))² + (-7 - (-6))²) = 5·√2

The length PM = MR and the lengths SM = MQ, therefore, the diagonals bisect each other.

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Answer: the product of the slopes of the diagonals PR and QS is equal to 1. Hence, the diagonals of parallelogram PQRS bisect each other.

Step-by-step explanation:

The first three terms of a sequence are given. Round to the nearest thousandth (if necessary).
16,48,144,...
Find the 6th term.

Answers

the 6th term of the geometric sequence will be 3888.

What is geometric sequence?

A geometric progression or geometric sequence is the order in which each word is different from the next by a common ratio. The next term in the series is produced by adding a constant (that is not zero) to the term that came before it. A, ar, ar2, ar3, ar4, etc. are used as substitutes.

Mathematicians use the term "Geometric Progression" (GP) to describe a type of sequence in which each word is formed by multiplying the previous term by a predetermined number, or "common ratio". An integer geometric series gets that follows a pattern is another name for this development.

The terms are given, 16,48,144

a = 16

r = 48/16 =3

The 6th term will be= a(r)[tex])^{5}[/tex]

= 16(3)[tex])^{5}[/tex]

= 16* 243

=3888

Hence, the 6th term of the geometric sequence will be 3888.

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eric walks around a man-made circular lake (pictured above) two times. how far (in miles) has he walked?

Answers

The amount of total distance walked (in miles) has walked by Eric in a man made circular lake is 62.8 miles.

A circle's circumference is the length of its perimeter. The circumference of a circle can be calculated using length units such as centimeters, meters, or kilometers by opening a circle and measuring the boundary in the same way that we would measure a straight line.

Let's now discover the components of circumference. The three most crucial components of a circle are those three.

Center: The circle's center is a point that is set apart from the other points on the circumference by a predetermined amount.Diameter: The diameter is the distance across the circle that must pass through the center; it is a line that must meet the circumference at both ends.Radius: The radius of a circle is the distance a circle's center is from any point along its perimeter.

Circumference = D*π

Total distance walked two times = 2.C

=2* D* π

=2x10x3.14

=62.8 miles

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find the solution to the differential equation dydt=y2(4 t), y=8 when t=1.

Answers

The solution to the differential equation is ln |y| = 4t - 4.

The given differential equation can be written as ∆y/∆t = y2(4t). This is a first-order separable differential equation, which can be solved using the separation of variables method. To do this, we first isolate the dy/dt term on one side of the equation. We can then separate the two variables (t and y) by moving all terms with t to one side and all terms with y to the other side. This gives us ∆y/y2 = 4∆t. We can then integrate both sides to find the solution.

Integrating both sides with respect to y gives us the following:

ln |y| = 4t + C,

where C is an integration constant. We can use the boundary condition to solve for C. The boundary condition states that y=8 when t=1, so substituting this into the equation above gives us ln |8| = 4 + C. Solving for C gives us C=-4.

Therefore, the solution to the differential equation is ln |y| = 4t - 4. Rearranging this equation gives us y = e4t-4, which is the solution to the differential equation.

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given the function f(x)=−x2−6x 13, determine the direction in which a parabola sketching this function would open.

Answers

The parabola f(x) = -x² - 6x - 13 is open downward.

Write the equation in vertex form f(x) = a(x-h)² + k

So, f(x) = -x² - 6x - 13

           = -(x² + 6x + 13)

           = -(x² + 6x + 9 + 4)

           = -((x + 3)² + 4)

           = -(x + 3)² - 4

Here, a = -1, h = -3 and k = -4

If a > 0, the parabola is open upward and if a < 0, the parabola is open downward.

In f(x) = -(x + 3)² - 4, a= -1 < 0. Therefore, the parabola is open downward.

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Simplify
5x5²
Leaving your answer in index form

Answers

Answer: 5^3

Step-by-step explanation: since we know that 5^2=5x5 (which equals to 5x5x5), our answer is 5^3

H
HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH

Answers

Answer:

what is this? ahdhdjxkdjjshsysusiiaiajsjjdyxc

David is tracking the amount of water he drinks each day.
On Thursday, he drinks 15 cups of water.
On Friday, he drinks 12 cups of water.

What is the percent change in his water intake?

What is the original amount of water in this problem?

What is the change in David's water intake in this problem?

What is the percent change?

Is it increasing or decreasing?

Answers

The original amount of water is  15 cups.

The change in David's water intake is  3 cups.

The percent change is 20 %. This is a percent  decrease.

What is percentage?

A percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100.

Given that, David is tracking the amount of water he drinks each day.

On Thursday, he drinks 15 cups of water.

On Friday, he drinks 12 cups of water.

The percent change in his water intake = (12 - 15) / 15 × 100 = -20%

Where minus sign is showing a decrease in the water intake,

The original amount of water is 15 cups.

The change in David's water intake is (15-12) = 13 cups.

Hence, the answers are;

The original amount of water is  15 cups.

The change in David's water intake is  3 cups.

The percent change is 20 %. This is a percent  decrease.

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I need the answer...

What is the average rate of change of F(x) over the interval of x=4 to x=8

Answers

The average change of the function is given by 3.5 over the interval [4,8]

What are functions?

Function, in mathematics, is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

Given here : The corresponding values of the functions to input x

now F(4)=-4 & F(8)=11

Thus the average rate of change of the function of the interval is given by

A= F(4)+F(8) / 2

  =-4+11 /2

  = 3.5

Hence,  The average change of the function is given by 3.5 over the interval [4,8]

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What are the coordinates for point C?

Responses

(2, 8)

(2, -8)

(-2, 8)

(-2, -8)

Answers

Answer: option c.

Step-by-step explanation:

It’s answer option c

what two types of feet are prominent in these lines

Answers

In poetry, the two main types of feet are iambic and trochaic.

Iambic foot: An iambic foot consists of two syllables, with the first syllable being unstressed and the second syllable being stressed. This creates a rhythmic pattern of "da-DUM" that gives iambic meter its characteristic "rising" sound. Iambic meter is one of the most common types of meter in English poetry, and is often associated with a natural, conversational rhythm.

Trochaic foot: A trochaic foot consists of two syllables, with the first syllable being stressed and the second syllable being unstressed. This creates a rhythmic pattern of "DA-dum" that gives trochaic meter its characteristic "falling" sound. Trochaic meter is less common than iambic meter in English poetry, but is still used in a variety of forms and styles. It can create a sense of tension or urgency, and is often used in poetry that is meant to be dramatic or suspenseful.

It's important to note that these two types of feet are just two of many possible metrical patterns that can be found in poetry. However, they are two of the most basic and widely recognized, and a good starting point for understanding the rhythms and structures of poetic meter.

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Connor bought a scarf that costs 13 dollars and a watch that costs 13 dollars. At the counter, he received a discount. If he only paid 18 dollars total, how many dollars was the discount?

Answers

$10.66 is dollars was the discount .

How does discount work?

Discount is a term that describes a sum or a percentage subtracted from a product's usual selling price. Make sure you need all of the items you plan to buy before waiting until after the holiday when you can frequently find them at a significant discount.

                           Discount is a term that refers to a reduction in the cost of a commodity or service.

Discount:

(13 x 18)/100 = $2.34

13 - 2.34 = $10.66

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prove that, given a nonnegative integer , there is a unique nonnegative integer such that m^2 < sqrt n < (m 1)^2 a letter 25 cm long the foot of the ladder is 7 m away from the base of the wall how far up the wall is the top of the ladder ric whitacre's choral music often does what? multiple choice alternates consonant chords with sweetly dissonant tone clusters uses african rhythmic patterns incorporates the rhythms of rock and roll quotes passages from the masters of european classical music an american investor owns stock in a spanish company and receives a dividend payment from that company. what interaction delivers no radiation dose to the patient? a patient with lateral epicondylitis of the left elbow presents to the outpatient surgery department of the hospital. under general anesthesia, manipulation of the elbow is performed in order for the patient to gain the loss of motion that has occurred due to his condition. the elbow is manipulated by stretching and rotating, until appropriate range of motion was achieved. the patient tolerated the procedure well. at discharge, the patient was scheduled to return for follow-up in one week. what is the appropriate procedure code to be reported? which option listed does not include conditions or supervision, but the judge retains the right to revoke the option for cause? group of answer choices federalism reflects the dynamic distribution of power between national and state governments. explain how societal needs affect the constitutional allocation of power between the national and state governments. be able to identify how this affects those looking to change policy. using ohm's law and ignoring units, if the voltage is 10 and the resistance of the circuit is 2, what is the current flow across the resister? The total rate of return earned on a stock is comprised of which two of the following?I. Current yieldII. Yield to maturityIII. Dividend yieldIV. Capital gains yielda. I and II onlyb. I and IV onlyc. II and III onlyd. II and IV onlye. III and IV only X^2+x-6=0Solve equation Please I NEED HELP ! function g can be thought of as translated (shifted) version of f(x) = x^2 wrote the equation for g(z) g(x) = Jacob rents tablecloths from two companies. The first company charges $18 for an initial fee and $3 per table cloth, and the second company charges $20 and $2 per table cloth. Hey I would really appreciate it if someone would help me with this math question!! 2 example of structurefunction complementarity in the human body A computer employs RAM chips of 512 x 8 and ROM chips of 2048 x 4. The computer system needs 2M x 16 of RAM, and 4K x 16 of ROM and two interface units with 256 registers each. A memory-mapped I/O configuration is used.a. How many RAM and ROM chips are needed?b. Tell, how many lines of the address bus must be used to access total memory?How many of these lines will be common to all chips?c. Propose, the number of lines must be decoded for chip select? Specify the size ofthe decoderd. Devise a memory-address map for the system and Give the address range inhexadecimal for RAM, ROM and interface.e. Design memory chip layout for that address map. If 4 is added to the quotient of 24 and -8, whatnumber results? Find the equation of the linear function represented by the table below in slope-intercept form. x y -2 -2 2 6 6 14 10 22 What does it mean open communication with example? if an individual performed 5,000 foot pounds of work in 1 minute, how many kilojoules of work were accomplished?