The vector field f(x, y) = (yex + sin y) i + (ex + x cos y) j is conservative because its curl vanishes everywhere. The function F such that ∇F = f is yex + cos y + C.
Explanation:To determine if the given function f(x, y) = (yex + sin y) i + (ex + x cos y) j is a conservative vector field, we need to see if the curl of the vector field vanishes everywhere. The curl of a vector field in two dimensions is the partial derivative of the j component with respect to x minus the partial derivative of the i component with respect to y.
For f(x, y), the partial derivative of the j component (ex + x cos y) with respect to x is ex + cos y, and the partial derivative of the i component (yex + sin y) with respect to y is ex + cos y. Both derivatives are equal, implying the curl is zero. Hence, f(x, y) is a conservative vector field.
Next, to find a function F such that ∇F = f, we need to integrate. The potential function, F, can be obtained by integrating the i component of f with respect to x (since it does not involve y), and the j component of f with respect to y (since it does not involve x): ∫yex dx + ∫sin y dy = yex + cos y + C. This is the function F such that ∇F = f.
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Tony and his three friends live in Albuquerque, New Mexico, but they all attend college in Boston, Massachusetts. Because they want to have a car at school this year, they are planning to drive Tony's car from Albuquerque to Boston at the beginning of the school year. Although they'll each pay for their own food during the road trip, the friends plan to split the costs for gas and hotels evenly between the four of them.
Estimate the total cost that each friend will have to pay for gas and hotels. Explain how you got your answer. Here are some figures that may help you out:
Tony's car can travel 28 miles for each gallon of gas.
The average fuel cost at the time of their trip is $3 per gallon.
They plan to drive about 650 miles each day.
They estimate the average cost of a hotel each night is $85.
They will drive approximately 2,240 miles to get from Albuquerque to Boston.
A road trip with these parameters
total distance 2240 midistance per day 650 micost per night for lodging $85mileage 28 mpggas price $3/galFind1/4 of the cost of gas and lodgingSolutionThe cost of gas is ...
... (2240 mi)/(28 mi/gal)·($3/gal) = $240
The cost of lodging is
... ($85/day)·floor(2240 mi/(650 mi/day)) = $85·3 = $255
Total cost of gas and lodging is $240 +255 = $495.
The cost for a 1/4 share is $495/4 = $123.75.
find the gradient of the line between each pair of points A(2,2) and B(6,6)
What is the prime factorization of 72?
A really bad carton of 18 eggs contains 9 spoiled eggs. an unsuspecting chef picks 6 eggs at random for his "mega-omelet surprise." find the probability that the number of unspoiled eggs among the 6 selected is
Final answer:
To find the probability that the number of unspoiled eggs among the 6 selected is X, we can use the hypergeometric probability formula. This formula is used when the sample drawn is without replacement from a finite population.
Explanation:
To find the probability that the number of unspoiled eggs among the 6 selected is X, we can use the hypergeometric probability formula. This formula is used when the sample drawn is without replacement from a finite population. In this case, we have 18 total eggs in the carton, with 9 spoiled eggs. So the probability that X unspoiled eggs are selected out of the 6 can be calculated as:
P(X) = (C(X,6) * C(9,6)) / C(18,6)
Where C(n,r) represents the combination of n things taken r at a time.
Jade spend $37.60 on groceries. 4/5 of that total was spent on vegetables. How much was spent on other items
divide the total spent by 5 to get what 1/5 is
37.60 / 5 = 7.52
they spent $7.52 on other items
simplify the expression 4y - 1 - 3y + 2
A person invests $4000 at 2% interest compounded annually for 4 years and then invests the balance (the $4000 plus the interest earned) in an account at 8% interest for 7 years. Find the value of the investment after 11 years.
(Hint: You need to break this up into two steps/calculations. Be sure to round your balance at the end of the first 4 years to the nearest penny so you can use it in the second set of calculations.)
which of the following is not a example of a molecule
A H2s
B Mn
C KOH
D O3
-5/11h=-5/9
Show your work please
Ann had 75 fliers to post around town. Last week,
she posted 2/5 of them. This week, she posted 2/3 of the remaining fliers. How many fliers has she still not posted?
jose rodriguez's checking account had a starting balance of 1,234.56. he wrote a check for 115.12 for plumbing supplies and check for 225.00 for a loan payment. Yesterday he deposited 96.75 in his checking account. what jose's current balance?
If a cheetah can run 96.5 km/h, what is its speed in m/s?
Which property is shown in the following equation?
20 x 1 = 20
A. Zero property of multiplication
B. Identify property of multiplication
C. Negative one property of multiplication
D. Identify property of division
Final answer:
The equation 20 x 1 = 20 demonstrates the Identity Property of Multiplication, which states that multiplying any number by one keeps its original value.
Explanation:
The property shown in the following equation 20 x 1 = 20 is the Identity Property of Multiplication. This property states that any number multiplied by one remains unchanged or that the number keeps its identity. The option B, Identify property of multiplication, is a typo, and it likely means to refer to the Identity Property of Multiplication. The options A, C, and D describe different properties that do not apply to this equation.
The Zero Property of Multiplication involves a multiplication where any number times zero is zero. The Negative One Property of Multiplication is not a standard mathematical property and seems to be a non-existent or misspelled option. The Identity Property of Division is not applicable as there is no division taking place in this equation.
Quadratic word problem:
A number minus 8 times its reciprocal equals 2. What is the number? (There may be more than one answer)
Final answer:
To solve the quadratic word problem, we set up the equation x - 8/x = 2 and simplify it to x^2 - 2x - 8 = 0. Using the quadratic formula, we find two possible values for x: -2 and 4.
Explanation:
To solve this quadratic word problem, we set up the equation: x - 8/x = 2. To simplify the equation, we multiply both sides by x to eliminate the denominator: x^2 - 8 = 2x. Rearranging the equation, we get x^2 - 2x - 8 = 0. To find the solutions, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). Substituting the values from our equation, we have x = (2 ± √(2^2 - 4(1)(-8))) / (2(1)). Solving the equation, we find two possible values for x: x = -2 or x = 4.
Luke can paint 91 portraits in 7 weeks.
How many portraits can Luke paint in 4 weeks?
portraits
graph the linear equation. Find 3 points to solve the equations. -5x+2y=11
What is the value of n in the equation –(2n + 4) + 6 = –9 + 4(2n + 1)?
5/2
Step-by-step explanation:Start by eliminating parentheses.
... 2n +4 +6 = -9 +8n +4
... 10 = -5 +6n . . . . subtract 2n
... 15 = 6n . . . . . . . add 5
... 15/6 = n = 5/2
_____
Check
(2·5/2 +4) +6 = -9 +4(2·5/2 +1)
5 +4 +6 = -9 +4(5 +1)
15 = -9 +24 . . . . . true, so the answer checks OK
Answer:
THE ANSWER IS 1
Step-by-step explanation:
is the square root of 10 greater than,less than or equal to 5
Calculate the second moment of area of a 4-in-diameter shaft about the x-x and yy axes, as shown.
To calculate the second moment of area for a 4-inch diameter shaft about the x-x and yy axes, we use the formula I = π * d^4 / 64 and find that I = 4 * π in´ for both axes.
Explanation:The student is asking for the calculation of the second moment of area, also known as the moment of inertia, for a shaft with a 4-inch diameter about both the x-x and yy axes.
The moment of inertia of a circular cross-section about its centroidal axis is calculated using the formula I = π * d^4 / 64, where d is the diameter of the shaft. In this case, the diameter d is given as 4 inches.
To calculate the second moment of area for the 4-inch diameter shaft, we substitute the diameter into the formula to get:
I = π * (4 in)^4 / 64 = π * 256 in´ / 64 = 4 * π in´.
So, the moment of inertia for both the x-x and yy axes would be the same and equal to 4 * π in´, since the shaft is symmetrical about these axes.
At State College, the cost of tuition is $6168, housing is $3280, the meal plan is $2190, and books cost $1200. How much would you need to pay to attend State College for 4 years given your family will contribute $17,600?
The function s(x)equals=startfraction 3600 over 60 plus x endfraction equals 3600 left parenthesis 60 plus x right parenthesis superscript negative 1 3600 60+x=3600(60+x)−1 gives a person's average speed in miles per hour if he or she travels one mile in x seconds more or less than 60 seconds. use a linear approximation to s at 0 to find a person's approximate average speed if he or she travels one mile in 5656 seconds. what is his or her exact speed?
The exact average speed when x = 56 seconds is approximately 30.86 mph.
To use linear approximation to approximate a person's average speed if they travel one mile in 56 seconds, we'll first find the derivative of the function [tex]\( s(x) = \frac{3600}{60 + x} \)[/tex] with respect to x.
[tex]\[ s'(x) = -\frac{3600}{(60 + x)^2} \][/tex]
Now, we'll evaluate the derivative at x = 0 to find the slope of the tangent line at that point, which will be our linear approximation.
[tex]\[ s'(0) = -\frac{3600}{(60 + 0)^2} = -\frac{3600}{3600} = -1 \][/tex]
So, the slope of the tangent line at x = 0 is -1.
Now, using the point-slope form of the equation of a line, we'll find the equation of the tangent line at x = 0:
y - s(0) = s'(0)(x - 0)
[tex]\[ y - s(0) = -1 \cdot x \][/tex]
y = -x + s(0)
We know that s(0) is the exact speed at x = 0, so we'll substitute x = 0 into the original function to find it:
[tex]\[ s(0) = \frac{3600}{60 + 0} = 60 \text{ mph} \][/tex]
So, the equation of the tangent line is:
y = -x + 60
Now, to approximate the average speed when x = 56, we'll substitute x = 56 into the equation of the tangent line:
y = -56 + 60 = 4
So, the approximate average speed when x = 56 seconds is 4 mph.
To find the exact speed, we'll substitute x = 56 into the original function s(x):
[tex]\[ s(56) = \frac{3600}{60 + 56} = \frac{3600}{116} \approx 30.86 \text{ mph} \][/tex]
So, the exact average speed when x = 56 seconds is approximately 30.86 mph.
While going on a field trip, a busload of 54 students will have to be split up into three groups. two thirds will go to lunch first, and one third will go visit the exhibits. how many will go to lunch first?
Every year, a teacher surveys his students about the number of hours a week they watch television. In 2002, his students watched an average of 12 hours of television per week. In 2012, the number of hours spent watching television decreased to five per week. What is the percent decrease in the hours of television watched, rounded to the nearest tenth? 5.8% 4.2% 41.7% 58.3%
Answer:
58.3%
Step-by-step explanation:
The percent decreased in the hours of television watched is 58.3%.
What is percentage?A percentage is a number or ratio that can be expressed as a fraction of 100. Also called per centum. one one-hundredth part; 1/100. percentage.
Given that, in 2002, the students watched an average of 12 hours of television per week. In 2012, the number of hours spent watching television decreased to 5 hours per week
We need to find the percent decrease in the hours of television watched,
Percent decreased = Difference in the initial and final quantity / initial quantity × 100
Percent decreased = 12-5 / 12 × 100 = 7/12 × 100
= 58.3%
Hence, the percent decreased in the hours of television watched is 58.3%
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What is the unit rate for 822.6 km in 18 h? Enter your answer, as a decimal, in the box. ______.
we know that
A unit rate is a ratio between two different units with a denominator of one
In this problem to find the unit rate divide the total distance by the total time
so
[tex]\frac{822.6}{18} \frac{Km}{hours}= 45.7\frac{Km}{hour}[/tex]
therefore
the answer is
The unit rate is equal to [tex] 45.7\frac{Km}{hour}[/tex]
During a certain week, a post office sold Rs.280 worth of 14-paisas stamps. How many of these stamps did they sell?
The post office sold 2000.
To find out how many 14-paisa stamps the post office sold to make Rs.280, we need to follow a step-by-step solution.
Convert the total amount from rupees to paisas:Since the stamp's value is given in paisas.There are 100 paisas in 1 rupee.Rs.280 = 280 x 100 = 28000 paisas.The value of one stamp is 14 paisas.To find the number of stamps sold, we divide the total amount in paisas by the value of one stamp:28000 paisas ÷ 14 paisas per stamp = 2000 stamps.
five quarts of latex enamel paint will cover about 100 square feet of wall surface. how many quart are needed to cover 71 square feet of kitchen wall and 69 square feet of bathroom wall ?
5x - 3(1 + 2x) =24 - 4x
Is it linear or not, the height of a person and the persons age?
Scott has 10 1/2 yd of fabric to make banners for the community fair. He needs 1 3/4 yd for each banner.
Answer:
6
Step-by-step explanation:
I took the k12 2.12 Quiz: Divide Fractions
4^2-6(2^x)-16=0
solve for x