First, we need to find the curl of F. Since the rectangle C can be divided into two regions by a horizontal line, we can split the integral into two parts.
curl(F) = (∂(x - y)/∂x - ∂(x^2 + y^2)/∂y)k
= (-2y)k
Now we can apply Green's Theorem:
∫C F·dr = ∫∫R curl(F) dA
where R is the region enclosed by C, and dA is the area element.
Since the rectangle C can be divided into two regions by a horizontal line, we can split the integral into two parts:
∫C F·dr = ∫∫R1 curl(F) dA + ∫∫R2 curl(F) dA
where R1 is the region above the line y = 4.5, and R2 is the region below.
In region R1, y > 4.5, so the curl is negative:
∫∫R1 curl(F) dA = ∫0^2 ∫4.5^9 (-2y) dy dx
= -81
In region R2, y < 4.5, so the curl is positive:
∫∫R2 curl(F) dA = ∫0^2 ∫0^4.5 (-2y) dy dx
= 81/2
Therefore, the total circulation is:
∫C F·dr = ∫∫R1 curl(F) dA + ∫∫R2 curl(F) dA
= -81 + 81/2
= -144
So the answer is D) -144.
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In Exercise 17 find the area of the regular polygon
According to the diagram, the area of the regular polygon is 144√3.
How to calculate area?To find the area of a regular polygon, use the formula:
Area = (1/2) × Perimeter × Apothem
In this case, given the length of one side of the polygon (12) and the apothem (2√3). The perimeter of a regular polygon is calculated by multiplying the number of sides (n) by the length of one side (s).
Plug in the values and calculate the area:
Perimeter = n × s = 12 × 12 = 144
Area = (1/2) × Perimeter × Apothem
Area = (1/2) × 144 × 2√3
Simplifying further:
Area = 72 × 2√3
Area = 144√3
Therefore, the area of the regular polygon is 144√3.
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if 0 is an eigenvalue of the matrix of coefficients of a system of n linear equations in n unknowns, then the system has infinitely many solutions.
The statement given "if 0 is an eigenvalue of the matrix of coefficients of a system of n linear equations in n unknowns, then the system has infinitely many solutions." is true because if 0 is an eigenvalue of the matrix of coefficients of a system of n linear equations in n unknowns, then the system has infinitely many solutions
If a matrix of coefficients of a system of n linear equations in n unknowns has 0 as an eigenvalue, it implies that the homogeneous version of the system (where all constant terms are 0) has non-trivial solutions. This is because the eigenvectors associated with 0 eigenvalue form the null space of the matrix, which represents the set of all solutions to the homogeneous system.
Since the homogeneous system has non-trivial solutions, this means that the original system of equations is linearly dependent, which in turn implies that there are infinitely many solutions. This is because there are linear combinations of the given solutions that are also solutions to the system. Therefore, the statement "if 0 is an eigenvalue of the matrix of coefficients of a system of n linear equations in n unknowns, then the system has infinitely many solutions" is true.
""
if 0 is an eigenvalue of the matrix of coefficients of a system of n linear equations in n unknowns, then the system has infinitely many solutions. true or false
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I WILL GIVE BRAINLIEST AND POINTS PLS HURRY A small tree that is 6 feet tall casts a 4-foot shadow, while a building that is 27 feet tall casts a shadow in the same direction. Determine the length of the building's shadow.
12 feet
14 feet
15 feet
18 feet
Answer:
18ft
Step-by-step explanation:
6/27= 4.5, 27/4. 4.5x4=18
Answer:
18ft
Step-by-step explanation:
I am taking the test right now and I think this would be the correct answer!
A simple way I found out: 6/4 = 1.5 so I took 1.5 and divided 27 by it. 27/1.5 = 18
Hope this helped!
A table increased in price by 2/5. After the increase it was priced at £133. What was the original
price of the table?
The table increased in price by 2/5, which means the new price is 2/5 more than the original price. Therefore: The original price of the table was £95.
New price = original price + 2/5 * original price
£133 = x + 2/5 * x
To solve for x, we can simplify the equation by multiplying both sides by the denominator of the fraction, which is 5:
665 = 5x + 2x
665 = 7x
Dividing both sides by 7, we get:
x = 95
Therefore, the original price of the table was £95.
To find the original price of the table, we'll first determine the amount of the price increase and then subtract it from the final price. Here are the steps:
1. Let the original price be x.
2. The table increased in price by 2/5, so the increase is (2/5)x.
3. After the increase, the table was priced at £133, so the equation is x + (2/5)x = £133.
Now we'll solve for x:
4. First, find a common denominator for the fractions. The common denominator for 1 (coefficient of x) and 5 is 5.
5. Rewrite the equation with the common denominator: (5/5)x + (2/5)x = £133.
6. Combine the terms with x: (5/5 + 2/5)x = (7/5)x = £133.
7. To solve for x, divide both sides by 7/5 or multiply by its reciprocal, 5/7: x = £133 * (5/7).
8. Perform the calculation: x = £95.
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a moving truck company salesperson rents moving trucks that have enclosed truck beds in the shape of right rectangular prisms. if a truck bed has dimensions of by by what is the volume of the truck bed?
The volume of the truck bed is simply the product of its three dimensions, which are given as length, width, and height. Therefore, the volume of the truck bed can be calculated as:
Volume = length x width x height
or
Volume = b x w x h
where b, w, and h represent the dimensions of the truck bed in feet, meters, or any other unit of length.
In summary, the volume of a right rectangular prism, such as a moving truck bed, can be obtained by multiplying the length, width, and height of the prism.
To provide further explanation, a right rectangular prism is a three-dimensional solid figure with six rectangular faces. The faces opposite each other are congruent, and the parallel faces have equal dimensions. The length, width, and height of the prism are perpendicular to each other, and the product of these dimensions gives the volume of the prism. In the context of a moving truck, the volume of the truck bed determines the amount of space available for loading and transporting goods.
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Use unit multipliers to convert 14 yards per minute to inches per second.
After change into inches per second, the number is,
= 16.8 inches per second.
We have to given that;
To convert 14 yards per minute to inches per second.
Since, We know that;
1 yards = 36 inches
1 minutes = 60 seconds
Hence, We can change as;
= 14 yards per minute
= 14 x 36 / 30 inches per second.
= 504/ 30 inches per second.
= 16.8 inches per second.
Thus, After change into inches per second, the number is,
= 16.8 inches per second.
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Assume that playing soccer requires 540 Calories per hour. On a particular day, you ate 2,000 Calories in food. You played soccer for 2.5 hours. Your body used 800 Calories in other activities. Did you use more energy than you consumed on this day?
Answer:
yes
Step-by-step explanation:
540 X 2.5 = 1350 calories burned at soccer.
1350 + 800 = 2150 total calories burned.
2150 > 2000.
yes, more energy was used than consumed
A professor has 10 similar problems to put on a test that has 3 problems. How many different tests can she design?
The number of different tests the professor can design is 120.
Since the professor has 10 problems and needs to choose 3 for each test, we can use the combination formula to calculate the number of different tests she can design.
The formula for combinations is n choose k = n! / (k! * (n-k)!) where n is the total number of items, and k is the number of items being chosen.
In this case, n = 10 and k = 3, so we have:
10 choose 3 = 10! / (3! * (10-3)!) = 120
Therefore, the professor can design 120 different tests.
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ammeters produced by a manufacturer are marketed under the specification that the standard deviation of gauge readings is no larger than .2 amp. one of these ammeters was used to make ten independent readings on a test circuit with constant current. if the sample variance of these ten measurements is .065 and it is reasonable to assume that the readings are normally distributed, do the results suggest that the ammeter used does not meet the marketing specifications? [hint: find the approximate probability that the sample variance will exceed .065 if the true population variance is .04.]
We do not have enough evidence to suggest that the ammeter used does not meet the marketing specifications.
Statistical inference:Statistical inference is the process of making conclusions or predictions about a population based on a sample.
Hypothesis testing:Hypothesis testing is a statistical method used to determine whether there is enough evidence in a sample to support a claim about a population.
To determine if the ammeter used meets the marketing specifications, we need to test if the sample variance is significantly larger than the acceptable standard deviation of 0.2 amp.
We can use a chi-square distribution to test this hypothesis.
The test statistic is given by:
=> x²= (n-1)× s² / σ²
Where n is the sample size, s² is the sample variance, and σ² is the true population variance.
We are given that n = 10, s² = 0.065, and we want to test if the ammeter does not meet the marketing specifications,
Which means that the true population variance is greater than 0.04.
We can calculate the test statistic as follows:
x² = (10-1) × 0.065 / 0.04 = 10.54
The critical value of the chi-square distribution with 9 degrees of freedom (n-1) and a significance level of 0.05 is 16.92.
Since our test statistic is less than the critical value, we fail to reject the null hypothesis that the true population variance is no larger than 0.04.
Therefore,
We do not have enough evidence to suggest that the ammeter used does not meet the marketing specifications.
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Line XC is a tangent to the circle and line XA is a secant of the circle. Given that arc BC measures 25 degrees and arc AC measures 115 degrees, what is the measure of Angle AXC?
The measure of the secant tangent angle m∠AXC is equal to 45°
What is the secant tangent angleThe secant tangent angle is the angle formed by a tangent and a secant that intersect outside of a circle. The measure of the secant tangent angle can be found using the following formula:
θ = 1/2 (arc EB - arc BD)
where arc EB and arc BD are the measures of the arcs intercepted by the secant and tangent, respectively.
m∠AXC = 1/2(arc AC - arc BC)
m∠AXC = 1/2(115 - 25)
m∠AXC = 1/2(90)
m∠AXC = 45°
Therefore, the measure of the secant tangent angle m∠AXC is equal to 45°
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find the x coordinate of the point of maximum curvature (call it x0 ) on the curve y=3ex and find the maximum curvature, κ(x0).
There is no maximum value of κ on the curve y=3e^x.
To find the point of maximum curvature on the curve y=3e^x, we need to first find the second derivative of y with respect to x, which will give us the curvature of the curve:
y = 3e^x
y' = 3e^x (since the derivative of e^x is e^x)
y'' = 3e^x (since the second derivative of e^x is also e^x)
Now, to find the point of maximum curvature, we need to set y'' equal to zero and solve for x:
y'' = 3e^x = 0
e^x = 0
This equation has no real solutions, which means that there is no point of maximum curvature on the curve y=3e^x.
To find the maximum curvature, we can use the formula:
κ = |y''| / (1 + y'^2)^(3/2)
Since we know that y'' = 3e^x, we can simplify this formula to:
κ = 3e^x / (1 + (3e^x)^2)^(3/2)
To find the maximum value of κ, we can take the derivative of κ with respect to x and set it equal to zero:
dκ/dx = 3e^x (9e^2x - 2) / (1 + 9e^2x)^(5/2) = 0
This equation has no real solutions, which means that there is no maximum value of κ on the curve y=3e^x.
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Two buses leave towns 492 miles apart at the same time and travel toward each other. One bus travels 11 mi/h slower than the other. If they meet in four hours, what is the rate of each bus? HELPPP ASAP
The faster bus is running at a velocity of 67 mph and the sluggish one is proceeding at a speed of 56 mph.
Determining the rate of each busLet "x" speed of the quicker bus
and the slower bus "x - 11",
since we comprehend that it is travelling 11 mph less than the faster one. When both meet, a total distance of 492 miles will have been covered (which is the distance between the two towns). We can utilize the formula:
distance = rate x time
Applicable to each auto:
distance = rate x time
distance = x (mph) x 4 (hours) (for the swifter motorcoach)
distance = (x - 11) (mph) x 4 (hours) (for the slower vehicle)
By adding those equations together, we are given:
492 = 4x + 4(x - 11)
After decreasing the equation, we acquire:
492 = 8x - 44
536 = 8x
Therefore, x = 67
So, the more rapidly running coach is going at an velocity of 67 mph and the sluggish one is proceeding at a speed of (67 - 11)
= 56 mph.
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Find a polynomial function whose graph passes through each set of points.
a. (-3, 15), (1, 11), and (0, 6)
b. (-2,-7), (-1, -3), (0, 3), (1, 5), and (2, -3)
c. (4,-1) and (-3, 13)
d. (-1,-6), (0, 2), (1, 8), and (2, 42)
Thank you!!
a. The polynomial function that passes through the points (-3, 15), (1, 11), and (0, 6) is y = -2x² - 3x + 6.
b. The polynomial function that passes through the points (-2,-7), (-1, -3), (0, 3), (1, 5), and (2, -3) is y = -1/2x⁴ - 3/2x³ + 3x² + 7/2x + 3.
c. The polynomial function that passes through the points (4,-1) and (-3, 13) is y = -3x + 11.
d. The polynomial function that passes through the points (-1,-6), (0, 2), (1, 8), and (2, 42) is y = 6x³ + 2x² - 18x
How to calculate the valuesa. Using the given points, we can create a system of three equations:
15 = 9a - 3b + c
11 = a + b + c
6 = c
Solving this system of equations gives us a = -2, b = -3, and c = 6
b. Using the given points, we can create a system of five equations:
-7 = 16a - 8b + 4c - 2d + e
-3 = -2a + b - c + d + e
3 = e
5 = 2a - b + c + d + e
-3 = 16a + 8b + 4c + 2d + e
Solving this system of equations gives us a = -1/2, b = -3/2, c = 3, d = 7/2, and e = 3.
c. Using the given points, we can create a system of two equations:
-1 = 4m + b
13 = -3m + b
Solving this system of equations gives us m = -3 and b = 11.
d. Using the given points, we can create a system of four equations:
-6 = -a + b - c + d
2 = b
8 = a + b + c + d
42 = 8a + 4b + 2c + d
Solving this system of equations gives us a = 6, b = 2, c = -18, and d = 4
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explain how the formulas V=Bh and V=lwh are related
The formulas V=Bh and V=lwh are related because the base area B equals lw
Explaining how the formulas V=Bh and V=lwh are relatedFrom the question, we have the following parameters that can be used in our computation:
V = Bh
V = lwh
The above formulas are formulas to calculate the volume of a rectangular prism
By substiution, we have
Bh = lwh
Divide both sides by h
B = lw
This means that the base area of the prism is lw
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Question I need help with:
Surface area of larger triangular pyramid is 49cm².
Given,
Altitude of smaller pyramid = 3 cm.
Altitude of larger pyramid = 7 cm.
Surface area of smaller pyramid = 9cm².
Now,
Relation between altitudes of similar pyramids and surface area :
Surface area of smaller pyramid / Surface area of larger pyramid = (altitude of smaller pyramid / altitude of larger pyramid)²
Let us assume the surface area of larger pyramid be x cm²
Substituting the given values in the relation,
9 cm²/x cm² = (3/7)²
x = 49 cm² .
Thus the surface area of larger pyramid is 49 cm².
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Please help me I don't understand this with steps please
The surface area of the rectangular prism is 1236 in².
We have,
The surface area of the rectangular prism.
= lower surface + top surface + back surface + front surface
+ 2 x side surface
Each surface is in the form of a rectangle.
So,
= 14 x 8 + 14 x 8 + 23 x 8 + 23 x 8 + 2 x (23 x 14)
= 112 + 112 + 184 + 184 + 2 x 322
= 112 + 112 + 184 + 184 + 644
= 1236 in²
Thus,
The surface area of the rectangular prism is 1236 in².
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Let m be a positive integer. Show that a ≡ b (mod m) if a mod m = b mod m. Drag the necessary statements and drop them into the appropriate blank to build your proof
If a and b have the same remainder when divided by m, then a is congruent to b modulo m.
We know that when a positive integer a is divided by a positive integer m, there is a unique quotient q and a remainder r such that a = mq + r and 0 ≤ r < m. This is called the Division Algorithm.
Now suppose a mod m = b mod m. This means that both a and b leave the same remainder when divided by m. So we can write a = mq + r and b = mq + r' for some integers q, r, and r' where 0 ≤ r, r' < m.
Then we have a - b = mq + r - mq - r' = (r - r') which is clearly divisible by m since m divides r - r'. Therefore, we have shown that m divides a - b, or equivalently, a ≡ b (mod m).
To summarize, if two integers have the same remainder when divided by a positive integer m, then they are congruent modulo m. This result is used frequently in number theory and modular arithmetic.
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Increase 600 by 8⅓%.
Answer:
650
Step-by-step explanation:
calculate 8 [tex]\frac{1}{3}[/tex]% of 600 then add this value to 600 for increase
8 [tex]\frac{1}{3}[/tex] % × 600 ← convert mixed number to improper fraction
= [tex]\frac{25}{3}[/tex] % × 600
= [tex]\frac{\frac{25}{3} }{100}[/tex] × 600 ( % is out of 100 )
= [tex]\frac{25}{300}[/tex] × 600
= 25 × 2
= 50
then increase is 50
so 600 increased by 8 [tex]\frac{1}{3}[/tex] % = 600 + 50 = 650
Roland works in a local factory
Here is the completed piecewise function that models Roland's pay:
[tex]\[f(x) = \begin{cases} 95x & \text{if } x \leq 100 \\1.25(x-100) + 95(100) & \text{if } 101 \leq x \leq 300 \\1.55(x-300) + 95(100) + 1.25(300-100) & \text{if } x > 300\end{cases}\][/tex]
This piecewise function represents Roland's pay based on the different pay rates for the respective ranges of units produced.
To create a piecewise function to model Roland's pay, we need to consider the different ranges of units produced and the corresponding pay rates.
Let's complete the missing portions of each expression:
[tex]\[f(x) = \begin{cases} 95x & \text{if } x \leq 100 \\1.25(x-100) + 95(100) & \text{if } 101 \leq x \leq 300 \\1.55(x-300) + 95(100) + 1.25(300-100) & \text{if } x > 300\end{cases}\][/tex]
In the piecewise function:
- For [tex]\(x \leq 100\)[/tex], Roland receives 95 cents for each unit, so the expression is [tex]\(f(x) = 95x\).[/tex]
- For [tex]\(101 \leq x \leq 300\),[/tex] Roland receives $1.25 for each unit between 101 and 300. The base pay for the first 100 units (at 95 cents each) is added, resulting in the expression [tex]\(f(x) = 1.25(x-100) + 95(100)\).[/tex]
- For [tex]\(x > 300\)[/tex], Roland receives $1.55 for each unit over 300. Both the base pay for the first 100 units and the additional pay for units between 101 and 300 are added, leading to the expression [tex]\(f(x) = 1.55(x-300) + 95(100) + 1.25(300-100)\).[/tex]
This piecewise function models Roland's pay based on the different pay rates for the different ranges of units produced.
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14. Find the area of the shaded region.
Answer:
13.5 cm^2
9*3 = 27
27/2=13.5
Two 5.0-cm-diameter metal disks separated by a0.61-mm-thick piece of Pyrex glass are charged to a potential difference of 1300V . (Dielectric constant of the Pyrex glass is Pkpyrex=4.7.)A) What is the surface charge density on the disks?= muC/m^2B) What is the surface charge density on the glass?= muC/m^2
A) The surface charge density on the metal disks is 2.45 μC/m^2.
B) The surface charge density on the Pyrex glass is -2.45 μC/m^2.
To determine the surface charge density on the disks and the glass, we need to use the formula for capacitance of a parallel plate capacitor with a dielectric between the plates:
C = ε0εrA/d
where C is the capacitance, ε0 is the permittivity of free space (8.85 x 10^-12 F/m), εr is the relative permittivity (dielectric constant) of the Pyrex glass, A is the area of the plates, and d is the distance between the plates. We can rearrange this equation to solve for the surface charge density:
σ = Q/A
where σ is the surface charge density and Q is the charge on the plates.
First, we need to calculate the capacitance of the capacitor:
C = ε0εrA/d = (8.85 x 10^-12 F/m)(4.7)(π(0.05 m)^2)/(0.00061 m) = 1.74 x 10^-11 F
The charge on each plate can be calculated using the potential difference:
Q = CV = (1.74 x 10^-11 F)(1300 V) = 2.26 x 10^-8 C
Now we can calculate the surface charge density on the disks:
σ = Q/A = (2.26 x 10^-8 C)/(π(0.05 m)^2) = 2.45 μC/m^2
The surface charge density on the glass is equal in magnitude but opposite in sign:
σ = -2.45 μC/m^2
This means that the disks have a positive surface charge density, while the glass has an equal but negative surface charge density.
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Suppose we are interested in the proportion of adults in the U. S. With a bachelor's degree or higher. We randomly select 5000 adults in order to estimate this proportion. Use this information to answer questions 1-4. 1) What is the population? 2) What is the sample? 3) What is the parameter? 4) What is the statistic?
The population of interest is all adults in the United States, the sample is 5000 randomly selected adults, the parameter is the proportion of adults with a bachelor's degree or higher, and the statistic is the proportion of adults in the sample with a bachelor's degree or higher.
1 - The population is the entire group of interest, which in this case is all adults in the United States.
2 - The sample is a subset of the population, selected in a random and representative way, in order to make inferences about the larger population. In this scenario, the sample consists of 5000 adults randomly selected from the population of all adults in the United States.
3 - The parameter is a numerical measurement that describes a characteristic of a population. In this case, the parameter of interest is the proportion of adults in the United States with a bachelor's degree or higher. This is because we are interested in making inferences about the entire population of adults in the United States.
4 - The statistic is a numerical measurement that describes a characteristic of a sample. In this scenario, the statistic of interest is the proportion of adults in the sample who have a bachelor's degree or higher. This is because the sample is used to estimate the parameter of the population.
To estimate the population parameter from the sample statistic, we can use statistical inference techniques such as confidence intervals or hypothesis testing. The accuracy of our estimates depends on the sample size, sampling method, and other factors that influence the quality of the data. It's important to ensure that the sample is representative of the population and that any bias or confounding factors are taken into account when making inferences.
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By visual inspection, determine the best-fitting regression model for the scatterplot.
A. Quadratic
B. No Pattern
C. Linear
D. Exponential
find f · dr c where c is a circle of radius 4 in the plane x y z = 3, centered at (1, 1, 1) and oriented clockwise when viewed from the origin. f = (z − y) i (x − z) j (y − x)
The line integral of f along the given circle is 0.
We need to evaluate the line integral of the vector field f = (z − y) i + (x − z) j + (y − x) k along the given path, which is a circle of radius 4 in the plane x y z = 3, centered at (1, 1, 1) and oriented clockwise when viewed from the origin.
To parameterize the circle, we can use the following parametric equations:
x = 1 + 4 cos t
y = 1 + 4 sin t
z = 3
where t varies from 0 to 2π as we traverse the circle once in the clockwise direction.
Taking the derivative of the parameterization with respect to t, we get:
dx/dt = -4 sin t
dy/dt = 4 cos t
dz/dt = 0
Now we can evaluate the line integral using the formula:
∫C f · dr = ∫[a,b] f(r(t)) · r'(t) dt
where C is the curve, r(t) = (x(t), y(t), z(t)) is its parameterization, and f(r(t)) is the vector field evaluated at r(t).
Substituting the parameterization and the derivative into the integral, we get:
∫C f · dr = ∫[0,2π] (3 - (1+4sin(t))) (-4sin(t)) + ((1+4cos(t)) - 3) (4cos(t)) + ((1+4sin(t)) - (1+4cos(t))) (0) dt
Simplifying, we get:
∫C f · dr = ∫[0,2π] (-16sin(t)cos(t) + 16cos(t)^2 + 4sin(t) - 4cos(t)) dt
Integrating each term, we get:
∫C f · dr = [-8cos(t)^2 + 16sin(t)cos(t) + 4cos(t) - 4sin(t)]|[0,2π]
Substituting the limits, we get:
∫C f · dr = [(-8cos(2π)^2 + 16sin(2π)cos(2π) + 4cos(2π) - 4sin(2π)) - (-8cos(0)^2 + 16sin(0)cos(0) + 4cos(0) - 4sin(0))]
Since cos(2π) = cos(0) = 1 and sin(2π) = sin(0) = 0, the expression simplifies to:
∫C f · dr = [(-8 + 0 + 4 - 0) - (-8 + 0 + 4 - 0)] = 0
Therefore, the line integral of f along the given circle is 0.
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6. You are making costumes for a play. You have sewn 9 costumes. If you are 30% finished, how many total costumes are you sewing?
Answer: 30 costumes
Step-by-step explanation:
Let x be the total amount of costumes.
In order to get 30% , we need to divide 9 by the total amount of costumes (x), but first we need to change 30% to its multiplier:
30% ÷ 100 = 0.3 (multiplier).
[tex]\frac{9}{x} =30[/tex]%
[tex]\frac{9}{x} =0.3[/tex]
[tex]x=\frac{9}{0.3}[/tex]
[tex]x=30[/tex] costumes
Or if that is confusing:
[tex]\frac{9}{x} =30[/tex]%
[tex]\frac{9}{x} =0.3[/tex]
[tex]\frac{x}{9} =\frac{1}{0.3}[/tex]
[tex]x=9 *\frac{1}{0.3}[/tex]
[tex]x=\frac{9}{0.3}[/tex]
[tex]x=30[/tex] costumes
The above equations are exactly the same, they are just written differently.
We can now check if our answer is correct:
[tex]\frac{9}{30} = 0.3\\0.3*100=30[/tex]%
Hope you understand!
(b) Explain why the following proportion would solve for the length of AC below.
The length of the arc is solved to get the proportion
= x / 12π = 130 / 360
this proves that the proportion would solve the arc length
How to find the length of arclength of arc is calculated using the formula given below
= (given angle) / 360 x 2 π r
Where
x is length or arc
r is radius = 6 in
given angle = 130 degrees
then substituting into the formula
x = (given angle) / 360 x 2 π r
x = 130 / 360 * 2 * π * 6 in
x = 130 / 360 * 12π
dividing both sides by 12π
x / 12π = 130 / 360 (this equals the given proportion)
but solving for x
x = 13.61 in
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(-3, -1) and (3, 3) into standard form
Answer: 6
Step-by-step explanation:
find the sum of the series. [infinity] 2n 9nn! n = 0
In conclusion, the series ∑ (n = 0 to ∞) 2^n * 9^n * n! diverges and does not have a finite sum.
To find the sum of the series ∑ (n = 0 to ∞) 2^n * 9^n * n!, we can start by analyzing the terms of the series.
Let's consider the nth term of the series:
Tn = 2^n * 9^n * n!
We notice that the term involves the exponential growth of 2^n and 9^n, as well as the factorial n! term. This suggests that the series may diverge since both exponential and factorial growth tend to increase rapidly.
To confirm this, let's examine the ratio of consecutive terms:
R = Tn+1 / Tn
R = (2^(n+1) * 9^(n+1) * (n+1)!) / (2^n * 9^n * n!)
Simplifying the expression, we get:
R = (2 * 9 * (n+1)) / n!
As n approaches infinity, this ratio does not tend to zero, indicating that the terms of the series do not converge to zero. Therefore, the series diverges, and we cannot find a finite sum for it.
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Find the coefficient of x5in the Maclaurin series generated by f(x) = sin 4x.
The coefficient of x^5 in the Maclaurin series generated by f(x) = sin(4x) is 256/15.
To find the coefficient of x^5 in the Maclaurin series generated by f(x) = sin 4x, we need to first find the derivatives of f(x) up to the fifth order, evaluate them at x=0, and then use the formula for the Maclaurin series coefficients.
The Maclaurin series of a function f(x) is an infinite series that represents the function as a sum of its derivatives evaluated at x=0, multiplied by powers of x. The formula for the Maclaurin series coefficients is given by:
an = (1/n!) * f^(n)(0)
where f^(n)(x) denotes the nth derivative of f(x), evaluated at x. To find the coefficient of x^5 in the Maclaurin series generated by f(x) = sin 4x, we need to find the fifth derivative of sin(4x), evaluate it at x=0, and then use the formula above.
We have:
f(x) = sin(4x)
f'(x) = 4cos(4x)
f''(x) = -16sin(4x)
f'''(x) = -64cos(4x)
f''''(x) = 256sin(4x)
f^(5)(x) = 1024cos(4x)
Therefore, the coefficient of x^5 in the Maclaurin series generated by f(x) = sin(4x) is given by:
a5 = (1/5!) * f^(5)(0) = (1/120) * 1024 = 256/15
Hence, the coefficient of x^5 in the Maclaurin series generated by f(x) = sin(4x) is 256/15.
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She put 4 sweets on top of each cake. (a) Write down an expression, in terms of x, for the number of sweets she used. . (1)
Paul made 3 more cakes than Jennifer. (b) Write down an expression, in terms of x, for the number of cakes Paul made. . (1)
Paul also put 4 sweets on each of his cakes. (c) Write down an expression, in terms of x, for the number of sweets Paul used
The expression is 4x.
The expression is x + 3.
The expression is 4(x + 3).
The number of sweets she used can be represented by the product of the number of cakes, x, and the number of sweets on each cake, which is 4.
The number of cakes Paul made can be represented by the sum of the number of cakes Jennifer made, x, and 3.
The number of sweets Paul used can be represented by the product of the number of cakes Paul made, which is x + 3, and the number of sweets on each cake, which is 4.
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