Use the first three non-zero terms of a Maclaurin series to estimate the integral from 0 to 1 of the cosine of x squared, dx.


0. 905


0. 904


0. 806


1. 16

We used the given sector area formula, 28.25 = ∅/360 × π × 12², to find the central angle (∅) by rearranging the equation

First, we used the given sector area formula, 28.25 = ∅/360 × π × 12², to find the central angle (∅) by rearranging the equation and solving for ∅, which resulted in ∅ ≈ 22.5 degrees.

Next, we applied the arc length formula, L = ∅/360 × 2 × π × r, and plugged in the values we had, including the calculated ∅ and the given radius (12 feet).

Finally, we calculated the arc length (L) to be approximately 4.71 feet, which is the length of steel that must be replaced.

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According to the model, each additional year of education adds $1200 to a person's yearly income in their first job after completing schooling.

We're given the equation y = 1200x + 7000, where x represents the number of years of education, and y represents the yearly income.

To find how much 1 year of education adds to a person's yearly income, we need to look at the coefficient of the variable x, which is 1200.

So according to the model, 1 year of education adds $1,200 to a person's yearly income.

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The cost of a bag of chips is $0.75 and the cost of a cheeseburger is $3.

The definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.

let the cost of a cheesburger be x.

let the cost of a bag of chips be y

therefore, it is given that x + y = 3.75                ...........(i)

it is also given that 2x + 3y = 8.25                     ............(ii)

multiplying the equation in( i) by 2 we get 2x + 2y = 7.50     ......(iii)

subtracting the equation in iii) from the equation in ii) we get y = $0.75

Therefore ,the cost of a bag of chips is $0.75

Substituting the value of y found in (ii) we get x = 3.

therefore ,the cost of a cheeseburger is $3.

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The complete question is ,

Nathan ordered 1 cheeseburger and 1 bag of chips for 3. 75 jack ordered 2 cheeseburgers and 3 bags of chips for 8. 25.find the value of one cheeseburger and one bag?

The rate of change of the revenue function R(x) is given by its derivative R'(x) = 190 - 0.8x. To find where the rate of change is equal to 0, we set R'(x) = 0 and solve for x:

190 - 0.8x = 0
0.8x = 190
x = 237.5

Therefore, the revenue function is constant when x = 237.5.
To find the value of x where revenue is constant, we need to find the point where the rate of change of the revenue function R(x) is equal to 0. This can be achieved by finding the derivative of R(x) with respect to x, and then setting it equal to 0.

R(x) = 190x - 0.4x^2

The derivative of R(x) with respect to x is:

R'(x) = dR(x)/dx = 190 - 0.8x

Now, set R'(x) equal to 0 and solve for x:

0 = 190 - 0.8x

0.8x = 190

x = 190 / 0.8

x = 237.5

The revenue function is constant when x = 237.5.

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To find the volume of a cone, we use the formula:

[tex]V = (1/3)\pi r^2h[/tex]

where V is the volume, r is the radius of the circular base, h is the height of the cone, and [tex]\pi[/tex] is approximately 3.14159.

In this problem, the height of the cone is given as 10 ft and the radius of the circular base is given as 6 ft.

First, we need to find the slant height of the cone. We can use the Pythagorean theorem:

[tex]l = \sqrt{(r^2 + h^2)[/tex]

[tex]l = \sqrt{(6^2 + 10^2)[/tex]

[tex]l = \sqrt{\\(36 + 100)[/tex]

[tex]l = \sqrt{136[/tex]

[tex]l = 11.66 ft[/tex]

Now we can substitute the values into the formula for the volume:

[tex]V = (1/3)\pi r^2h[/tex]

[tex]V = (1/3)\pi (6^2)(10)[/tex]

[tex]V = 120\pi /3[/tex]

[tex]V = 40\pi[/tex]

[tex]V= 125.6 cubic feet[/tex]

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Answer:

28 degrees and 62 degrees

Step-by-step explanation:

Set up your equation like this:

x+(x-34)=90

2x-34=90

2x=56

x=28

28+34=62

So the smaller angle is 28 degrees and the larger angle is 62 degrees.

Hope this helps!! :D

To solve this integral, we'll use a trigonometric substitution. Let x = (1/6)tan(θ), which implies dx = (1/6)sec^2(θ)dθ.

Now, we can rewrite the integral as:

∫√(1 + 36(1/6tan(θ))^2) (1/6)sec^2(θ)dθ

Simplify the expression inside the square root:

∫√(1 + 6^2tan^2(θ)) (1/6)sec^2(θ)dθ

Now, recall the trigonometric identity: 1 + tan^2(θ) = sec^2(θ). Using this identity, we have:

∫√(sec^2(θ)) (1/6)sec^2(θ)dθ

Simplify and integrate:

(1/6)∫sec^3(θ)dθ

Unfortunately, the integral of sec^3(θ) is non-elementary, so we cannot find a closed-form expression for it. However, you can look up the techniques used to evaluate this integral, such as integration by parts or reduction formulas, if you need a more detailed solution.

Remember to convert the result back to the original variable x using the substitution x = (1/6)tan(θ), and don't forget to add the constant of integration, C, at the end.

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The algebraic expressions representing the books Jan and Jackie have left are J - 4 and J - 2 - 4, respectively, where J is the initial number of books checked out by both.

Let's use "J" to represent the number of books Jan and Jackie checked out from the library. After 3 weeks, Jan turned in 4 books, so she has J - 4 books left.

Jackie returned 2 books after the first week, so she has J - 2 books left. When she returned 4 more books later, she had J - 2 - 4 = J - 6 books left. Therefore, the algebraic expressions for the number of books Jan and Jackie have left are

Jan: J - 4

Jackie: J - 6

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Answer:

Step-by-step explanation:

18 students don't walk to school

6 boys, 12 girls

12 students walk to school (30-18)

3/4 of students who walk to school are boys = 9 boys, 3 girls

There would be 26,790 books. of the 30,000 books in the collection should Mrs. Ross expect to have no damage

Out of the random sample, 902 books had no damage. To estimate the number of books with no damage in the entire collection of 30,000 books, we can use the proportion of books in the sample that had no damage.

The proportion of books in the sample with no damage is:

902 / (902 + 80 + 43) = 0.893

Therefore, we can expect approximately:

30,000 x 0.893 = 26,790

books in the collection to have no damage. Rounded to the nearest whole number, the answer is:

26,790 books.

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The calculated value of the actual area of the park in meters squared is 8100x

From the question, we have the following parameters that can be used in our computation:

A scale drawing of a rectangular park had a scale of 1 cm = 90 m.

This means that

Scale factor = 90/1

Evaluate

Scale factor = 90

The actual area of the park in meters squared is calculated as

Area = Area of scale * Scale factor^2

Substitute the known values in the above equation, so, we have the following representation

Area = Area of scale * 90^2

Evaluate

Area = Area of scale * 8100

Let Area of scale = x

So, we have

Area = 8100x

Hence, the actual area of the park in meters squared is 8100x

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The function given is y = 18000(0.76)^t, where y represents the value of the car and t represents the time in years.

This is an exponential decay function, meaning that the value of the car decreases over time. To determine if the value of the function will ever be 0, we would need to find if there exists a time t when y = 0. Let's analyze the function:

0 = 18000(0.76)^t

In an exponential decay function, the base (0.76 in this case) is between 0 and 1, so as time (t) increases, (0.76)^t will approach 0, but it will never actually reach 0. Thus, the value of the car will keep decreasing over time but will never be exactly 0.

In summary, the value of the function, which represents the car's value, will never be 0, but it will get infinitely close to 0 as time progresses. This is a characteristic of exponential decay functions, where the value never reaches 0 but approaches it as time goes on.

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The student does 18 J of work on the textbook. Their power output is 0.0228 W. Work is calculated by multiplying force and displacement, while power is the rate at which work is done and is calculated by dividing work by time.

To calculate the work done by the student on the textbook, we use the formula

work = force x distance x cos(theta)

where force is the weight of the textbook, distance is the height it is lifted, and theta is the angle between the force and distance vectors, which is 0 degrees since the force is acting vertically upward and the distance is upward as well. Thus, we have

work = 12 N x 1.5 m x cos(0) = 18 J

To calculate the power output of the student, we use the formula:

power = work / time

where work is the work done by the student on the textbook, and time is the total time it took to lift and carry the book. Thus, we have:

time = 1.5 s + 7 s = 8.5 s

power = 18 J / 8.5 s = 2.12 W

Therefore, the student did 18 J of work on the textbook, and had a power output of 2.12 W.

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--The given question is incomplete, the complete question is given

" a student lifts a 12 N textbook 1.5 m in 1.5 s and carries the books 5 m across the room in 7 s. how much work does the student do on the textbook? and what is the power output of the student.​"--

Answer:

568.2 in

Step-by-step explanation:

To find this we first have to divide 1894/10 then we get 189.4 which we multiply by 3 to find how much more sand we need.

(a) The three different trigonometric ratios that could have been used to find the distance x between Emma and the base of the tree are sin(60°) = 43.3/50, cos(60°) = x / 50, and tan(60°) = 43.3 / x. The distance is 25 ft.

(b) Using the Pythagorean theorem, the distance x between Emma and the base of the tree is 25 ft.

(a) We can use sine, cosine, and tangent trigonometric ratios to find the distance x between Emma and the base of the tree.

1. Sine:
sin(60°) = opposite/hypotenuse = (tree height) / 50
tree height = 50 * sin(60°) = 43.3 ft (since it's given that the tree is 43.3 ft tall)

2. Cosine:
cos(60°) = adjacent/hypotenuse = x / 50
x = 50 * cos(60°) = 25 ft

3. Tangent:
tan(60°) = opposite/adjacent = (tree height) / x
x = (tree height) / tan(60°) = 43.3 / tan(60°) ≈ 25 ft

(b) To find the distance x between Emma and the base of the tree using the Pythagorean theorem, we can consider the triangle formed by Emma, the base of the tree, and the top of the tree.

Let's call the distance from the base of the tree to the top of the tree (tree height) y.

Emma's distance to the cat (50 ft) is the hypotenuse, the distance x is one leg, and the tree height y (43.3 ft) is the other leg of the right triangle.

Using the Pythagorean theorem: x² + y² = hypotenuse²
x² + 43.3² = 50²
x² + 1874.89 = 2500
x² = 625.11
x ≈ 25 ft

So, the distance between Emma and the base of the tree is approximately 25 ft.

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Sine and cosine are both equivalent to x/y, depending on the angle of the point on the circle.

To find two trigonometric functions equivalent to the ratio x/y, we can use the unit circle. Let's assume that x and y are the coordinates of a point on the circle, where x represents the horizontal displacement, and y represents the vertical displacement.

From this, we can find that the sine of the angle that x and y make with the x-axis is y, and the cosine of that angle is x. Therefore, the two trigonometric functions equivalent to the ratio x/y are sine and cosine.

We can write this as sin(angle) = y/y and cos(angle) = x/y. It's important to note that the value of the angle depends on the position of the point on the unit circle.

So, sine and cosine are both equivalent to x/y, depending on the angle of the point on the circle.

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It will take 15 months for the number of basic subscriptions to match the number of deluxe subscriptions.

Let's denote the number of months passed by "m".

In m months, the number of basic subscriptions will be 288 - 5m (since 5 basic subscriptions are decreasing each month), and the number of deluxe subscriptions will be 93 + 8m (since 8 deluxe subscriptions are increasing each month).

We want to find out when the number of basic subscriptions will match the number of deluxe subscriptions, so we can set the two expressions equal to each other:

288 - 5m = 93 + 8m

We can then solve for m:

288 - 93 = 8m + 5m

195 = 13m

m = 15

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Answer:

The pattern is you are subtracting by 8. The subsequent three terms are 51, 43, and 35.

Step-by-step explanation:

First, you can subtract the first value from the second to find the common difference. Then you continue on this pattern. Simple!

The equation is Gross Income = Taxes Owed / 0.09.

To solve for the gross income for each family based on their taxes owed and the fact that this represents 9% of each family's gross income, follow these steps:

1. Write the equation: Taxes Owed = 0.09 * Gross Income
2. Rearrange the equation to solve for Gross Income: Gross Income = Taxes Owed / 0.09
3. Substitute the Taxes Owed value for each family into the equation and calculate their Gross Income.

For each family, input their taxes owed into the equation and you will find their gross income. Remember, the equation is Gross Income = Taxes Owed / 0.09.

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The measures of the angles JIK and JIL are 67 degrees and 157 degrees

From the question, we have the following parameters that can be used in our computation:

The inscribed angle opposite to the same arc is half of the external angle

Using the above as a guide, we have the following:

JIK = 90 - 46/2

JIK = 67 degrees

Also, we have

JIL = 90 + JIK

So, we have

JIL = 90 + 67

JIL = 157 degrees

Hence, the measure of the angle JIL is 157 degrees

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The indicated coefficients are:

[tex]c_2 = -(-2) = 2[/tex]

[tex]c_4 = 5/2[/tex]

[tex]c_5 = -22[/tex]

To find the power series solution of the differential equation about x = 0, we assume that the solution has the form:

y(x) = ∑(n=0 to infinity) [tex]c_n x^n[/tex]

where [tex]c_n[/tex] are the coefficients of the power series.

Differentiating y(x), we get:

y'(x) = ∑(n=1 to infinity) [tex]n c_n x^{(n-1)}[/tex]

Next, we substitute y(x) and y'(x) into the differential equation:

([tex]x^2[/tex] - x + 1)y' - y + 8y = 0

([tex]x^2[/tex] - x + 1) ∑(n=1 to infinity)[tex]n c_n x^{(n-1)}[/tex] - ∑(n=0 to infinity)[tex]c_n x^n[/tex] + 8∑(n=0 to infinity)[tex]c_n x^n[/tex] = 0

Simplifying this expression and grouping the terms with the same power of x, we get:

∑(n=1 to infinity) [tex]n c_n x^n (x^2 - x + 1)[/tex]+ ∑(n=0 to infinity) [tex](8c_n - c_{(n+1)}) x^n[/tex] = 0

Since this equation holds for all values of x, we must have:

[tex]n c_n (n+1) - (n+2) c_(n+2) + 8c_n - c_(n+1) = 0[/tex]

for all n ≥ 0, where we have set [tex]c_{(-1){ = 0[/tex]and [tex]c_{(-2)}[/tex]= 0.

Using the initial conditions y(0) = 0 and y'(0) = 4, we have:

[tex]c_0 = 0[/tex]

[tex]c_1 = y'(0) = 4[/tex]

Substituting these values into the recurrence relation, we can recursively find the coefficients of the power series solution:

[tex]n = 0: 0 c_0 - 2 c_2 + 8 c_0 - c_1 = 0 = > c_2 = (4-8c_0+c_1)/(-2) = -2[/tex]

[tex]n = 1: 1 c_1 - 3 c_3 + 8 c_1 - c_2 = 0 = > c_3 = (9c_1-c_2)/3 = 6[/tex]

[tex]n = 2: 2 c_2 - 4 c_4 + 8 c_2 - c_3 = 0 = > c_4 = (10c_2-c_3)/(-4) = 5/2[/tex]

[tex]n = 3: 3 c_3 - 5 c_5 + 8 c_3 - c_4 = 0 = > c_5 = (11c_3-c_4)/5 = -22/15[/tex]

[tex]n = 4: 4 c_4 - 6 c_6 + 8 c_4 - c_5 = 0 = > c_6 = (9c_4-c_5)/(-6) = -64/45[/tex]

Hence, the power series solution of the differential equation about x=0 is:

[tex]y(x) = 4x + 2x^2 - 4x^3 + 23x^4 - 44/9 x^5 + 24/5 x^6 - 326/315 x^7 + ...[/tex]

Therefore, the indicated coefficients are:

[tex]c_2 = -(-2) = 2[/tex]

[tex]c_4 = 5/2[/tex]

[tex]c_5 = -22[/tex]

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Answer:We can start by resolving the forces acting on the ornamental star in the vertical direction and in the horizontal direction. Since the star is stationary, these forces must balance each other out.

Let's call the tension in the higher chain T1 and the tension in the lower chain T2. We can use trigonometry to find the vertical and horizontal components of each tension force.

For T1:

The vertical component is T1cos(40°) and the horizontal component is T1sin(40°).

For T2:

The vertical component is T2cos(30°) and the horizontal component is T2sin(30°).

Now, resolving the forces in the vertical direction, we have:

T1cos(40°) + T2cos(30°) - mg = 0

where m is the mass of the star and g is the acceleration due to gravity.

Substituting the values:

T1cos(40°) + T2cos(30°) - (40 kg)(9.81 m/s^2) = 0

T1cos(40°) + T2cos(30°) = 392.4 N

Next, resolving the forces in the horizontal direction, we have:

T1sin(40°) = T2sin(30°)

Now we have two equations with two unknowns:

T1cos(40°) + T2cos(30°) = 392.4 N

T1sin(40°) = T2sin(30°)

Solving these equations simultaneously, we get:

T1 = 266.4 N

T2 = 205.1 N

Therefore, the tension in the higher chain (T1) is 266.4 N and the tension in the lower chain (T2) is 205.1 N.

Step-by-step explanation:

Answer:

Step-by-step explanation:

Car are the preferred transportation for 168 residents.Together, Walk and Streetcar are the preferred transportation for 55 residents.Bus is the preferred transportation for 45 residents.Bicycle is the preferred transportation for 50 residents.Question 2(Multiple Choice Worth 2 points)(Appropriate Measures MC)The box plot represents the number of tickets sold for a school dance.A horizontal line labeled Number of Tickets sold that starts at 8, with tick marks every one unit up to 30. The graph is titled Tickets Sold for A Dance. The box extends from 17 to 21 on the numb

The area of the semi-circular window in Sandra's dining room by rounding to nearest tenth is approximately 100.5 square inches.

To find the area of a semi-circle, we need to first calculate the area of a full circle and then divide it by 2. The formula for the area of a circle is

A = π * r²,  where A is the area and r is the radius.

Rounding to the nearest tenth, the area of the window in the shape of semi-circle in Sandra's dining room is approximately 100.5 square inches.

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Step-by-step explanation:

area= base*height (8+10)*16=288

perimeter=2L+2B

to find L we will use Pythagorean theorem(check attachment for the solving to find L)

L=17.8

perimeter= 2(17.8)+2(18)

=71.77

=71.8

The survey results show that the estimates for the number of students in the school who prefer rap music range from 110 to 154. The center of the estimates is around 121, with a variation of about 66.

To estimate the number of students who prefer rap music, multiply the proportion who prefer rap in each sample by the total number of students in the school (440)

Sample 1: 0.25 x 440 = 110

Sample 2: 0.20 x 440 = 88

Sample 3: 0.30 x 440 = 132

Sample 4: 0.35 x 440 = 154

The center of the estimates is the average of the four estimates: (110 + 88 + 132 + 154) / 4 = 121. Variability can be described using the range (154 - 88 = 66) or the standard deviation of the estimates, which requires additional calculations.

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Driver's data set that has a mode that is greater than the mean or median is Fabian and a median with the lowest value of the three measures is Kadisha.

Data of Fabian: 7, 12, 11, 23, 13, 23, 30

Mean = sum of all observation / total no. of observation

Mean = (7+ 12+ 11+ 23+ 13+ 23+ 30) / 7

Mean = 17

Mode = most repeating observation

Mode = 23

For median we have to write observation in ascending order

7,11,12,13,23,23,30

Median = (N+1)/2

Where N = No. of observation

Median = (7+1)/2

Median = 4th observation

Median = 13

Here mode that is greater than the mean or median.

similarly for,

Kadisha: 8, 17, 23, 16, 17, 18, 125

Mean = 32

Median = 17

Mode = 17

Here median with the lowest value of the three measures.

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The derivative of f(x) is -3/4.

To find the derivative, use the definition of a derivative:

f'(x) = lim h→0 [f(x + h) - f(x)] / h

Substitute f(x) = √(4x + 3) into this definition:

f'(x) = lim h→0 [√(4(x + h) + 3) - √(4x + 3)] / h

Multiplying by the conjugate of the numerator:

f'(x) = lim h→0 [(√(4(x + h) + 3) - √(4x + 3)) * (√(4(x + h) + 3) + √(4x + 3))] / [h * (√(4(x + h) + 3) + √(4x + 3))]

Expanding the numerator, we get:

f'(x) = lim h→0 [(4(x + h) + 3) - (4x + 3)] / [h * (√(4(x + h) + 3) + √(4x + 3)) * (√(4(x + h) + 3) + √(4x + 3)))]

f'(x) = lim h→0 [4h] / [h * (√(4(x + h) + 3) + √(4x + 3)))]

Canceling out the h terms, we get:

f'(x) = lim h→0 4 / (√(4(x + h) + 3) + √(4x + 3)))

Now, we can evaluate the limit as h approaches 0:

f'(x) = 4 / (√(4x + 3) + √(4x + 3))

f'(x) = 4 / (2√(4x + 3))

f'(x) = 2 / √(4x + 3)

Therefore, the derivative of f(x) is -3/4.

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The equation that could be used to calculate the average speed of each vehicle is C. 140x = 95x + 40, where x represents the average speed of the helicopter in miles per hour and 140x represents the distance traveled by the small private airplane.

The equation that can be used to calculate the average speed of each vehicle is:

C. 140x = 95x + 40

Let's break it down:

Since the time taken by both vehicles is the same, the distances covered by each vehicle can be equated, giving us the equation 140x = 95x + 40.

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In statistical analysis, the mean absolute deviation (MAD) is a measure of the average distance between each data point and the mean of the data set. For Doctor A's data set on corrective lenses, the MAD is calculated as 0.42, while for Doctor B's data set, it is calculated as 0.38.

This shows that Doctor B's data set has a slightly smaller variation compared to Doctor A's data set.

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