1) Consider the second-order linear DE (1- tcot t) y - ty y 0 for 0 < t < T- Let Y1 (t)-t and y2 (t) sin t. a) Are y1 and y2 both solutions to this DE? b) Are yi and y2 linearly independent? If so, then find the general solution of the DE. If not, then find constants A and B, not both zero, such that Ay? By 0. 2) Suppose that is a constant and {31 , y2} is a set of solutions to the DE Find the Wronskian W(x) of {31 , y, given that W(0) = 1.

The line of best fit is given as follows:

y = 0.25x + 72.25.

The correct option regarding the slope is given as follows:

The slope of the line of best fit is 0.25. Each time the shoe size increases by one, the height increases by 0.25 inches.

To find the regression equation, also called line of best fit or least squares regression equation, we need to insert the points (x,y) in the calculator.

The points for this problem are given as follows:

(7.5, 63), (8, 80.5), (12, 68), (7, 71), (9, 69.5), (10, 70), (12, 75), (12.5, 63), (14,72)

Inserting these points into a calculator, the equation is given as follows:

y = 0.25x + 72.25.

The slope of 0.25 represents the rate of change of the output variable(height) relative to the input variable(shoe size).

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if cot(θ)<0 and cos(θ)<0, then we can conclude that θ lies in the second quadrant.

What is trigonometry functions ?

Trigonometry functions are a set of mathematical functions that relate the angles of a triangle to the lengths of its sides. The three main trigonometric functions are sine, cosine, and tangent, which are commonly abbreviated as sin, cos, and tan. These functions can be defined as ratios of the sides of a right triangle.

In a right triangle, the side opposite an acute angle is called the "opposite" side, the side adjacent to the angle is called the "adjacent" side, and the hypotenuse is the side opposite the right angle. With this terminology, we can define the following trigonometric functions:

Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In other words, sin(θ) = opposite/hypotenuse.

Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In other words, cos(θ) = adjacent/hypotenuse.

Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In other words, tan(θ) = opposite/adjacent.

There are also three reciprocal trigonometric functions: cosecant (csc), secant (sec), and cotangent (cot). These functions are the reciprocals of sine, cosine, and tangent, respectively.

Trigonometric functions have many practical applications in fields such as engineering, physics, and astronomy, where they are used to solve problems involving angles and distances. They also have many applications in pure mathematics, such as in the study of periodic functions and Fourier analysis.

According to given information :

To understand why θ must be in the second quadrant, we need to recall the signs of the trigonometric functions in each quadrant.

In the first quadrant, all trigonometric functions are positive.

In the second quadrant, sine is positive and cosine is negative.

In the third quadrant, both sine and cosine are negative.

In the fourth quadrant, sine is negative and cosine is positive.

Since we know that cos(θ) is negative, we can conclude that θ lies in either the second or third quadrant. However, since cot(θ) = cos(θ)/sin(θ) and cot(θ) is negative, we know that cos(θ) and sin(θ) must have opposite signs. In the third quadrant, both cos(θ) and sin(θ) are negative, so their quotient cot(θ) would be positive, which is not the case.

Therefore, we can conclude that θ lies in the second quadrant.

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Answer:

If  cot(θ)<0 and cos(θ)<0, then we can conclude that θ lies in the second quadrant.

What are trigonometry functions ?

Trigonometry functions are a set of mathematical functions that relate the angles of a triangle to the lengths of its sides. The three main trigonometric functions are sine, cosine, and tangent, which are commonly abbreviated as sin, cos, and tan. These functions can be defined as ratios of the sides of a right triangle.

In a right triangle, the side opposite an acute angle is called the "opposite" side, the side adjacent to the angle is called the "adjacent" side, and the hypotenuse is the side opposite the right angle. With this terminology, we can define the following trigonometric functions:

Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In other words, sin(θ) = opposite/hypotenuse.

Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In other words, cos(θ) = adjacent/hypotenuse.

Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In other words, tan(θ) = opposite/adjacent.

There are also three reciprocal trigonometric functions: cosecant (csc), secant (sec), and cotangent (cot). These functions are the reciprocals of sine, cosine, and tangent, respectively.

Trigonometric functions have many practical applications in fields such as engineering, physics, and astronomy, where they are used to solve problems involving angles and distances. They also have many applications in pure mathematics, such as in the study of periodic functions and Fourier analysis.

According to given information :

To understand why θ must be in the second quadrant, we need to recall the signs of the trigonometric functions in each quadrant.

In the first quadrant, all trigonometric functions are positive.

In the second quadrant, sine is positive and cosine is negative.

In the third quadrant, both sine and cosine are negative.

In the fourth quadrant, sine is negative and cosine is positive.

Since we know that cos(θ) is negative, we can conclude that θ lies in either the second or third quadrant. However, since cot(θ) = cos(θ)/sin(θ) and cot(θ) is negative, we know that cos(θ) and sin(θ) must have opposite signs. In the third quadrant, both cos(θ) and sin(θ) are negative, so their quotient cot(θ) would be positive, which is not the case.

Therefore, we can conclude that θ lies in the second quadrant.

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Answer:

We have to find co-ordinates of the fourth vertex.

Let fourth vertex be D(x,y)

Since, ABCD is a parallelogram, the diagonals bisect each other.

Si, co-ordinate of mid-point of AC= Co-ordinate of mid-point of BD.

∴ The fourth vertex is D(−b,b)

Step-by-step explanation:

The value of the composition function ( f + g ) ( x ) = x²³ - 3x + 2

Evaluation of a function at the value of another function is known as Composition of function. A function composition is a process in which two functions, f and g, form a new function, h, in such a way that h(x) = g(f(x)). This signifies that function g is being applied to the function x. So, in essence, a function is applied to the output of another function.

Given data ,

Let the first function be represented as f ( x )

Now , the value of f ( x ) = -3x + 2

Let the second function be represented as g ( x )

Now , the value of g ( x ) = x²³

On simplifying , we get

The value of ( f + g ) ( x ) = f ( x ) + g ( x )

Substituting the values of f ( x ) and g ( x ) , we get

( f + g ) ( x ) = x²³ - 3x + 2

Hence , the composition function is ( f + g ) ( x ) = x²³ - 3x + 2

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Answer:

<1=26

<2=154

<3=26

<4=26

<5=154

<6=154

<7=26

The area of the figure can be found to be 53 cm ²

The figure given is a composite shape so to find the area of the figure, you need to divide the shape into two other shapes.

The area of the first shape is :

= Length x Width

= 5 x 4

= 20 m ²

The area of the second shape would be :

= Length x Width

= ( 3 + 5 + 3 ) x 3

= 11 x 3

= 33 cm ²

The area of the shape is :

= 20 + 33

= 53 cm ²

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Answer: Here you go.

Step-by-step explanation:

The worker makes 4 ornaments every 15 minutes, which is equivalent to 16 ornaments per hour (since 60 minutes / 15 minutes = 4, and 4 x 4 = 16).

Therefore, the worker can make 48 ornaments in 3 hours (since 3 hours x 16 ornaments per hour = 48 ornaments).

If he is paid $1.15 for each ornament he makes, then he earns $55.20 for making 48 ornaments (since 48 ornaments x $1.15 per ornament = $55.20).

Therefore, if the worker works for 3 hours making ornaments, he would earn $55.20.

Answer: If each drink is 8.64 fluid ounces and Gwen is pouring 48 drinks, then the total amount of fluid ounces she poured is 8.64 fluid ounces * 48 drinks = 413.12 fluid ounces.

Step-by-step explanation:

The range of flight time is 22 minutes, and the sample standard deviation is 8.24 minutes.

To compute the range, we simply subtract the smallest value from the largest value: Range = 282 - 260 = 22.

To compute the sample standard deviation, we first calculate the mean flight time by summing the values and dividing by the sample size:

Mean = (282 + 270 + 260 + 266 + 260 + 267) / 6 = 266.17

Then we subtract the mean from each flight time, square the differences, sum the squared differences, divide by the sample size minus one, and take the square root of the result:

Standard Deviation = √((1/5) * [(282-266.17)² + (270-266.17)² +

(260-266.17)² + (266-266.17)² + (260-266.17)² + (267-266.17)²]) = 8.24.

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Complete Question:

The following data represent the flight time (in minutes) of a random sample of six flights from Las Vegas, Nevada, to Newark, New Jersey, on United Airlines. 282, 270, 260, 266, 260, 267 Compute the range and sample standard deviation of flight time. The range of flight time is minutes.

The Price for one slice pizza is $1.25 and for one drink is $2.25.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given:

Jim ordered a sports drink and three slices of pizza for $8.50.

His friend ordered two sports drinks and two slices of pizza for $8.00.

let the price for pizza be $ and for one drink is $y.

So, the system of equation is

x + 3y = 8.5....(1)

2x+ 2y= 8......(2)

Solving above two equations we get

2(8.5- 3y) + 2y= 8

17 - 6y + 2y= 8

-4y = -9

y= 9/4

y= 2.25

and, x= 8.5- 3(2.25)= 8.5- 6.75 = $1.25

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Answer: See attached.

Step-by-step explanation:

   We will input the given x-values and solve for y by substituting x for the values in the original function, h(x)=2x-2.

h(-5) = 2(-5)-2 = -12

h(-1) = 2(-1)-2 = -4

h(1) = 2(1)-2 = 0

h(2) = 2(2)-2 = 2

We can use algebraic equations to solve this kind of problems. The correct answer is 50.00( 0.25p+ 1.00). So option number 3 is correct.

The cost of printing a single page = 0.25

The cost of printing p pages = 0.25p

Cost of print 50 booklets with p pages = 50× 0.25p = 12.50p

Cost of binding 1 booklet = $1

Cost of binding 50 booklet = 50

Cost of  50 booklets = Cost of printing 50 booklets with p pages + cost for binding 50 booklets

   = 12.5p + 50.00 = 50.00(0.25p+1.00)

So the correct answer is the third option , 50.00 ( 0.25p + 1.00)

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Step-by-step explanation:

27cups of flour got to multiple

The lateral area and the surface area of the right prism are 540 square yards and 1152 square yards, respectively.

Herein we find the case of a right prism with a triangular base, whose lateral area (A) and surface area (A'), both in square yards, must be determined.

The lateral area is the sum of the areas of three rectangular faces and the surface area is the sum of the lateral area and the area of the two triangles. Area formulas for rectangles and triangles are shown below:

Rectangle

A = w · l

Triangle

A = 0.5 · w · l

Where:

Now we determine the lateral area and the surface area of the prism:

Lateral area

A = (37 yd) · (5 yd) + (20 yd) · (5 yd) + (51 yd) · (5 yd)

A = 540 yd²

Surface area

A' = 540 yd² + 2 · 0.5 · (51 yd) · (12 yd)

A' = 1152 yd²

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Answer:

makes no sense

Step-by-step explanation:

Answer:5/4

Step-by-step explanation:

2/3k=8/15

then,

2*15=8*3k

30=24k

k=30/24=15/12=5/4

The appropriate null hypothesis for a random sample of car's color and speeding tickets is equals to the observed counts are equal to the expected counts.

The curious newspaper columnist tabulate the car color on a random sample. The above table showed car color and their corresponding speeding tickets. Sample size, n = 120

Registed red colour cars in his state

= 14%

Registed white/silver colour cars in his state= 35%

Registed gray/black colour cars in his state = 23%

Registed others colour cars in his state

= 28%

We habe to determine the right choice for null hypothesis from the options. For this, here we have to use chi square test of goodness of fit to test this hypothesis. We know that in chi square test, they arevalready given the observed frequencies. Then we have to find out the expected frequencies using the given percentages. So here we have to findout whether the expected frequency and observed frequency are equal or not. So the null hypothesis is The observed counts are equal to the expected counts.

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For each variable, the possible values are:

a) T = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

b) X = {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

c) U = {0, 1, 2, 3, 4, 5, 6}

d) Z = {0, 1, 2}

Item a:

In total, there are 10 pumps, hence the possible values are all integers from 0 to 10, that is:

T = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Item b:

In the first, there are 6 pumps, and in the second 4, hence, the difference ranges from -4 to 6, that is:

X = {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

Item c:

The maximum number in a station is 6 pumps, hence this variable ranges from 0 to 6.

U = {0, 1, 2, 3, 4, 5, 6}

Item d:

Either none of the stations has exactly two pumps in use, or one has or both, hence the variable ranges from 0 to 2.

Z = {0, 1, 2}

The question is incomplete. The complete question is:

"The number of pumps in use at both a six-pump station and a four-pump station will be determined. Give the possible values for each of the following random variables. (Enter your answers in set notation.)

a. T = the total number of pumps in use

b. X = the difference between the numbers in use at stations 1 and 2.

c. U = the maximum number of pumps in use at either station

d. Z = the number of stations having exactly two pumps in use"

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The zeroes of the polynomials will be x = (-1 ± 7) / 4.

A polynomial in mathematics is an expression made up of coefficients and indeterminates and involves only the operations of multiplication, addition, subtraction, and non-negative integer exponentiation of variables.

To represent the quadratic polynomial 2x²+ x - 6 using algebra tiles, we can use the following tiles:

Two squares with side length x, representing 2x²

One rectangle with length x and width 1, representing x

Six small squares, representing -6

We can arrange these tiles to form a rectangle as shown below:

A rectangle made of algebra tiles with 2x² squares, x rectangle, and -6 small squares]

The area of this rectangle is the same as the value of the polynomial, which is 2x²+ x - 6.

To determine the equivalent factored form of this polynomial, we need to find two binomials that, when multiplied together, give us the original polynomial. We can start by using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

where a = 2, b = 1, and c = -6, since our polynomial is 2x^2 + x - 6.

Plugging in these values, we get:

x = (-1 ± √(1² - 4(2)(-6))) / 2(2)

x = (-1 ± √(49)) / 4

x = (-1 ± 7) / 4

So, the two roots of the quadratic polynomial are x = 3/2 and x = -2.

Therefore, we can write the factored form of the quadratic polynomial as:

2x² + x - 6 = 2(x - 3/2)(x + 2)

To determine the number of zero pairs needed to model this polynomial, we can count the number of negative tiles (small squares) left after forming the rectangle. In this case, we have six negative tiles, which means we need six zero pairs to model this polynomial.

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Answer:

Step-by-step explanation:

Answer:

$84.00

Step-by-step explanation:

$80.00 x .05 = $4.00

$80.00 + $4.00 = $84.00

Answer:

the cost is $84.00

Step-by-step explanation:

you take 80.00 x 5% tax that gives you $4.00 so the total is $84.00

Answer: A, i think

Step-by-step explanation: edge 2020

The measure of angle x is given as follows:

a. x = 30º.

Angle y is complementary with the angle of 60º, meaning that the sum of their measures is of 90º, hence it's measure is given as follows:

y + 60 = 90

y = 30º.

Angle y and angle x form vertical angles, meaning that they are congruent, that is, they have the same measure, hence the measure of angle x is given as follows:

x = y = 30º.

Hence the correct option is given by option a.

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A basis for R3 that includes vectors in the set is {[1 0 -2], [3 2 -4], [0 0 1]}. This is achieved by removing the linearly dependent vector [1 0 -2] and adding the vector [0 0 1] to the set to obtain a set that spans R3.

A basis for R3 that includes vectors in the set can be found by removing vectors from a linearly dependent set and/or adding vectors to a set that does not span R3.

The set given is {[1 0 -2], [3 2 -4]}.

To check if the set spans R3, we need to calculate the determinant of the matrix created by the two vectors. The determinant of the matrix is 0, which tells us that the set does not span R3. To create a basis for R3, we need to add a third vector that is not linearly dependent on the other two. We can add the vector [0 1 1] to the set, creating the following set:

{[1 0 -2], [3 2 -4], [0 1 1]}

To check if this set spans R3, we calculate the determinant of the matrix created by the three vectors. The determinant of the matrix is -6, which is not 0.

Therefore, the set {[1 0 -2], [3 2 -4], [0 1 1]} is a basis for R3.

The complete question: For the matrix below. Find a basis for R3 that includes vectors in the set by removing vectors from a linearly dependent set and/or adding vectors to a set that does not span R3.

( matrix attached below )

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The capacity of the tank is 7234.56litres

Capacity refers to the amount of fluid a container can hold. It expressed in units like litres (l), millilitres (ml). Capacity is also known as volume of a substance.

The capacity of the cylindrical pipe is

V = πr²h

r = 8cm

if it takes the pipe 2m per second

then it's volume in one second is

V = 3.14 ×8 × 2× 100

V = 5024cm³/s

if it takes 24 minutes

24 minutes = 24 × 60 s

= 1440s

therefore volume of the tank = 1440 × 5024

= 7234560cm³

in litres = 7234560/1000

= 7234.56litres

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The total area of the banner is approximately 800.54 square inches.

A quadrilateral is a four-sided polygon, having four edges and four corners

The area of the square is side length squared, or 24² = 576 square inches.

The diameter of the semicircle is also the side length of the square, so it is 24 inches.

Radius = 24/2=12 inches

The area of a semicircle is half of the area of a circle with the same radius. The area of a circle is π times the radius squared, so the area of the semicircle is:

1/2×π×12²=72π

Add Area of the square and the area of the semicircle:

576 + 72π = 800.54

Therefore, the total area of the banner is approximately 800.54 square inches.

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Answer:

The equation for the parabola (quadratic function) that passes through the points (-1, 1), (1, -5), and (2, 1) is y = (6/49)(x - 2/3)^2 - 1.

Step-by-step explanation:

To write the equation of a quadratic function (parabola) given three points, we can use the vertex form of the equation:

y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola and a is a scaling factor that determines the shape of the parabola.

To find the vertex of the parabola, we can use the formula:

h = (x1 + x2 + x3) / 3

k = (y1 + y2 + y3) / 3

where (x1, y1), (x2, y2), and (x3, y3) are the given points.

Substituting the given points into these formulas, we get:

h = (-1 + 1 + 2) / 3 = 2/3

k = (1 - 5 + 1) / 3 = -1

So the vertex of the parabola is (2/3, -1).

Now, we can substitute this vertex and one of the given points (e.g. (-1, 1)) into the vertex form equation and solve for a:

1 = a(-1 - 2/3)^2 - 1

2 = a(7/3)^2

a = 6/49

So the equation of the parabola is:

y = (6/49)(x - 2/3)^2 - 1

Answer: y = -2x^2 + 6x - 4

Step-by-step explanation:

The equation for this parabola/quadratic function is y = -2x^2 + 6x - 4. To arrive at this equation, we use the fact that the equation for a parabola/quadratic function is y = ax^2 + bx + c, where a, b, and c are constants. We can then substitute in the coordinates of each point to solve for a, b, and c.

For the point (-1,1), we have 1 = -2(-1)^2 + 6(-1) + c, so c = 5.

For the point (1,-5), we have -5 = -2(1)^2 + 6(1) + 5, so -2 = -2.

For the point (2,1), we have 1 = -2(2)^2 + 6(2) + 5, so -4 = -4.

And therefore, the equation is y = -2x^2 + 6x - 4.

ΔABC and ΔDCB would be congruent triangles by the AAS postulate.

In congruent geometry, the shapes that are so identical. can be superimposed on themselves.

Here,
In a square, all sides are equal in length and all angles are right angles (90 degrees). Additionally, the diagonal of a square bisect each other at 90 degrees.

The lengths of AB, BC, CD, and DA are equal.

The angles ABC and CDA are both right angles (90 degrees).

The angles BAC and DCB are both 45 degrees.

So ΔABC and ΔDCB became congruent triangles by the AAS postulate.

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To calculate the property tax rate, you need to divide the property taxes by the appraised value of the property and then multiply by 100 to express the result as a percentage.

In this case, the property tax rate can be calculated as follows:

Property tax rate = (Property taxes / Appraised value) * 100%

= ($1,650.00 / $112,000.00) * 100%

= 0.014732142857142857 * 100%

= 1.47%

So, the property tax rate in your neighborhood is 1.47%, rounded to the nearest hundredth of a percent.

Answer: Since IK and LN are parallel lines, then mIJM and mNMJ are supplementary, meaning that they add up to 180 degrees. Therefore, mIJM = 180° - mNMJ = 180° - 116° = 64°.

Step-by-step explanation: