Evaluate the expression.

|-2| – |-1|

Answer: Permutations that preserve distances are also known as isometries or distance-preserving transformations.

Step-by-step explanation:

Permutations that preserve distances refer to a type of mathematical transformation that preserves the distances between pairs of points in a geometric space. In other words, if you have a set of points arranged in a particular way and you apply a permutation that preserves distances, the resulting arrangement of points will have the same distances between each pair of points as the original arrangement. This type of permutation is important in geometry and can be used to study properties of geometric objects such as polyhedra, graphs, and other structures. Permutations that preserve distances are also known as isometries or distance-preserving transformations.

Answer:

find how many inches are in one square yard. if it's more than 296, then it doesn't cover it. If it's less, then it does.

Step-by-step explanation:

Answer:

The cyclist has travelled a distance of 931.888 meters.

Matrix B and AB both are invertible implies that A is invertible as a matrix C such that AC = CA = I.

For matrix A to be invertible,

Show that there exists a matrix C such that AC = CA = I,

where I is the identity matrix.

Since B and AB are both invertible,

There exist matrices D and E such that ,

BD = DB = I

And ABE = EAB = I.

Multiplying both sides of the equation ABE = I by D on the left and E on the right, we get,

ADEBE = DE

Since BD = I, we can simplify this to,

ADE = DE

Multiplying both sides of this equation by B on the left and B^(-1) on the right, we get,

AD = DB^(-1)

Now, let C = DB^(-1).

Then we have,

AC

= ADB^(-1)

= ABEB^(-1)

= AI

= A

and

CA

= DB^(-1)A

= DB^(-1)ABE

= DI

= I

Therefore, A is invertible as a matrix C such that AC = CA = I.

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For the equations 2xy - y^2 = 1 and xy + x + y = x^2y^2 using implicit differentiation the value Dxy is given by Dxy = (1 - 2xy + 3y^2)/(x - y)^3 and Dxy = (2y^2 - 2xy - 3y - 1)/(x - 2xy + 1)^3  respectively.

Equation 2xy - y^2 = 1,

Differentiate both sides of the equation with respect to x,

Treating y as function of x and then differentiate again with respect to x.

Using implicit differentiation,

First, differentiate both sides with respect to x,

2y + 2xy' - 2yy' = 0

Next, solve for y',

⇒2xy' - 2yy' = -2y

⇒y' (2x - 2y) = -2y

⇒y' = -y/(x - y)

Now, differentiate again with respect to x,

y''(x - y) - y'(2x - 2y) = y/(x - y)^2

Substitute the expression we obtained for y' in terms of y and x,

y''(x - y) - (-y/(x - y))(2x - 2y) = y/(x - y)^2

Simplify and solve for y'',

y''(x - y) + (2xy - 3y^2)/(x - y)^2 = 1/(x - y)^2

The expression for Dxy is,

Dxy = (1 - 2xy + 3y^2)/(x - y)^3

For the equation xy + x + y = x^2y^2,

Differentiate both sides of the equation with respect to x,

Using implicit differentiation,

First, differentiate both sides with respect to x,

⇒y + xy' + 1 + y' = 2xyy'

Solve for y',

⇒xy' - 2xyy' + y' = -y - 1

⇒y' (x - 2xy + 1) = -y - 1

⇒y' = -(y + 1)/(x - 2xy + 1)

Now, differentiate again with respect to x,

y''(x - 2xy + 1) - y'(2y - 2x y' + 1) = (y + 1)/(x - 2xy + 1)^2

Substitute the expression we obtained for y' in terms of y and x,

y''(x - 2xy + 1) - (-y - 1)/(x - 2xy + 1)^2 (2y - 2x y' + 1) = (y + 1)/(x - 2xy + 1)^2

Simplify and solve for y''

y''(x - 2xy + 1) - (2y^2 - 2xy - 2y)/(x - 2xy + 1)^2 = (y + 1)/(x - 2xy + 1)^2

The expression for Dxy is,

Dxy = (2y^2 - 2xy - 3y - 1)/(x - 2xy + 1)^3

Therefore , the value of Dxy using implicit differentiation for two different functions is equal to

Dxy = (1 - 2xy + 3y^2)/(x - y)^3 and Dxy = (2y^2 - 2xy - 3y - 1)/(x - 2xy + 1)^3

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On solving the question we can say that so the other side of triangle is [tex]B = \sqrt324[/tex], therefore the angle will be [tex]cos^{-1} (0.38)[/tex].

A triangle is a closed two-dimensional geometric object consisting of three line segments, called edges, that intersect at three places called vertices. Triangles are distinguished by their sides and angles. A triangle can be equilateral (all sides equal), isosceles, or odd, depending on the sides. Triangles are classified as acute (any angle less than 90 degrees), right (angles equal to 90 degrees), or obtuse (any angle greater than 90 degrees). The area of ​​a triangle can be calculated using the formula A =​​ (1/2)bh. where A is the area, b is the base of the triangle, and h is the height of the triangle.  

here two sides of the triangle are given that are 19.5 and 7.5

so by

[tex]A^2 = B^2 + C^2\\B^2 = 19.5^2 - 7.5^2\\B^2 = 380.25 - 56.25\\B^2 = 324\\B = \sqrt324[/tex]

so the other side is [tex]B = \sqrt324[/tex], therefore the angle will be [tex]cos^{-1} (0.38)[/tex].

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On Thursday, 12 students brought their lunch at school.

Using a number out of 100, a percentage is a technique to indicate a fraction or piece of a total. The word "percent" means "per hundred."

If 4% of the students bring their lunch to school every day, we can find the number of students who brought their lunch on Thursday by multiplying the total number of students by the percentage that brought their lunch:

Number of students who brought their lunch = (4/100) x 300

Number of students who brought their lunch = 12

Therefore, On Thursday, 12 students brought their lunch at school.

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The objective function that must be minimized is:

z = 40x + 50y

subject to the constraints:

0x + 4y ≥ 4 (protein constraint)

3x + 2y ≥ 11 (carbohydrate constraint)

2x + 4y ≤ 16 (fat constraint)

We want to find the number of ounces of fruit (x) and nuts (y) that will meet the requirement with the least number of calories.

Solving the system of inequalities, we get:

x = 2 ounces

y = 2 ounces

Therefore, the dietician should use 2 ounces of fruit and 2 ounces of nuts. These amounts will have a total of 180 calories (402 + 502).

Answer:

Step-by-step explanation:

Let's assume we need x ounces of fruit and y ounces of nuts to meet the requirements with the least number of calories. Then, the problem can be expressed as an optimization problem:

Minimize: 40x + 50y (since we want to minimize the number of calories) Subject to:

0x + 4y ≥ 4 (we need at least 4 units of protein)3x + 2y ≥ 11 (we need at least 11 units of carbohydrates)2x + 4y ≤ 16 (we cannot have more than 16 units of fat)

To solve this problem, we can use the simplex method. First, we convert the problem to standard form by introducing slack variables:

Minimize: 40x + 50y Subject to:

0x + 4y + s1 = 43x + 2y + s2 = 112x + 4y + s3 = 16

Now we can create the initial simplex tableau:

xys1s2s3RHSs1041004s23201011s32400116z-40-500000

We want to find the minimum value of z, so we need to choose the variable with the most negative coefficient in the bottom row as the entering variable. In this case, that is y. We then choose the variable with the smallest non-negative ratio between the right-hand side and the coefficient of the entering variable in its row as the leaving variable. In this case, that is s3, since 16/4 = 4 is the smallest non-negative ratio.

We then perform the pivot operation to eliminate the coefficient of y in the other rows:

     x  y  s1s2s3RHSs1001-214y3/2101/2-1/24s2-1001-1/25z-100025-15200

We repeat this process until all the coefficients in the bottom row are non-negative. The final tableau is:

x

In this case, we'll have to carry out several steps to find the solution.

Step 1:

Data:

camera:

list price (before tax) = $459.99

sales tax = 7.25%

Step 2:

percentage:

[tex]sales \ tax = 7.25\% = 7/100 = 0.07[/tex]

[tex]total \ cost = \$459.99 + \$459.99 \times (0.07) = \$459.99 + \$32.1993 = \$492.1893[/tex]

The answer is:

$492.19

Here is a step-by-step explanation for your problem:

Step 1: Calculate the amount of the first deposit after one year

First deposit: $20,839

Interest earned on first deposit: (20,839 x 10.91%) = $2,269.82

Total amount after one year: 20,839 + 2,269.82 = $23,108.82

Step 2: Calculate the amount of the second deposit after one year

Second deposit: $22,872

Interest earned on second deposit: (22,872 x 10.91%) = $2,511.33

Total amount after one year: 22,872 + 2,511.33 = $25,383.33

Step 3: Calculate the amount of the final deposit after one year

Final deposit: $20,217

Interest earned on final deposit: (20,217 x 10.91%) = $2,214.93

Total amount after one year: 20,217 + 2,214.93 = $22,432.93

Step 4: Calculate the total amount available after four years

Total amount available after four years = 23, 108.82 + 25,383.33 + 22,432.93 = $71,925.08

Answer:

32 units square

Step-by-step explanation:

Area of trapezoid = 1/2 x h x (a +b) {a and b are parallel sides}

1st parallel side = 2 + 6 + 2

                          = 10 units

2nd parallel side = 6 units

height = 4 units

Area = 1/2 x 4 x (10+6)

         = 2x16

         = 32 units square

The average rate of change is 5π; for r changing from 2 to 2.5, it is 2.5π, and for r changing from 2 to 2.1, it is 4.1π.

The area of a circle is given by the formula A = πr². To find the average rate of change of A with respect to r, we can take the derivative of A with respect to r:

dA/dr = 2πr

This tells us how much the area changes for a small change in the radius. To find the average rate of change over a larger interval, we can use the formula:

ΔA/Δr = (A2 - A1)/(r2 - r1)

where A1 and A2 are the areas at the initial and final radii, and r1 and r2 are the initial and final radii.

(i) For r changing from 2 to 3:

ΔA/Δr = (π(3)² - π(2)²)/(3 - 2) = 5π

The average rate of change of the area with respect to the radius is 5π.

(ii) For r changing from 2 to 2.5:

ΔA/Δr = (π(2.5)² - π(2)²/(2.5 - 2) = 2.5π

The average rate of change of the area with respect to the radius is 2.5π.

(iii) For r changing from 2 to 2.1:

ΔA/Δr = (π(2.1)² - π(2)²)/(2.1 - 2) = 4.1π

The average rate of change of the area with respect to the radius is 4.1π.

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The value of the triple integral[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex] by using spherical coordinates [tex]2\pi(e^{-1}-e^{-9})[/tex].

Given that the triple integral is-

[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex]

E is the region bounded by the spheres which are,

[tex]x^2+y^2+z^2=1\\\\x^2+y^2+z^2=9[/tex]

In spherical coordinates we have,

x = r cosθ sin ∅

y = r sinθ sin∅

z = r cos∅

dV = r²sin∅ dr dθ d∅

E contains two spheres of radius 1 and 3 () respectively, the bounds will be like this,

1 ≤ r ≤ 3

0 ≤ θ ≤ 2π

0 ≤ ∅ ≤ π

Then

[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex]

[tex]\int\int\int _{E} \frac{e^{-r^2}}{r}r^2Sin\phi drd\phi d\theta\\\\2\pi \int_{0}^{\pi} \int_1^3 re^{-r^2} dr d\phi\\\\2\pi \int_1^3 re^{-r^2} dr\\\\2\pi(e^{-1}-e^{-9})[/tex]

The complete question is-

Use spherical coordinates to evaluate the triple integral ∭ee−(x2 y2 z2)x2 y2 z2−−−−−−−−−−√dv, where e is the region bounded by the spheres x2 y2 z2=1 and x2 y2 z2=9.

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[tex]53 - 3^2 * 3 = 35[/tex]

The ice cream scoop will fit inside the cone without overflowing, as shown by the volume comparison, which reveals that V ice cream > V cone.

A cone is a smooth-tapering, three-dimensional geometric shape with a flat base and a pointed tip or vertex. A cone is made up of a collection of line segments, half-lines, or lines that link the apex—the common point—to every point on a base that is in a plane other than the apex. The base can be any shape, but is most often a circle. Cones are frequently used in science and mathematics, as well as in commonplace items like ice cream cones, party hats, and traffic cones.

given

We need to compare their volumes to see if the ice cream scoop will fit inside the cone or spill out.

The quantity of the ice cream scoop could be determined by applying the following formula for the volume of a sphere:

[tex]V ice cream = (4/3)\pi r^3 \\= (4/3)\pi (3 cm)^3 \\= 113.1 cm^3[/tex]

[tex]V cone = (1/3)\pi r^2h \\= (1/3)\pi (3 cm)^2(13 cm) \\= 122.7 cm^3[/tex]

The ice cream scoop will fit inside the cone without overflowing, as shown by the volume comparison, which reveals that V ice cream > V cone.

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1. The diagonals are equal. Rectangle

2. All sides are equal, and one angle is 60°. Rhombus

3. All sides are equal, and one angle is 90°. Square

4. It has all the properties of parallelogram, rectangle, and rhombus. Square

5. It is an equilateral parallelogram. Rhombus

A rectangle is a four-sided figure with two sets of parallel sides, with each side being a different length. The opposite sides of a rectangle are always equal in length, so the angles of a rectangle are all 90 degrees. A rectangle can also be referred to as a quadrilateral.

A rhombus is a four-sided figure with all sides the same length. The angles of a rhombus are not all 90 degrees, but the opposite sides of a rhombus are equal in length. A rhombus can also be referred to as a diamond.

A square is a four-sided figure with all sides being the same length and all angles being 90 degrees. A square can also be referred to as a regular quadrilateral.

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The complete question is:

Identify whether the following statements describe a rectangle, rhombus or square.

1. The diagonals are equal. ____________

2. All sides are equal, and one angle is 60°. ____________

3. All sides are equal, and one angle is 90°. ____________

4. It has all the properties of parallelogram, rectangle, and rhombus. ____________

5. It is an equilateral parallelogram. ____________

Answer:

Step-by-step explanation:

5m(4m) = 20m^2

The number of fractions between 0 and 1 (inclusive) with a denominator of 15 can be found using the formula (n-1)/n, where n is the denominator.

So, to answer your question, we can use the formula and plug in 15 for the value of n:

(15-1)/15 = 14/15

Therefore, there are 14 fractions between 0 and 1 (inclusive) with a denominator of 15.

Answer:

it's 3.9

Step-by-step explanation:

Assume Samir has total 100 songs and use combined mean formula

Answer:

Step-by-step explanation:

To find the greatest profit, we need to determine the fare that will maximize revenue, while also considering the decrease in ridership due to the fare increase.

Let's assume the initial fare is $2, and the number of passengers is 800 per day. So, the initial revenue is:

$2 x 800 = $1600 per day

Now, let's say we increase the fare by 4 cents to $2.04. According to the problem, for each 4 cents increase in fare, there will be 5 fewer passengers. So, the number of passengers will decrease to:

800 - (5 x 4) = 780 passengers per day

The new revenue at this fare will be:

$2.04 x 780 = $1591.20 per day

By increasing the fare, the revenue decreased. This means that we may have increased the fare too much. Let's try another fare.

If we increase the fare by 2 cents to $2.02, the number of passengers will decrease by:

800 - (5 x 2) = 790 passengers per day

The new revenue at this fare will be:

$2.02 x 790 = $1595.80 per day

This is more revenue than the initial fare of $2 per person. Let's continue this process:

If we increase the fare by another 2 cents to $2.04, the number of passengers will decrease by:

790 - (5 x 2) = 780 passengers per day

The new revenue at this fare will be:

$2.04 x 780 = $1591.20 per day

This is less revenue than the $2.02 fare, so we can stop here.

Therefore, the greatest profit can be earned by charging $2.02 per person per day, and the maximum revenue will be:

$2.02 x 790 = $1595.80 per day

This is a bit less than the initial daily revenue of $1600, but it is the most revenue we can get by increasing the fare without causing a significant reduction in ridership.

Answer:

  $2205

Step-by-step explanation:

You want the greatest profit that can be earned by a commuter railway that has 800 passengers per day at a fare of $2, and 5 fewer for each 4¢ increase in the fare.

The number of riders (q) as a function of price (p) can be described by ...

  q = 800 -5(p -2)/0.04

  q = 1050 -125p . . . . . . . simplified

The daily revenue is the product of price and the number of riders who pay that price.

  r = pq

  r = p(1050 -125p)

  r = 125p(8.40 -p)

This function describes a parabola that opens downward. It has zeros at p=0 and p=8.40. The vertex of the parabola is on the line of symmetry, halfway between the zeros:

  pmax = (0 +8.40)/2 = 4.20

The maximum revenue is ...

  r(4.20) = 125·4.20(8.40 -4.20) = 125(4.20²) = 2205

The maximum revenue that can be earned is $2205.

__

Additional comment

The ridership at that fare is 125(4.20) = 525.

Profit is the difference between revenue and cost. Here, we have no information about the cost function, so we cannot predict the maximum profit. The question seems to assume that profit is equal to revenue.

The appropriate inference from the data is (B) Since [tex]20%[/tex] of this group read below grade level, we cannot draw any generalizations about the population. Thus, option B is correct.

A representative sample is a subset of data, often drawn from a wider population, that can show qualities that are similar.

Because the data produced contains more manageable, smaller representations of the larger group, representative sampling aids in the analysis of bigger groups.

Although the sample may be representative of seventh-grade American students, it is not necessarily representative of all seventh-graders or all American children. Hence, without additional data or research, we cannot extrapolate the sample's results to the overall population.

Therefore, 20%20, percent of all American students in seventh grade were reading below grade level.

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The correct option on Hash 1 and Hash 2 is C. Hash 1 is designed to work with small input data, while Hash 2 is optimized for processing large input data.

The main difference between Hash 1 and Hash 2 is the amount of input data they can process. Hash 1 can process input data that is 2 characters long, while Hash 2 can handle input data that is up to 300,000 characters long. This means that Hash 2 is better suited for processing larger datasets than Hash 1.

In terms of which hash function is better suited for different types of data, it depends on the specific application and the characteristics of the data being processed. For example, if the data being hashed is small and of low complexity, Hash 1 may be a good choice due to its speed and simplicity.

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The options for this question include:

The area of this triangle is 30 m².

Area is a surface measure, that is, it is the amount of space that a geometric figure occupies on a flat surface.

To calculate the area of a triangle, we can use the formula:

Area = (base x height) / 2

In the case of the given triangle, we can choose the measure of 5m as the base and the measure of 12m as the height, since the height forms a right angle with the base and is perpendicular to it.

So, we have:

Area = (b*h)/2

Area = (5m * 12m) / 2

Area = 30m²

The Euler's method was used to approximate the solution of two initial value problems at various time intervals with different step sizes. For problem, the solution is decreasing and converges to 1.

We will use the following formula for Euler's method:

y_{n+1} = y_n + h*f(t_n, y_n)

y' = 5 – 3sqrt(y), y(0) = 2

Using h = 0.1, we get:

t=0, y=2

t=0.1, y=1.738

t=0.2, y=1.508

t=0.3, y=1.303

t=0.4, y=1.119

t=0.5, y=0.953

Using h = 0.05, we get:

t=0, y=2

t=0.05, y=1.837

t=0.1, y=1.695

t=0.15, y=1.568

t=0.2, y=1.452

t=0.25, y=1.346

t=0.3, y=1.248

t=0.35, y=1.158

t=0.4, y=1.076

t=0.45, y=0.999

t=0.5, y=0.929

Using h = 0.025, we get:

t=0, y=2

t=0.025, y=1.861

t=0.05, y=1.737

t=0.075, y=1.622

t=0.1, y=1.516

t=0.125, y=1.418

t=0.15, y=1.328

t=0.175, y=1.246

t=0.2, y=1.17

t=0.225, y=1.101

t=0.25, y=1.038

t=0.275, y=0.98

t=0.3, y=0.927

t=0.325, y=0.878

t=0.35, y=0.833

t=0.375, y=0.791

t=0.4, y=0.753

t=0.425, y=0.718

t=0.45, y=0.685

t=0.475, y=0.655

t=0.5, y=0.627

Using h = 0.01, we get:

t=0, y=2

t=0.01, y=1.88

t=0.02, y=1.764

t=0.03, y=1.652

t=0.04, y=1.544

t=0.05, y=1.44

t=0.06, y=1.34

t=0.07, y=1.244

t=0.08, y=1.151

t=0.09, y=1.062

t=0.1, y=0.976

t=0.11, y=0.893

t=0.12, y=0.813

t=0.13, y=0.736

t=0.14, y=0.662

t=0.15, y=0.591

t=0.16, y=0.523

t=0.17, y=0.458

t=0.18, y=0

So, we can say that the step size had to be decreased to achieve more accurate approximations

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_____The given question is incomplete, the complete question is given below:

In each of Problems 9 and 10, use Euler's method to find approximate values of the solution of the given initial value problem at t = 0.5, 1, 1.5, 2, 2.5, and 3: a. With h=0.1. b. With h = 0.05. c. With h= 0.025. d. With h=0.01.

9. y' = 5 – 3 sqrt y,      y(0) = 2

(a) Therefore after t = 1.75 seconds the diver will hit the water.

(b) The velocity the diver hit the water is 56 ft/s.

From the given condition we have d(t) = 16t²

and the height is 49ft

(a) Now when the diver hit the water the equation become

16t² = 49

t² = 49/16

t = ±7/4

t = ±1.75

since time can not be negative so t = 1.75

Therefore after t = 1.75 seconds the diver will hit the water.

(b)

Now differentiating d(t) with respect to t we get

d'(t) = 32t

now putting t=7/4 we get

the velocity d'(7/4) = 32*7/4

d'(7/4) = 56ft/s

Therefore the velocity the diver hit the water is 56 ft/s.

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The complete question is :

A cliff diver plunges from a height of 49ft above the water surface. The distance the diver falls in t seconds is given by the function d(t)=16t²ft

(a) After how many seconds will the diver hit the water?

(b) With what velocity (in ft/s ) does the diver hit the water?

The probability mass function (PMF) for z, the number of ones at the channel output, can be expressed using the binomial distribution where p is the probability of transmitting a one and n is the total number of bits transmitted.

A probability mass function (PMF) is a function that assigns probabilities to each possible outcome in a discrete probability distribution. It describes the probability distribution of a discrete random variable, which takes on a finite or countably infinite number of possible values. The PMF is defined as the probability of each possible outcome, with the sum of all probabilities equal to 1. It is typically denoted as P(X = x), where X is the random variable and x is a possible value that it can take. The PMF is used to calculate various properties of the probability distribution, such as the expected value, variance, and higher moments.

The probability mass function (PMF) for z, the number of ones at the channel output, can be expressed using the binomial distribution formula:

[tex]$p(z) = \binom{n}{z} p^z (1-p)^{n-z}$[/tex]

where p is the probability of transmitting a one, n is the total number of bits transmitted, and [tex]$\binom{n}{z}$[/tex] is the binomial coefficient which counts the number of ways to choose z ones from n bits. The PMF specifies the probability of observing each possible value of z, ranging from 0 to n.

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Answer:

Total cost for groceries = ($3.75, $1.09,

$4.50, $3.25, $2.58, $4.71, $5.19, $0.89, and $5.34. add them all). = $ 31.3

the amount she paid= $ 35

balance =$ 3.7

therefore she have enough money

Answer:

True

Step-by-step explanation:

A parallelogram has four sides so it's a quadrilateral

Answer: 314 square meters

Step-by-step explanation:

The formula for the area of a circle is given by A = πr^2, where r is the radius of the circle. Since the diameter of the circular patch is given as 20 meters, the radius would be half of that or 10 meters.

So, using the formula, we can calculate the area of the circular patch as follows:

A = πr^2

A = π(10)^2

A = 3.14(100)

A = 314 square meters

Therefore, the area of the circular patch is 314 square meters.

so as we speak, the subfunctions are discontinued, the 1st goes close to 2 and who knows what happens it goes somewheres, the 2nd one makes it to 2.

we know that since the 2nd one makes to 2, to x = 2 that is, well, f(2) = kx, well, let's make f(2) for the 2nd one be equal to the 1st one then, if both they equate each other, that's where they meet, at x = 2.

[tex]f(x)= \begin{cases} k^2-24x,&x > 2\\\\ kx,&x\leqslant 2 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ k^2-24x~~ = ~~kx\hspace{5em}\stackrel{\textit{now let's go to f(2)}}{k^2 - 24(2)~~ = ~~k(2)}\implies k^2-48=2k \\\\\\ k^2-2k-48=0\implies (k-8)(k+6)=0\implies \boxed{k= \begin{cases} 8\\ -6 \end{cases}}[/tex]