What is the length of the unknown leg in the right triangle?

In Fluid Mechanics, we describe the motion of fluids using two different frames of reference- the Eulerian( fixed frame of reference wrt fluid particles; describes the rate of change of fluid flow property at a fixed position in space) and the Lagrangian (follows individual fluid particles and describes the rate of change of fluid flow properties of the particle as it passes through space).

A Material derivative is basically a Lagrangian way of expressing the rate of change of fluid flow property. It expresses the rate of change of fluid property wrt time and also the rate of change wrt space because of its movement at some velocity. It is best explained by John D Anderson jr. in his book "Computational Fluid Dynamics". He explains that the material derivative can be best understood by considering an example of imagining oneself entering a cold cave. The material derivative is the change in temperature you would feel as if you would just enter into the cave from the outside and precisely at the same time if someone throws snowballs at you. The rate of change of temperature because of your movement into the cave is the rate of change wrt space and that due to the snowball effect would be the rate of change w.r.t time.

Coming to the Reynolds Transport theorem is a powerful tool in fluid mechanics that relates the change in properties of a system to change in properties in a control volume. Since laws of thermodynamics are applicable to a system, the Reynolds transport theorem allows it to be applied to a control volume (fixed or moving, or deformable).

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The simplification by combining like terms of 5f - (5d - f) is 6f -5d.

According to the given question.

We have an expression 5f - (5d - f).

As we know that, like terms are terms whose variables (and their exponents such as the 2 in x2) are the same. When we combine like terms, such as 2x and 3x, we add or subtract their coefficients.

Since, we have to combine the like terms of the given expression

5f -(5d -f).

Here 5f and -f are the like term.

So, we subtract coeffcients of f to simplify the given expression.

Therefore, the simplification of 5f - (5d -f) is given by

5f - (5d - f)

= 5f - 5d + f

= 6f - 5d

Hence, the simplification by combining like terms of 5f - (5d - f) is 6f -5d.

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Answer:

Y=5

Step-by-step explanation:

1) find gradient

2) find equation of line using the formula y-y1=m(x-x1) where m is the gradient.

The expression 3-14x is equivalent to -5+8-6x-8x.

The phrase has the same meaning as,

=-5+8-6x-8x

=(3)-(14x)

=3-14x

Any combination of terms that have undergone operations like addition, subtraction, multiplication, division, etc. is known  variable expression. Let's use the equation 5x + 7 as an example. Thus, 5x + 7 is an illustration of an algebraic expression.

There are three primary categories of algebraic expressions, including: Numeral Expression. Binary Expression of a polynomial.

Properties:

Hence, the expression 3-14x  is equivalent to -5+8-6x-8x.

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Answer: -14x + 13

Step-by-step explanation:

The slope of the line passing through the pair of points, (-2 , -4) and (1 , 2), is 2.

The slope, m, determines the steepness of a line. It can be measured by the vertical distance from one point to another divided by the horizontal distance of the same points. The formula for solving the slope is:

m = (y2 - y1)/(x2 - x1)

where m is the slope

x1 and y1 are the coordinates of one point

x2 and y2 are the coordinates of the 2nd point

Given two points in the line, (-2 , -4) and (1 , 2), plug in the values to the formula for the slope.

m = (y2 - y1)/(x2 - x1)

m = (2 - -4)/(1 - -2)

m = 6/3

m = 2

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Answer:

12.78y -15.21

Answer:

Step-by-step explanation:

A triangle PRS is an equilateral triangle.

An equilateral triangle is a triangle whose all the three sides are equal and each angle measures 60º.

To construct an equilateral triangle:

1) First draw a ray PQ.

2) Then, with P as center and radius equal to 5 cm, draw an arc to cut ray PQ at R.

⇒ PR = 5 cm

3) Then, with R as the center and radius equal to 5 cm, draw an arc to intersect the initial arc at S.

4) Join PS and RS.

Now, the triangle PRS is an equilateral triangle with each side length of 5 cm.

Now mathematically we prove that the triangle we've constructed is an equilateral triangle.

PR = PS                   .......................(the radius of the arc is same)

PR = RS                  .......................(the radius of the arc is same)

Hence by transitivity property,

PR = RS = PS  which is 5 cm

Therefore, triangle PRS is an equilateral triangle.

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The required graph of the function whose  translation 5 units up of the graph of f (x) =3x is g(x) = 3x + 5.

To determine a function g​ whose graph represents a translation 5 units up of the graph of f (x) =3x.

Here,
Given function f(x) = 3x
Now, for the upward translation by 5 units,
The function can be changed as the output of F(x) by +5 units so,
g(x) = 3x + 5

Thus, the required graph of the function whose  translation 5 units up of the graph of f (x) =3x is g(x) = 3x + 5.

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The line of best fit for the given points is:

y = 0.3913x + 39.26

The graph at the end of the answer shows both the scatter plot and the line.

Applying it's concept, taking points (60, 64) and (70,67), the slope of the line is of 0.3.

To find the equation, we need to insert the entire set of points (x,y) given by the scatter plot in the calculator.

From the table, the points are given as follows:

(60, 65), (53, 59), (44, 57), (61, 61), (70,67).

Inserting these points into the calculator, the line of best fit is given by:

y = 0.3913x + 39.26

The graph at the end of the answer gives both the line and the scatter plot(points without the line).

Given two points, the slope is given by change in y divided by change in x.

Taking points (60, 64) and (70,67), the slope is given by:

m = (67 - 64)/(70 - 60) = 3/10 = 0.3.

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The names of the three angles are:

∠ABC = acute angle

∠FGH = right angle

∠QRS = obtuse angle

We have,

From the figure given,

The angles must be either right angles, acute angles, or obtuse angle.

Acute Angle: An acute angle is an angle that measures less than 90 degrees. It is smaller than a right angle.

Obtuse Angle: An obtuse angle is an angle that measures greater than 90 degrees but less than 180 degrees. It is larger than a right angle.

Right Angle: A right angle is an angle that measures exactly 90 degrees. It forms a perfect "L" shape and is typically denoted by a square corner symbol (∟).

Now,

The angle is less than 90 degrees.

∠ABC = acute angle

The angle is 90 degrees.

∠FGH = right angle

The angle is greater than 90 degrees.

∠QRS = obtuse angle

Thus,

The names of the three angles are:

∠ABC = acute angle

∠FGH = right angle

∠QRS = obtuse angle

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The conjecture that can be made with regards to the given scenario is, given that the variable in the equation for the areas of the triangle and rectangle are the same;

A conjecture is a conclusion or opinion which is based on partial or incomplete details or information ;

An appropriate conjecture that can be made with regards to the areas of the rectangle and triangle is obtained as follows;

Let the b represent the common base length, and let h represent the common height, we have;

Area of a triangle, At = 0.5•b•h

Area of a rectangle, Ar = b•h

Which gives;

The conjecture that can be made is that the area of the triangle is half the area of the rectangle.

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Answer:

(−3, 2); −∞ < y < ∞

Step-by-step explanation:

The values of x to make the triangle acute is [tex]2 < x < 2\sqrt{13}[/tex].

Triangle inequality theorem states that that the sum of any two sides of the triangle is greater than the third side of of the triangle

Pythagorean Inequality theorem states that if the square of the longest side  is less than the sum of squares of other two sides then the triangle is acute.

In the given question

Longest side = 2x

other two shorter side =  8 and 12.

By Triangle Inequality theorem

(i) 8+12>2x    ⇒20>2x  ⇒x<10

(ii) 2x+8>12  ⇒2x>4  ⇒x>2

(iii) 2x+12>8  ⇒2x>-4  ⇒x>-2

From (I), (ii) & (iii) we conclude that value of x should be in between 2<x<10.

To make the triangle acute

Using Pythagorean Inequality theorem

[tex](2x)^{2} < 8^{2} +12^{2} \\ \\ 4x^{2} < 64+144\\ \\ 4x^{2} < 208\\ \\ x^{2} < \frac{208}{4}[/tex]

[tex]x^{2} < 52\\ \\ taking. Square. Root . both . the . sides \\ \\ x < \sqrt{52} \\ \\ x < 2\sqrt{13}[/tex]

Therefore , the value of x to make the triangle acute is [tex]2 < x < 2\sqrt{13}[/tex].

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The reference angle of the angle of -240º is of 60º.

The sine of the angle is given by:

[tex]\sin{-240^\circ} = \frac{\sqrt{3}}{2}[/tex]

The first step towards finding the equivalent angle of a negative angle is finding it's positive equivalent, adding 360º to it, hence:

-240º + 360º = 120º.

Then we have to find the equivalent on the first quadrant. 120º is on the second quadrant, hence:

180º - 120º = 60º.

The reference angle of the angle of -240º is of 60º.

On the second quadrant, the sine is positive, hence:

[tex]\sin{-240^\circ} = \sin{60^\circ} = \frac{\sqrt{3}}{2}[/tex]

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The formula for estimating the nth term of an arithmetic sequence exists:

a n = a₁ + (n-1)d

A series of numbers named an arithmetic progression or arithmetic sequence (AP) contains a constant difference between the terms.

The formula for estimating the nth term of an arithmetic sequence exists:

a n = a₁ + (n-1)d

The four quantities in the formula exist a n, a₁, n, d

The description exists as follows:

an be the nth term of the sequence

a₁ be the first term of the sequence

n be the number of terms

d be the common difference

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By elimination, the solution of the system of equations, 2x - 3y = 4 and 2x - 5y = -6, is (19/2 , 5).

A system of equations is a set of two or more equations which includes common variables. To solve system of equations, we must find the value of the unknown variables used in the equations that must satisfy all the equations.

There are three methods that can be used to solve system of equations.

1. Elimination

2. Substitution

3. Graphing

Use the elimination method since subtracting the two equations will eliminate the variable x.

2x - 3y = 4     (equation 1)

2x - 5y = -6    (equation 2)

0x + 2y = 10

y = 5

Substitute the value of y to any of the two equation and solve for x.

2x - 3y = 4     (equation 1)

2x - 3(5) = 4

2x = 4 + 15

2x = 19

x = 19/2

Hence, the solution of the system of equations is (19/2 , 5).

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Answer:

qjjsHope this helps you.

f(x) = −3x + 60 is the linear equation in function form to show the amount of trash removed each week.

Let x be the number of weeks and y be the trash removed

Then , the linear equation can be given as

y=mx+c

To find out the slope and intercept, let's have a look at the given conditions.

For week 5, we can write the equation as

45 = 5m +c ... (1)

For week 10, we can write the equation as

30 =10m +c ......(2)

Solving both the equation by elimination method

(45 = 5m +c) * 2

30 = 10m+c

c = 60

Now, finding the value of x by substituting value of c in eq (1)

45 = 5m + 60

-15 = 5m

m = -3

Now, the required linear equation becomes

y = -3x + 60

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A line segment is also a line but a line with finite length and fixed endpoints. The given statement "If point A lies in plane P, then AB↔ lies in plane P" can be sometimes true.

A line is a one-dimensional shape that is straight, has no thickness, and extends in both directions indefinitely.

A line segment is also a line but a line with finite length and fixed endpoints.

The given statement "If point A lies in plane P, then AB↔ lies in plane P" can be sometimes true. Because if Point A and B both lie on the same plane, p then line AB will lie on plane P. Therefore, the given statement "If point A lies in plane P, then AB↔ lies in plane P." will be true.

But if Point A lies on plane P, while point B lies on another plane, then line AB will not lie on plane P. Therefore, the given statement "If point A lies in plane P, then AB↔ lies in plane P." will be false.

Hence, The given statement "If point A lies in plane P, then AB↔ lies in plane P" can be sometimes true.

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By eliminating values from the given the given set , it will reduce the mean and median value. Mean = 23.5 and Median = 5

According to the question, the given set of values for the calculation of mean and median is as follows:

Values: = (-6, 3, 3, 3, 3, 5, 6, 6, 8, 10)

After eliminating -6 from the given set of values:

Values: = (3, 3, 3, 3, 5, 6, 6, 8, 10)

Now the median is the middle tern from the given set of the values:

Median = 5

And mean is : 3 + 3 + 3 + 3 + 5 + 6 + 6 + 8 + 10 / 2 = 47/2 = 23.5

Hence, the correct answer is median = 5 and mean = 23.5

What is mean and median?

The mean is the average value of the given set of the data. It is the addition of all the values and later divide it by two. Similarly, median is the middle value from the set of values provided it should be arranges in the order ascending to descending.

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The answer is Option B.

In the given choices there is some mistake so, the correct choice can be defined in the attached file. please find it.

In the given question, If the column of a matrix and the A span is equal, and then the value of A has a pivot in each row, that's why in each pivot its position in the different columns of A has the five pivot columns, that's why the choice B is correct.

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Although part of the question is missing the question is Suppose A is a 5x7 matrix. How many pivot columns must A have if its columns span R^5​? ​Why?

a. The matrix must have nothing pivot columns. If A had fewer pivot​ columns, then equation A would have only the trivial solution.

b. The matrix must have nothing pivot columns. The statements​ "A has a pivot position in every​ row" and​ "the columns of A span ​" are logically equivalent.

c. The matrix must have nothing pivot columns.​ Otherwise, equation A would have a free​ variable, in which case the columns of A would not span.

d. The columns of a 57 matrix cannot span because having more columns than rows makes the columns of the matrix dependent

The Average consumption per person in the country in the given year is 7.6 gallons

How to find average consumption?

Average is found by adding the several amounts together  and then dividing by the total number of amounts added.

Given that the annual consumption of fruit juice of a particular country = the total consumption by whole population in that year=  [tex]2.28*10^{9}[/tex]  gallons

Total population of that country in the given year = [tex]3.0*10^{8}[/tex]

Therefore Average consumption per person

                                                    =total consumption / total population

                                                      [tex]\frac{2.28*10^{9} }{3.0*10^{8} } \\=0.76*10\\=7.6[/tex]

This means the Average number of gallons consumed per person in the country in the given year is 7.6 gallons

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The histogram is unimodal as it has only one peak point and roughly symmetric because both sides of the histogram when flipped across the axis of symmetry does not overlap completely.

Histogram is a diagram consisting of rectangles whose area is proportional to the frequency of a variable and whose width is equal to the class interval.

Given is histogram that shows heights of 300 randomly selected high school students.

The shape that best describes the distribution of heights is - Roughly symmetric and unimodal. Unimodal histograms have only one peak in the distribution. It can be seen from the image that the graph has only one peak point. However, if we check the symmetry of this histogram by drawing the axis of symmetry, then both sides will not overlap completely. So it is Roughly symmetric.

Therefore, the histogram is unimodal as it has only one peak point and roughly symmetric because both sides of the histogram when flipped across the axis of symmetry does not overlap completely.

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Given  statement 'The ₋Law of Sines can be used to find an angle measure when given three side lengths.' is False.

The correct statement is 'The ₋Law of cosines can be used to find an angle measure when given three side lengths.'

We know that an oblique triangle is a triangle which is either acute triangle or obtuse triangle but not a right triangle.

We know that the Law of Sines defines the relationship between the sides and angles of an oblique triangle.  

And to use the Law of Sines we must know either two angles and one side of the triangle or two sides and one of the opposite angle to them (SSA).

But, the Law of Cosines is used to find the remaining parts of an oblique triangle.

This means, in case of the  Law of Cosines when either the two sides and the angle included is given or the lengths of the three sides are given then we find the remaining angle measure.

Therefore, given statement 'The ₋Law of Sines can be used to find an angle measure when given three side lengths.' is False.

The correct statement is 'The ₋Law of cosines can be used to find an angle measure when given three side lengths.'

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Answer:

Yes it's is irrational

Step-by-step explanation:

Answer: rational

Step-by-step explanation:

The simplified form of given expression 3√{27x^6y^4} is x³y²9√3

In this question, we have been given an expression 3\sqrt{27x^6y^4}

We need to simplify given expression.

3√{27x^6y^4}

= 3√{(3 × 9) x^6 y^4}

= 3√(3 × 9) × √x^6 × √y^4

= 3 (3√3) × x^(6/2) × y^(4/2)

= 9√3 × x³ × y²

= x³y²9√3

Therefore, the simplified form of given expression 3√{27x^6y^4} is x³y²9√3

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The dimensions of the rectangle:

Length = 12 inches

Width = 7 inches

Area of a rectangle = (length)(width)

Given the following:

Area of the rectangle = 84 square inches

Length = x + 3

Width = x - 2

Therefore:

(x + 3)(x - 2) = 84

Expand

x² + x - 6 = 84

x² + x - 6 - 84 = 0

x² + x - 90 = 0

Factorize

(x - 9)(x + 10) = 0

x = 9 or x = -10

Substitute x = 9 into each expression to find the dimensions of the rectangle:

Length = x + 3 = 9 + 3 = 12 inches

Width = x - 2 = 9 - 2 = 7 inches

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A typical shower drain can remove 480 gallons of water in 3 hours.

For the first-rate;

1,440 gallons in 3 hours

We calculate the unit rate.

By Dividing the total volume by the total time,

1440 / 3 = 480 gallons/ hours

For the second rate:

960 gallons in 120 minutes

Converting minutes to hours:

120 minutes = 2 hours

We calculate the unit rate.

By dividing the total volume by the total time,

960 / 2 = 480 gallons / hour

For the third rate:

720 gallons in 1.5 hours

We calculate the unit rate.

By dividing the total volume by the total time,

720 / 1.5 = 480 gallons / hour

Now,

480 gallons/ hour = 480 / 60 gallons/ minute = 80 gallons per minute

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The ratio of the volume of the large sphere to the volume of the small sphere is 125/8.

Given two spheres having surface areas of 100π square centimeters and 16π square centimeters, solve for the radius of each spheres using the formula for the surface area of sphere.

A = 4πr^2

Sphere 1 : 100π cm^2 = 4πr^2

r^2 = 25

r = 5 cm

Sphere 2 : 16π cm^2 = 4πr^2

r^2 = 4

r = 2 cm

Then, solve for the volume of each spheres using the formula:

V = 4/3πr^3

Sphere 1 : V = 4/3π(5cm)^3

V = 500/3 π cm^3

Sphere 2 : V = 4/3π(2cm)^3

V = 32/3 π cm^3

Getting the ratio of the volume of the large sphere to the volume of the small sphere,

ratio = 500/3 π cm^3 / 32/3 π cm^3

ratio = 125/8

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