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The locus of point P such that PA²+PB² = 16 is [tex]x^{2} +y^{2} + 4x - 4y + 5 =0[/tex]

A locus is the collection of all points that satisfy a condition, typically forming a curve or surface. For instance, the set of points in three-space that are equally spaced out from a given point is a sphere, and the locus of points in the plane that are equally spaced apart from a given point is a circle.

Given that A and B are two fixed points on a plane locus of P such that PA²+PB² =16

Let point P be (x,y) then [tex]{(x-0)^{2} +(y-1)^{2} }+{(x-(-4))^{2}+(y-3)^{2} } = 16[/tex]

is the Locus of the given point P

On simplyfying we  get

[tex]{x^{2}+(y-1)^{2} } +{(x+4)^{2}+(y-3)^{2} } = 16[/tex]

[tex]2x^{2} +2y^{2} +8x-8y+10=0\\x^{2} +y^{2} + 4x - 4y + 5 =0[/tex]

This is the Locus of point P where PA²+PB²  = 16.

The locus of point P such that PA²+PB² = 16 is [tex]x^{2} +y^{2} + 4x - 4y + 5 =0[/tex]

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Answer:

The answer is D.

Step-by-step explanation:

Test each multiply choice to see if given the x-axis points.

(x+1)(x+2)(x-3)=0

(x+1) = 0 , (x+2) = 0 , (x-3) = 0

x = -1 , x = -2 , x = 3

(-1 , 0) , (-2 , 0) , (0 , 3)

Answer:

Step-by-step explanation:

take your fingers, raise 2 of them now raise 2 more how many do you see? 4.

[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ A(\stackrel{x_1}{4}~,~\stackrel{y_1}{1})\qquad B(\stackrel{x_2}{x}~,~\stackrel{y_2}{y}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ x +4}{2}~~~ ,~~~ \cfrac{ y +1}{2} \right) ~~ = ~~\stackrel{\textit{\LARGE M}}{(1~~,~~3)} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ x +4}{2}=1\implies x+4=2\implies \boxed{x=-2} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{ y +1}{2}=3\implies y+1=6\implies \boxed{y=5}[/tex]

Answer:

Step-by-step explanation:

Solution

verified

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Correct option is D)

p p

∼p⇒q  ∼(∼p⇒q)

T  T  T  F

T  F  T  F

F  T  T  F

F  F  F  T

p  q  p∧q  p∧∼q ∼p∧q   ∼p∧∼q

T  T  T  F  F  F

T  F  F  F  F  F

F  T  F  T  T  F

F  F  F  F  F  T

So we see the given relation is logically equivalent to option D, as shown in last column.

The value of x from the given equation is 6

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

The given equation is 4x=24.

The solution of an equation is the set of all values that, when substituted for unknowns, make an equation true.

Here, 4x=24

x=24/4

x=6

Therefore, the value of x is 6.

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"Your question is incomplete, probably the complete question/missing part is:"

Solve 4x=24.

Answer:

full answer part(A,B,C)

Part A: An experiment would be the most appropriate design for investigating the effectiveness of a new formula of fertilizer.

The reason for this is that an experiment allows for the manipulation of one or more variables in order to study their effects on a particular outcome. In this case, the orchard owner could manipulate the use of the new formula of fertilizer and compare the results to the use of the current popular brand. This would allow for a more controlled and systematic investigation of the effects of the new formula on fruit production, as compared to other designs such as observational studies or surveys, which do not involve manipulation of variables.

Part B: To implement the experiment, the orchard owner could divide the orchard into two sections, with one section receiving the new formula of fertilizer and the other section receiving the current popular brand. The orchard owner could then measure the fruit production in both sections over a period of time, taking into account factors such as the type of fruit being grown, the age of the trees, and the weather conditions during the study period. It would also be important to ensure that the two sections are as similar as possible in terms of the type of soil, the amount of sunlight they receive, and other relevant factors that could affect fruit production.

Part C: One benefit of this design is that it allows for a more precise and accurate assessment of the effectiveness of the new formula of fertilizer, as compared to other designs such as observational studies or surveys. By manipulating the use of the fertilizers and controlling for other variables, the orchard owner can be more confident that any differences in fruit production are due to the use of the new formula, rather than other factors that might confound the results. This can help the orchard owner make more informed decisions about whether to adopt the new formula for use on their orchard.

Answer:

[tex]-17.2y-54.4[/tex]

Step-by-step explanation:

Use the distributive property.

Answer:

A

Step-by-step explanation:

The correct answer is A. −17.2y − 13.6.

To find the equivalent expression, we can first distribute the 4 to the terms inside the parentheses:

4(−4.3y − 13.6) = −17.2y − 54.4

Then, we can combine like terms to simplify the expression:

−17.2y − 54.4 = −17.2y − 13.6

Therefore, the equivalent expression is −17.2y − 13.6.

Answer:

what the shapes mg me down..

The number of minutes of practice in two weeks is 658.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

Total minutes of practice in one week = 329.

Total minutes of practice in two weeks.

= 2 x 329

= 658 minutes.

Thus,

658 minutes was the practice in two weeks.

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The quadratic equation has 2 non-real numbers as solutions.

Here we have the graph of a quadratic equation of the form:

y = a*x^2 + b*x + c

And the solutions of the quadratic equation are the values of x such that:

0 =  a*x^2 + b*x + c

To identify these values, we need to see the values of x such that the graph intercepts the x-axis. But here we can see that the graph never intercepts the x-axis.

Then the solutions are complex numbers, and because it is a quadratic, there are two.

So the correct option is the 5th one, 2 non-real numbers.

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The graph of the parabola is shown with a vertex, while the root of the equation is x = 7 and -1.

A parabola is a cross-section cut out of the cone and represented by an equation

Here,
y = 29x² - 174x - 203
Roots the equation,
29x² - 174x - 203 = 0
x² - 6x - 7 = 0
(x - 7)(x + 1) = 0

Roots are x = 7 and -1
Now illustrating the graph of the parabola,

Thus the graph of the parabola is shown with a vertex, while the root of the equation is x = 7 and -1.

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Answer:

The answer is √84.

Step-by-step explanation:

To find the value of p, we need to square the expression 2√21 to get rid of the square root. When we do this, we get 4 * 21 = 84, so the value of p is 84. This means that 2√21 can be written as √84.

Answer:

p = 84

Step-by-step explanation:

2[tex]\sqrt{21}[/tex] ( place the 2 inside the radical by squaring it , that is

= [tex]\sqrt{2^2(21)}[/tex]

= [tex]\sqrt{4(21)}[/tex]

= [tex]\sqrt{84}[/tex]

The general solution to the given system of equations  dx/dt = 4x − y,  dy/dt = 16x − 4y is y = 4x + C

In this question we need to find the general solution of the given system. dx/dt = 4x − y              ............(1)

dy/dt = 16x − 4y           .............(2)

Divide equation (2) by equation (1),

(dy/dt)/(dx/dt) = 16x − 4y /  4x − y

(dy/dt)/(dx/dt) = 4(4x − y) / (4x − y)

(dy/dt)/(dx/dt) = 4

(dy/dt)* (dt/dx) = 4

dy/dx = 4

dy = 4 * dx

Now integrate both sides of above expression.

∫dy = ∫4 * dx

y = 4x + C

Therefore, the general solution is y = 4x + C

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A car travels along the x-axis with increasing speed. Among the given graphs. the graph given in (a) most closely represents the motion of the car.

The slope of graphs (b) and (e) remains constant.

Here, the distance increases with time.

Graph (c) shows decreasing the velocity of the vehicle.

In graph (d) the velocity of the vehicle increases and then slows down.

In graph (a), speed is increased which means that as time passes the slope of the graph becomes steeper. here, the slope of the graph increases with time.

Hence, the speed of the car increases.

Therefore, the correct representation of the motion of the car is the graph (a).

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The question diagram is missing in the question,

The diagram is like (As attached below)

After solving, the expression as per the given statement will be equal to 135.

If a mathematical operation includes at least two words that are connected by an operator and either comprise numbers, variables, or both, it is referred to as an expression.

The operations with reflection coefficients include adding, subtracting, multiplying, and dividing. To include terms in an expression, a mathematical operation like reduction, addition, multiplication, or division is used.

As per the given information in the question,

The given statement is, the product of fifteen and negative three decreased by twelve.

Let the other number be x.

Negative 3 is decreased by 12.

x = 12 - 3 = 9

Then, the product of 15 and another number will be,

= 15 × 9

= 135

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Answer:

D. 60

Step-by-step explanation:

A triangle has 180 degree

We know that the three angles of this triangle are corresponding, which means all three angles are equal to each other.

Take 180 divided by 3 = 60 degree

So, the answer is D

Answer:

D, 60°

Step-by-step explanation:

These lines on all angles of the triangle mean all of the angles are equal to each other!

Remember, ALL triangles equal 180°

There are 3 angles that share 180° degrees!

Let's divide 180° by 3 to see what EACH ANGLE equals

180 ÷ 3 = 60

All of these angles are equal to 60°

Or, in other words, angle 1 = 60°

One circled number equals 10. Another circled number equals 30. The sum of those numbers is the third number because 10 + 30 is equal to 40.

In Mathematics, a numerical data is sometimes referred to as a number and it can be defined as a data set that is primarily expressed in numbers only. This ultimately implies that, a numerical data simply refers to a data set consisting of numbers rather than words.

Based on the information provided in the multiplication table, we can logically deduce the following three (3) numbers that are circled:

First circled number = 10

Second circled number = 30

Third circled number = 40

In this context, the sum of the first circled number and the second circled number is equal to the third circled number;

Third circled number = 10 + 30

Third circled number = 40.

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Complete Question:

Three numbers are circled in the multiplication table. Use the drop-down menus to explain why the sum of two of the numbers is the third number.

Click the arrows to choose an answer from each menu.

One circled number equals ____. Another circled number equals _____. The sum of those numbers is the third number because ____.

The volume of the soup pot is 226.08 cube inches.

Jayla bought the ingredients to make chicken soup, and wanted to make a double batch, which would be 12 cups of soup.

A quick search told her that this was 173.3 cubic inches. she hoped the soup pot below would be big enough.

The soup pot is 8 inches tall with a radius of 3 inches.

We have to find the volume of the soup pot.

Since the cup is in the shape of cylinder.

So the volume of cylinder = πr^2h

from the question, r = 3 inches and h = 8 inches

π = 3.14

The volume of cylinder = πr^2h

The volume of cylinder = 3.14 * (3)^2*(8)

The volume of cylinder = 226.08 cube inches

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The speed of the current and his paddling speed in still water will be 5.33 miles per hour and 1.33 miles per hour, respectively.

Speed is defined as the length traveled by a particle or entity in an hour. It is a scale parameter. It is the ratio of length to duration.

We know that the speed formula

Speed = Distance/Time

Let 'x' be the paddling speed in still water and 'y' be the current of water. Then the equations are given as,

x - y = 10 / 2.5

x - y = 4                  ...1

x + y = 10 / 1.5

x + y = 6.667          ...2

Add both equations, then we have

x + x = 4 + 6.667

x = 5.33 miles per hour

Then the value of 'y' is given as,

x - y = 4

5.33 - y = 4

y = 1.33 miles per hour

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[tex]\huge \boxed{\sf x < -3} \\\\\\ \displaystyle \sf 15 < -5x \\\\ Divide\ each\ side\ by\ -5\ and\ switch\ the\ symbol. \\\\ \frac{15}{-5} > \frac{-5x}{-5} \\\\-3 > x[/tex]

Answer:

3/4

Step-by-step explanation:

First way:

We can find a common denominator for the fractions and then add the fractions:

½ + ¼ = (2/4) + (1/4)

= (2 + 1)/4

= 3/4

Second way:

We can convert the fractions to decimals and then add the decimals:

½ + ¼ = 0.5 + 0.25

= 0.75

In both methods, the result is 3/4 or 0.75.

the number of tickets sold in advance to the number of tickets needed to be sold at the door is 88 .The club needs to sell at least 160 tickets at the door to reach their goal of $400.

1. Calculate the total amount of money raised from the tickets sold in advance: 40 tickets x $4 = $160

2. Subtract the total amount from the goal of $400: $400 - $160 = $240

3. Calculate the number of tickets needed to be sold at the door to reach the goal: $240 / $5 = 48 tickets

4. Add the number of tickets sold in advance to the number of tickets needed to be sold at the door: 40 + 48 = 88

Therefore, the club needs to sell at least 88 tickets at the door to reach their goal of $400.

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The intervals in which both functions f(x) and g(x) are increasing are given as follows:

(-∞, -3) U (-1, ∞).

The behavior of a function is classified as either increasing, decreasing or constant, as follows:

From these bullet points and the functions given at the end of the answer, we have that the behavior of each function is given as follows:

Thus, both functions are increasing on the interval, (-∞, -3) U (-1, ∞), which is the first option.

The problem is given by the image shown at the end of the answer.

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Answer:

y=2x+8 and y=-2x+7

Step-by-step explanation:

The attached graphic shows the steps.  Two distinctive features of a straight line in the form of y=mx+b are its slope, m, and its y-intercept, b.  A quick look at the equations reveals that their slopes form unique pairs.  The slopes in the first option are 2 and -2.  The other slope pairs are b)  4 and -2, c) -3 and 3, and d)  -2,-6.  This means that if we calculate the slopes of lines p and r from the given points, there should only be one result that matches the combinations provided.

Set up a table for the points (see worksheet) and then pick any two points for each line to calculate slope.  The slope is the Rise/Run of the line:  the change in y for a change in x.  The table uses the first and last points for each line, but any two points will work (for a straight line).  

Lines p and r are found to have slopes of 2 and -2, respectively.  That means their equations will take the form of:

p:  y = 2x + b

r:   y= -2x + b

We could (and will) work further to find the y-intercepts (b), but just looking at the answer options, it is clear that the first set of equations match the slope calculations exactly.

 y = 2x+8, and y = -2x+7

None of the others come close.  This is the answer.  If this is a test, or you are missing a more interesting activity, stop here.  But just to insure we are doing the work correctly, and that the ideas are understood, lets plot the two given equations and add the points, as shown on the worksheet.  It is satisfying to find that the graph confirms our choice.  The points all fall on the two lines.  Another useful check is to find the y-intercepts (the value of y when x=0).  Those are noted on the graph (0,7) for r, and (0,8) for p, and lend further confimration to our choice.  Add these intercepts to the first equations that contain the y-intercept as b.  b for line p is 8,  b for line r is 7:

p:  y = 2x + b

r:   y= -2x + b

p:  y = 2x + 8

r:   y= -2x + 7

A perfect match to our selection.  Calculating b is not needed based on the options presented, but a tougher question may require finding the y-intercept before making a choice.

The simplified expression of 3x⁻⁴y⁴ is 3(y/x)⁴

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

3x^-4y^4

Rewrite properly

So, we have the following representation

3x⁻⁴y⁴

Express x⁻⁴ as a fraction

So, we have the following representation

3y⁴/x⁴

x⁴ and y⁴ have the same exponent

So, they can be combined as follows

3(y/x)⁴

Hence, the equivalent is 3(y/x)⁴

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The probability that the proportion surviving for at least five years will exceed 80% is 0.00084.

It is given to us that -

65% of the total dialysis patients can survive for at least 5 years

A simple random sample consists of 100 new dialysis patients

We have to find out the probability that the proportion surviving for at least five years will exceed 80%.

From the given information, we have -

Probability, p = 65% = 0.65

=> Population mean, μ = 0.65

Number of patients in the sample, n = 100

The raw data, x = 80% = 0.80

Now, we know the standard deviation of a given sample can be calculated as -

σ = [tex]\sqrt{\frac{p(1-p)}{n} }[/tex]

=> σ = [tex]\sqrt{\frac{0.65(1-0.65)}{100} }[/tex]

=> σ = [tex]\sqrt{\frac{0.2275}{100} }[/tex]

=> σ = 0.04770 ----- (1)

We also know that the z-score can be calculated by the formula -

z = (x-μ)/σ

Substituting all the values in the above formula for z-score, we have -

z = (x-μ)/σ

=> z = (0.80-0.65)/0.04770

=> z = 3.14

From the normal distribution table, we can find out the corresponding area for the value of z = 3.14 which is equal to 0.99916.

For a value of the raw data, x to be more than 80%, the area to the right of normal distribution table for z = 3.14 is required.

=> P(x>80%) = 1-0.99916

=> P(x>80%) = 0.00084

Therefore, the probability that the proportion surviving for at least five years will exceed 80% is 0.00084.

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In order to display the distribution of grades (a, b, c, d, f) in a course, it would be correct to use a bar chart.

To create a bar chart to display the distribution of grades in a course, you would first need to gather the data on the number of students who received each grade. Once you have gathered this data, you can use a computer software tool or graph paper and a pencil to create the bar chart.

When creating the chart, you would label the x-axis with the different grades (a, b, c, d, f) and the y-axis with the number of students. Then, you would draw a bar for each grade, with the height of the bar representing the number of students who received that grade. You can also add a legend to the chart to provide more information about the data, such as the percentage of students who received each grade.

Thus, a bar chart would be the best to use for dispaying the distribution of grades.

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