slope: 3; y-intercept: -8

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:

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.

We need to use the concept of Percentile Rank. The Percentile Rank for score 71 is   33.33   and the score of Viktor is   84.

According to the pupils in the particular norm group, a student's performance is shown by their percentile rank. Let's try to put the percentile rank in a tidy mathematical equation now that you are clearer on what it actually is. A collection of values A = x1, x2, x3,..., xn with N values will be used in our attempt to calculate the percentile rank of the data point xm. In this scenario:

PR = L / N × 100 where:

PR stands for percentile rank and has a range of 0 to 100;

L is the quantity of values in set A that are less than or equal to the value of your data, xm; and

The entire collection of values in A has a total of N values.

Given some of the test scores. Arrange the test Scores in ascending order.

A={58,64,66,70,71,75,77,80,84,85,87,90,93,95,96}

N=15

Now find out the values in the set which are less than or equal to 71.

L=5

Apply, PR=L/N ×100

⇒PR=5/15 × 100

PR=33.33 is the Percentile Rank of score 71.

Viktor is in 60th percentile, so find out L.

⇒60=L/15 ×100

⇒L=60×15/100

L=9.

So 84 is Viktor Score.

We need to use the concept of Percentile Rank. The Percentile Rank for score 71 is   33.33   and the score of Viktor is   84.

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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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The system of equations have Unique solution.

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The system of equation is,

⇒ 3x - y = 13

⇒ y + 3 = 9x

Now,

Since, The system of equation is,

⇒ 3x - y = 13

⇒ y + 3 = 9x

It can be written as,

⇒ 3x - y = 13

⇒ 9x - y = 3

Clearly, We have;

⇒ 3/9 = - 1 / - 1

⇒ 1/3 ≠ 1

Thus, It has Unique solution.

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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 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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The equations of the asymptotes of the hyperbola is 6y = -7x,

There are two asymptotes on each hyperbola. A hyperbola with a horizontal transverse axis and centre at (h, k) contains one asymptote with equation y = k + (x - h) and the other with equation y = k - (x - h). A straight line that continuously approaches a certain curve without ever meeting it is an asymptote. In other words, an asymptote is a line that a curve travels towards as it approaches infinity. The two vertices, the two focus points, and the centre are all on the same horizontal line and equally spaced from one another. This is a horizontal hyperbola because of this. Our graph of the hyperbola follows the lines of the asymptotes.

Given,

Equations of hyperbola,

(y+4)². (x-6)²

——— - ———. =. 1

36. 49

a² = 36

a = 6

b² = 49

b = 7

Asymptotes of the hyperbola:

y = ±b/a × x

y = 7/6x

6y = 7x

y = -7/6 x

6y = -7x

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The system of equations  presented in the graph is written as

Linear function in the slope intercept formula is written in the form

y = mx + c

definition of variable

y = output variable

m = slope

x = input variable

c = y intercept

The dashed line represents inequality without "equal to" hence less than or greater then only. shading above the line is greater than

the line passed through points (0, 2) and (3, 0)

the slope, m is = (2 - 0) / (0 - 3) = -2/3

the y intercept = 2

y > -2x/3 + 2

The solid line means the inequality has equal to hence it either greater than or equal to OR less than or equal to. no shading means it is only equal to

the line passed through points (0, 0) and (2, 2)

the slope, m is = (0 - 2) / (0 - 2) = 1

the y intercept = 0

y = x

The equation of the graph is

y > -2x/3 + 2

y = x

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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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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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There are 5x6=30 one-to-one functions from a set containing 5 elements to a set containing 6 elements.

A one-to-one (or injective) function is a type of mathematical function that maps each element of one set to one, and only one, element of another set. In other words, a one-to-one function is a function that has a unique output for each input. To calculate the number of one-to-one functions from a set containing 5 elements to a set containing 6 elements, we can use the formula f(x)=y, where x is the number of elements in the original set and y is the number of elements in the target set. In this case, x = 5 and y = 6, so f(x) = 6. Since each element in the original set must be mapped to a unique element in the target set, the total number of one-to-one functions is equal to the product of x and y, or 5 x 6 = 30.

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By solving a system of equations we can find the two numbers:

y = 5 + √15

x = 5 - √15

We can define the two numbers as x and y, and we know that:

x*y = -10

x  + y = 10

So we have a little system of equations here.

We can isolate x on the second equation to get:

x = 10 - y

And now replace it on the first equation to get:

(10 - y)*y = -10

10y - y^2 = 10

Then we have a quadratic equation:

y^2 - 10y + 10 = 0

The solutions are given by:

[tex]y = \frac{10 \pm \sqrt{(-10)^2 - 4*10*1} }{2} \\\\y = \frac{10 \pm \sqrt{60} }{2}\\\\y = 5 \pm \sqrt{15}[/tex]

So the two numbers are:

y = 5 + √15

x = 5 - √15

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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:

i think ur answer is c

Step-by-step explanation:

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.

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.

-$650, Spent -$842, Earned +$522 and Earned +$650 are required integers and Spending $650 and earning $650 are opposite transactions but they have the same absolute value.

A number system is defined as a system of writing to express numbers.

Spent $650

Spent $842

Earned $522

Earned $650

Spent -$650, Spent -$842, Earned +$522 and Earned +$650.

Spending $650 and earning $650 are opposite transactions but they have the same absolute value because the absolute value of a number n is written as |n|.

The absolute value of 0 is 0. Two numbers are opposites if they have the same absolute value but different signs. Opposites are the same distance from 0 on a number line, and they are on opposite sides of 0.

Hence, -$650, Spent -$842, Earned +$522 and Earned +$650 are required integers and Spending $650 and earning $650 are opposite transactions but they have the same absolute value.

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Answer:

Step-by-step explanation:

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

Answer:

what the shapes mg me down..

Answer:

-------------------------------------------

See the steps below, they should be self-explanatory.

Let me know if anything is unclear.

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 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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[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

Verified by Toppr

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.

Answer:

A

Step-by-step explanation:

To solve this problem, we need to first find out how much interest the account earns each year. If the account earns 2.5% simple interest per year, and it starts with an initial balance of $260, then the interest earned each year is 0.025 * $260 = $6.50.

Over 5 years, the account will earn a total of 5 * $6.50 = $<<5*6.5=32.50>>32.50 in interest.

To find the total amount of money in the account at the end of the 5 years, we need to add the interest earned to the initial balance of the account. The total amount of money in the account will be $260 + $32.50 = $<<260+32.5=292.50>>292.50.

Therefore, the total amount of money in the account at the end of the 5 years is $292.50.

Answer:

A)  $292.50

Step-by-step explanation:

Simple Interest Formula

A = P(1 + rt)

where:

Given:

Substitute the given values into the formula and solve for A:

⇒ A = 260(1 + 0.025 · 5)

⇒ A = 260(1 + 0.125)

⇒ A = 260(1.125)

⇒ A = 292.50

Therefore, the total amount of money in her savings account at the end of 5 years is $292.50.

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°