Three friends live in houses with backyards adjacent to a neighborhood bike path. They decide to have a round-robin party using their three homes, inviting their friends to start at one house and then move to each of the other two. If one friend's house is centered at the origin, then the location of the other homes are (5,12) and (13,0) . Write a coordinate proof to prove that the triangle formed by these three homes is isosceles.

The simplification of difference of the given two rational expressions [tex]\frac{x}{x^{2} -4}+\frac{1}{x+2}[/tex] is equal to 2(x-1)/ (x-2)(x+2).Restriction on the variables is given by x≠ -2,  x≠ 2.

As given in the question,

Given expression is [tex]\frac{x}{x^{2} -4}+\frac{1}{x+2}[/tex]

Simplify the given two rational expressions by factorizing them,

[tex]\frac{x}{x^{2} -4}+\frac{1}{x+2}\\\\= \frac{x}{(x-2)(x+2)}+\frac{1}{x+2}\\\\= \frac{x+x-2}{(x-2)(x+2)}\\\\= \frac{2x-2}{(x-2)(x+2)}\\\\= \frac{2(x-1)}{(x-2)(x+2)}[/tex]

Restriction on the variables is given by  x≠ -2,  x≠2.

Therefore, the  simplification of difference of the given two rational expressions [tex]\frac{x}{x^{2} -4}+\frac{1}{x+2}[/tex] is equal to 2(x-1)/ (x-2)(x+2). Restriction on the variables is given by x≠ -2,  x≠ 2.

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Answer:

8th grader

Step-by-step explanation:

They walked the the fastest since they walked 50 feet more than the 6th grader in 5 more seconds ( im sorry if you dont understand! i tried my best explaining it! hope this helps :) )

Answer: x=

7.5

Step-by-step explanation:

Answer:7.5

Step-by-step explanation:

move all unknowns to one side by either adding or subtracting from both sides

so, subtract 4x from both sides and you get 2x-10=5

now mode -10 to the other side by adding 10 to both sides. you get 2x=15

divide both sides by 2 to get x by itself

you get x=7.5

The required solution for the given arithmetic problems are,
A) 2/5
B) 72 / 125
C) 6, 3/4, 12/5, and 3/10
D) 4, 1/6 and 6

A) 12/25 times 15/18.
Division of two fractions performed as multiplication of numerator of one fraction with a denominator of another function over a denominator of one function with a numerator of another function.
= 12 / 25  * 15 / 18 = 2 / 5

Similarly

b) 12/25 divided by 15/18.
= [12 / 25 ] / 15 / 18
= 12 / 25 * 18 / 15
= 72 / 125

C) 6 and 3/4 times 1 and 2/5
 6 * 1 = 6
3 / 4 * 1 = 3/4
6 * 2/5 = 12/5
3/4 * 2/5 = 3 / 10

D) 4 and 1/6 divided by 1 and 2/3.
= 4 / 1 = 4
= 1/6  / 1 = 1/6
= 4 / 2/3 = 6
= 1/6 / 2/3 = 1/4

Thus, the required solution for the given arithmetic problems is mentioned above.

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The number of pensioners in the sample is 40.

Given,

Visitors to a cinema are either adults, children, or pensioners.

The ratio of children to adult visitors is 1:2.

The ratio of adults to pensioners is 3:4.

A stratified sample of 85 is to be taken from the cinema visitors.

We need to find out how many pensioners should be in the sample.

Example:

Out of 10 pens, the ratio of the red and blue pens is a:b

The number of blue pens = [ b / (a + b) ] x 10

The number of red pens = [ a / (a + b) ] x 10

We have,

Total number of visitors = 85

children:adult  = 1:2

adults:pensioners = 3:4

Here,

Let,

The number of children be C.

The number of adults be A.

Number of Pensioners be P.

Now,

1C : 2A

    : 3A : 4P

3C : 6A : 8P

Now,

3C + 6A + 8P = 85

This means 8 is the ratio of pensioners.

= 8 / (3 + 6 + 8) x 85

= (8 / 17)x 85

= 8 x 5

= 40

Thus the number of pensioners in the sample is 40.

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An equation in slope intercept form the line which goes through the points (0,3) and (4,-5) is;

⇒ 2x + y = 3

Here,

The line passes through the points (0,3) and (4,-5).

We have to find an equation in slope intercept form the line which goes through the points (0,3) and (4,-5).

What is Equation of line?

The equation of line passes through the point (a, b) with slope m is;

y - b = m (x - a)

Now,

The line passes through the points (0,3) and (4,-5).

Hence, The slope is given as;

[tex]m = \frac{y_{2}-y_{1} }{x_{2}- x_{1} } = \frac{-5-3}{4-0} = -2[/tex]

Therefore, the equation of line is;

y - y₁ = m (x - x₁)

y - 3 = -2 (x - 0)

y - 3 = -2x

2x + y = 3

Hence, An equation in slope intercept form the line which goes through the points (0,3) and (4,-5) is;

⇒ 2x + y = 3

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The value of s in the given figure is 28

The given parameters are:

m<TUW = m<VUW =62 degrees

TW = s + 56 and VW= 3s

From the figure, we have the following congruent equation

TW = VW

This means that

s + 56 = 3s

Evaluate the like terms

So, we have

2s = 56

Divide both sides of the equation by 2

So, we have:

s = 28

Hence, the value of s in the given figure is 28

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The correct answer from each drop-down menu is as follows:

A quadratic function can be defined as a mathematical expression that can be used to define and represent the relationship that exists between two or more variable on a graph.

In Mathematics, the graph of any quadratic function is parabolic because it is a u-shaped curve. For the given quadratic function, the graph is a downward parabola because the coefficient of x² is negative.

Therefore, function p is a quadratic function.

When the length of the tomato patch is 8 feet, the area of the bell pepper patch is given by:

p(x) = -0.5x² + 6x

p(8) = -0.5(8)² + 6(8)

p(8) = -0.5(64) + 48

p(8) = -32 + 48

p(8) = 16 square feet.

From the graph, we can logically deduce that the maximum possible area occurred at x = 6 feet. Thus, maximum possible area is given by:

p(x) = -0.5x² + 6x

p(6) = -0.5(6)² + 6(6)

p(6) = -18 + 36

p(6) = 18 square feet.

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Answer:

The water cooler can hold approximately 23,6L

I'll do problem 6 to get you started

x = number of people training all 3 sports.

This is because all 3 sports are needed for the triathlon

There are 35 swimmers total. 9 of which are also cyclers (but not runners), and there are 13 swimmers who are also runners (but not cyclers)

That leaves 35-9-13-x people who are swimmers only. I'll leave this expression un-simplified so you can see where the numbers are coming from. Take note of where I'm placing this in the venn diagram below.

There are 32 cyclers. Of this group, 9 are swimmers (but not runners) and 11 are runners (but not swimmers). That leaves 32-9-11-x people who are cyclers only.

There are 38 runners. Of this group, 13 are swimmers (but not cyclers) and 11 are cyclers (but not swimmers). That leaves 38-13-11-x people who are runners only.

All of these values are honestly hard to keep track of. This is why a visual approach using a Venn Diagram is recommended. See the diagram below.

The idea is that every value in the diagram then adds up to the 58 total attendees. This helps form the equation needed to solve for x.

Such an equation is possible because each attendee participates in at least one of the sports mentioned. No values will be outside of all three circles at the same time.

(35-9-13-x)+(32-9-11-x)+(38-13-11-x)+(9)+(13)+(x)+(11)=58

-2x+72 = 58

-2x = 58-72

-2x = -14

x = -14/(-2)

x = 7

Answer:

the 3 steps

Step-by-step explanation:

1. Rearrange the equation so "y" is on the left and everything else on the right.

2. Plot the "y=" line (make it a solid line for y≤ or y≥, and a dashed line for y< or y>)

3. Shade above the line for a "greater than" (y> or y≥) or below the line for a "less than" (y< or y≤).

Answer: False

Step-by-step explanation:

Because The letter "x" is often used in algebra to mean a value that is not yet known. It is called a "variable" or sometimes an "unknown". In x + 2 = 7, x is a variable, but we can work out its value if we try!

Step-by-step explanation:

12 degres below 0 is -12°C ..so the temperature rose from -12°C to 24°C

The difference now from Morning to afternoon is

=24°C-(-12°C)

=24°C+12°C

=36°C

Explicit formulas for arithmetic sequences are derived from terms in arithmetic sequences. It helps to find each term in arithmetic progression easily. The arithmetic progression is a1, a2, a3, ..., an. where the first term is denoted as 'a', we have a = a1, and the tolerance is denoted as 'd'. The tolerance formula is d = a2 - a1 = a3 - a2 = an - an - 1. The nth term of the arithmetic progression represents the explicit formula for the arithmetic progression.

Explicit formula: an= a + (n − 1) d

Explicit formula: Sn = n/2 [2a+(n-1) d]

Where,

nth term in the arithmetic sequence

a = first term in the arithmetic sequence

d = difference (each term and its term difference) previous term, i.e., d = an-an-1

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Answer:

There are actually 20 occurrences of the digit "2" in the numbers {1,2,⋯,100}, though there are 19 numbers in the same range where the digit "2" appears.

Step-by-step explanation:

Hope this will be helpful for you.

Before the demolition, since the RCA dome was the home to the Indianapolis Colts,   and the attendance in 2001 and 2005 was 450,746, 457,373, the attendance will not continue to increase indefinitely.

Before it was demolished, the RCA Dome was home to the Indianapolis Colts. The attendance in 2001 was 450,746, and attendance in 2005,  the rate of change in attendance 2001 to 2005 was 1657, the attendance will not continue to increase indefinitely until it will continue to increase when the capacity of the stadium was reached, and the building new stadium will help them to accommodate more fans, since if the attendance is high.

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Answer:C

Step-by-step explanation:

c

The equation that represents the situation is

11 (p - 382) = 506

where p represents the amount in dollars that Esmeralda earns each week.

Option C is the correct answer.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

We have,

Weekly expenses = $382

Money saved after 11 weeks = $506

The equation that represents the given above situation:

Let p = amount earned per week.

We can write,

11p - 11 x 382 = 506

Taking 11 as common.

11 (p - 382) = 506

Now,

We can say that,

11 (p - 382) = 506, where p represents the amount in dollars that Esmeralda earns each week.

Thus,

The equation that represents the situation is

11 (p - 382) = 506

where p represents the amount in dollars that Esmeralda earns each week.

Option C is the correct answer.

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Answer: (1, 34) and (2, 17)

Step-by-step explanation:

Step 1: Divide 34 by the smallest prime factor.

34/2 = 17

Step 2: Since 17 itself is a prime number, therefore, it is divisible by 17 only.

17/17 = 1

Step 3: Now further division is not possible. Thus, we will consider 2 and 7 as the prime factors of 34.

Answer:

5(2) + 5(-3)

Step-by-step explanation:

10 + -15 = -5

Finding the equation of the parabola, it is found that the height of the diving board is of 24 feet.

The equation of a quadratic function, of vertex (h,k), is given by:

y = a(x - h)² + k

In which a is the leading coefficient.

From the maximum height, it is found that the vertex is at (1,25), hence:

h = 1, k = 25.

Hence:

y = a(x - 1)² + 25.

You hit the water 6 feet from the diving board, hence y(6) = 0, meaning that we use it to find a as follows:

0 = a(6 - 1)² + 25

25a = -25.

a = -1.

Hence:

y = -(x - 1)² + 25.

The height of the diving board is y(0), hence:

y = -(0 - 1)² + 25 = -1 + 25 = 24 feet.

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The point-slope form be y - 7= 0.4(x - 6) then 0.4x - y = -4.6 be the standard form.

Given:

(x1, y1) = (6, 7)

The value of m = 2/5

The point-slope form is given by, y - y1 = m(x - x1)

substitute the values in the above equation, we get

y - 7 = 2/5(x - 6)

simplifying the equation,

y - 7 = 0.4(x - 6)

Therefore, the point-slope form is y - 7= 0.4(x - 6)

Standard form is Ax + By = C

Since, slope-intercept form is y = 0.4x + 4.6

⇒ 0.4x - y = -4.6

Therefore, 0.4x - y = -4.6 is the standard form.

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The word “quantity” indicate to use a variable

The phrase is given as:

quantity

In an algebraic expression, phrases are used to indicate variables

So, anywhere the word is used, it implies that we make use of a variable

Take for instance:

A quantity is more than 2 means x > 2

Hence, the word “quantity” indicate to use a variable

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The given expression:

(3/8)*( 1 + √49)^2 - (4 - 1)^3

Can be simplified to -3.

So we have the equality:

(3/8)*( 1 + √49)^2 - (4 - 1)^3 = -3

Which means that the first option is the correct one.

Here we have the expression:

(3/8)*( 1 + √49)^2 - (4 - 1)^3

(I think is something like that)

And we want to simplify it, so first let's simplify the things inside the two parentheses.

The first one is:

1 + √49

We know that 7*7 = 49, then the square root of 49 is 7.

1 + √49 = 1 + 7 = 8

The other one is:

4 - 1 = 3

Replacing that in the expression we get:

(3/8)*( 1 + √49)^2 - (4 - 1)^3 = (3/8)*(8)^2 - (3)^3

Now we can solve the exponents:

8^2 = 8*8 = 64

3^3 = 3*3*3 = 27

Replacing that we get:

(3/8)*64 - 27 = 3*8 - 27 = 24 - 27 = -3

We conclude that the given expression:

(3/8)*( 1 + √49)^2 - (4 - 1)^3

Can be simplified to -3.

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Answer: take a picture of it please

Step-by-step explanation:

Answer:

4.5

Step-by-step explanation:

To answer the problem:

PK * PN = PL^2

Let PN = x

8x = 10^2

8x = 100

x = 100/8

x = 12.5

KN = PN - PK

12.5 - 8 = 4.5 would be the length of segment KN.

The situation can be represented by a geometric sequence with a(1) = 5000 and ratio or r = 3/4. At the start of the 10th day, the remaining helium volume is 375.42 cm³

Let a(n) be the volume of helium left in the balloon at the nth day.

Day 1: The balloon is filled with 5000 cm³ of helium. We can conclude the volume of  helium in day 1 is:

     a(1) = 5000.

Day 2: The volume of helium loses 1/4 from its initial volume. Hence,

     a(2) = (1 - 1/4) . a(1)

            = 3/4 . 5000 = 3/4 . 5000 = 3750

Day 3: The volume of helium loses 1/4 from its volume in day 2. Hence,

      a(3) = (1 - 1/4) . a(2)

             = 3/4 . 3750 = 2812.5

Continue the process, it will form a geometric sequence with a(1) = 5000 and ratio or r = 3/4.

To find the 10th term, use the nth term formula:

a(n) = a(1) . r^(n-1)

a(10) = 5000. (3/4)¹⁰⁻¹

        = 5000. (3/4)⁹ ≈ 375.42

Hence, at the start of the 10th day, the remaining helium volume is  375.42 cm³

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The sum of the given finite series is -5050

The summation notation is used to expressed a long summation into a single formula.

Expand the sum of the given finite series:

[tex]\sum\limits^{100}_{n=0} {(-1)n} =0-1-2-3-...-100[/tex]

The right side is an arithmetic series. We can ignore the 0 in the first term, so the arithmetic series is:

-1 -2 - 3 -4 - ... - 100

We get:

a(1) = -1

d = -2 - (-1) = -1

a(n) = -100

n = 100

Substitute these parameters into the sum formula:

S(n) = n/2 [a(1) + a(2)]

S(100) = 100/2  [-1 + (-100)]

S(100) = 50 . (-101) = -5050

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The lengths of the diagonals are 8 and 24

Area of kite :

Area of kite is the space enclosed by a kite. A kite is a quadrilateral in which two pairs of adjacent sides are equal. The elements of a kite are its 4 angles, its 4 sides , and 2 diagonals.

In the given statement is :

Suppose the length of one diagonal of a kite is three times the length of the other diagonal. If the area of the kite is 96 square inches.

And, to find the lengths of the diagonals.

Now, Area of the kite is 96 sq. inches.

The length of one diagonal of a kite is x

and, other diagonal is three times to other diagonal

So, d2 = 3x

The area of a kite = 1/2 x (d1) x (d2)

96 = 1/2 × x × 3x

96 = [tex]\frac{3x^{2} }{2}[/tex]

192 = [tex]3x^{2}[/tex]

64 = x^2

x = [tex]\sqrt{64}[/tex]

x = 8

Therefore, The length of one diagonal is 8

and, other length of diagonal is 3x = 3x 8 = 24

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the order of titles according to the correct description boxes is  (0, 4.9), (-3,10), (-9, -3) and then (4.9, 10) respectively.

We first look at the graph, and then, we analyze it, completing the tiles.

From the graph, the function is increasing in two intervals (-10,-9) and (4.9,10). The function is increasing over two interval, (-10, -9) and (4.9,10) , so the corresponding tile is (4.9, 10) for increasing in two intervals.

Between -9 and 4.9 the function is decreasing, so the corresponding tile is (0, 4.9).

An interval where the function is above the x-axis, so between -9 and -3, and then above 10, so the corresponding tile is (-9,-3) for interval over which the difference is positive.

Interval over which the difference is negative is when an interval where the function is below the x-axis, so before -10, and then between -3 and 10. Thus the corresponding tile is (-3,10).

Therefore, we can conclude the following relations

interval in which difference of the number of sales is decreasing→ (0, 4.9)

interval over which difference of the number of sales is negative→ (-3,10)

interval over which difference of the number of sales is positive→  (-9, -3)

interval in which difference of the number of sales is increasing→ (4.9, 10)

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The smallest domain element of the relation is 2

The domain of the function is the set of input values of the graph

The relation is given as

R = {(2, 3), (3, 3), (4, 4), (5, 7), (7, 9), (12, 11), (14, 13)}.

Remove the y values to determine the domain

So, we have

Rx = {2, 3, 4, 5, 7, 12, 14}

The smallest element in the above list is 2

So, we have

Smallest Rx = 2

Hence, the smallest domain element of the relation is 2

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Noah and Lin each need 6 cups of sugar

Noah and Lin buy 12 cup bag of sugar and divide it evenly to make their recipe

If the entire sugar is consumed, the number of flour required by each person can be calculated by dividing 12 by 2

= 12/2

= 6

Hence each person will need 6 cups of flour for the recipe

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