Do these pairs of values (x and y) represent two quantities that are proportional?​

Mathematical induction is a method used to prove statements for all positive integers.

To prove the formula 1 + 2 + 3 + ... + (n-1) + n = n(n+1)/2 using induction,

we first show it holds for the base case n = 1.

When n = 1, the formula becomes 1 = 1(1+1)/2, which is true.

Then, we assume the formula holds for some integer k, and prove it holds for k+1.

Inductive step: Assume that the formula holds for some integer k, i.e., 1 + 2 + ... + (k-1) + k = k(k+1)/2. We need to prove that the formula also holds for k+1.

So, 1 + 2 + ... + (k-1) + k + (k+1) = k(k+1)/2 + (k+1) = (k+1)(k+2)/2, which shows that the formula holds for k+1 as well.

By repeating this process, we conclude the formula holds for all positive integers.

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The system of linear inequality to model the situation is as follows:

A system of linear inequality is two or more inequalities that can be solved simultaneously to make the inequalities valid.

Inequalities must have the inequality symbols, including:

The cost of soil per bag = $4.00

The cost of plants each = $10.00

The total number of plants ≥ 4

The total budget ≤ $80

Let the number of plants = p

Let the number of soil bags = b

p ≥ 4... Inequality 1

10p + 4b ≤ 80... Inequality 2

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Write the system of linear inequality to model the situation.

Answer:

9/64

Step-by-step explanation:

[tex]=0.75\times \frac{\left(3.75+-3.5\right)}{1.3}\\0.75\times \frac{\left(3.75+-3.5\right)}{1.3}\\=0.75\times \frac{0.25}{1.3}\\=0.75\times 0.1875(\left a.k.a\:\frac{3}{16}\right)\\\frac{\left(3\times 0.1875\right)}{4}\\\mathrm{\left(Clarification:\:The\:\:math\:\:above\:\:is\:\:a\:\:broken\:\:down\:version\:of\:the\:original\:problem.\right)}\\=\frac{0.5625\left(\frac{9}{16}\right) }{4}\\=0.1406\left(\frac{9}{64}\right)[/tex]

Hope this helps!

The carrying capacity is impacted by both biotic and abiotic processes. The carrying capacity develops as these conditions improve. The carrying capacity decreases as the factors grow scarcer. When resources are depleted faster than they are replaced, the species has reached its carrying capacity.

The average population size of a species in a given habitat is referred to as carrying capacity.

Environmental considerations such as appropriate food, shelter, water, and mates limit the species population size.

The carrying capacity is impacted by both biotic and abiotic processes. The carrying capacity develops as these conditions improve.

The carrying capacity decreases as the factors grow scarcer. When resources are depleted faster than they are replaced, the species has reached its carrying capacity.

Thus, in this way abiotic factors and biotic factors affect population growth.

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The fact that and allows you to conclude that the system is consistent because if two vectors are equal, then their components must be equal.

Since and , we can set up the following linear system of equations:

Ax = b

where A is the matrix and x and b are the vectors. Since we know that and , then the system of equations is consistent. This is because if two vectors are equal, then their components must be equal. Therefore, if and , then

a1x1 + a2x2 = b1

a3x1 + a4x2 = b2

Since the two vectors are equal, then the components must be equal as well, so

a1x1 + a2x2 = a3x1 + a4x2

which can be rearranged to

(a1 - a3)x1 + (a2 - a4)x2 = b1 - b2

Since b1 - b2 is a constant, then the coefficient of x1 and x2 must both be 0, meaning that the system is consistent.

The fact that and allows you to conclude that the system is consistent because if two vectors are equal, then their components must be equal. This means that if and , then we can rearrange the system of equations to get (a1 - a3)x1 + (a2 - a4)x2 = b1 - b2, which implies that the coefficients of x1 and x2 must both be 0, meaning that the system is consistent.

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F(x)=a(x−h)2+k , where (h,k) is the Vertex of the parabola. X=h is the axis of symmetry

A quadratic function’s Vertex Form is provided by: f(x)=a(xh)2+k, where (h,k) is the parabola’s Vertex. The axis of symmetry is x=h. To convert f(x) to Vertex Form, use the completing the square approach.

In vertex form, the quadratic equation is y = a(x-h)2+k. The axis of symmetry is defined as the point at where the vertex crosses the parabola (h, k). The x coordinate is denoted by h. The axis of symmetry is the vertical line that passes through the vertex of a parabola, ensuring that the left and right sides are symmetric. To put it simply, this line divides the graph of a quadratic equation into two mirror images.

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a) The equation with infinite number of solutions is 3x + 9 = 3x + 9

b) The equation with no solution is 3x + 9 = 3x

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

Substituting the values in the equation , we get

3x + 9 = 0   be equation (1)

Now , when the value of an equation is 0 , we can solve for x

Subtracting 9 on both sides of the equation , we get

3x = -9

Divide by 3 on both sides , we get

x = -3

So , the value of x = -3 and has only one solution

Now , when the coefficients and the variables of the equation are same on both sides , we get

3x + 9 = 3x + 9

So , 3x = 3x

x = x

And , the equation have infinite number of solutions

Now , when the value of the equation is not defined or does not equates , we get

3x + 9 = 3x

Subtracting 3x on both sides of the equation , we get

9 = 0

So , the equation is false and has no solutions

Hence , the equations are solved

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How can AA criterion be used to show that ΔPOR and ΔXYZ are similar: 2. Rotate the image of ΔPQR so that the image of side QP aligns with side YX and the image of side QR aligns with side YZ.

In Mathematics, the AA criterion states that the third angle is equal on two (2) similar triangles when two (2) angles of one of the triangle are respectively equal to the two (2) angles of the other triangle.

In Mathematics, a rotation can be defined as a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or an anticlockwise (counterclockwise) direction.

By applying AA criterion to triangle PQR (ΔPQR) and triangle XYZ (ΔXYZ), we have the following:

side QP ≅ side YX.

side QR ≅ side YZ.

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Answe:

Step-by-step explanation:

Answer:

[tex]6\frac{1}{5}[/tex] or [tex]\frac{31}{5}[/tex]

Step-by-step explanation:

Answer:

31/5 OR 6 1/5  (Depends if your teacher wants an improper fraction or a mixed number.)  

Step-by-step explanation:

First, rewrite the decimal as a fraction with 1 as the denominator.  

6.2 = 6.2/1

Nest, multiply to remove one decimal place so it's easier to solve.  

6.2 x 10 = 62 and 1 x 10 = 10 so... that would be 62/10

Find to GCF of both 62 and 10, or in other words something that they are both divisible by.  That would be 2, and now you can simplify the fraction further.  

62/2 = 31 and 10/2 = 5 so that would be 31/5, which is 6 1/5 as a mixed number.  

I hope this was useful!! :)

The proof of the trigonometry identity is shown below

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

Prove: (sec (x) + csc (x))^2 = 2 sec (x) csc (x) +(cot (x) + tan (x))^2.

So, we have

(sec (x) + csc (x))^2 = 2 sec (x) csc (x) +(cot (x) + tan (x))^2.

Open the bracket

sec^2(x) + csc^2(x) + 2 sec(x) csc(x) = 2 sec (x) csc (x) +(cot (x) + tan (x))^2.

Evaluate the like terms

sec^2(x) + csc^2(x) = (cot (x) + tan (x))^2.

Expand

sec^2(x) + csc^2(x) = cot^2(x) + tan^2(x) + 2

So, we have

sec^2(x) + csc^2(x) = cot^2(x) + 1 + tan^2(x) + 1

As a general rule:

tan^2(x) + 1 = sec^2(x)

csc^2(x) = cot^2(x) + 1

This means that

sec^2(x) + csc^2(x) = sec^2(x) + csc^2(x)

Hence, the identity has been proved

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In A bar graph, The Verical axis should start at 0 rather than 100 .

What does a bar graph explain?

Graphs are used to represent information in bar graphs. To represent value, bars that stretch to various heights are used. Bar graphs can be made using stacked bars, horizontal bars, vertical bars, or grouped bars (several bars that compare values in a category) (bars containing multiple types of information).

                            An easy way to compare several sets of data among various groups is with the aid of a bar graph. Two axes are used to depict the relationship, with the discrete values on one axis and the categories on the other.

The Verical axis should start at 0 rather than 100 .

Reason: For Accurate presentationa and comparision the verical axis should start at 0 rather than 100

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The probability of at most 4 threes appearing when 20 dice are tossed is 0.0756 and The probability of at most 2990 threes appearing when 12000 dice are tossed is 0.054.

a. The probability of at most 4 threes appearing when 20 dice are tossed can be calculated using the cumulative distribution function of the binomial distribution with parameters n = 20 and p = 1/4 (the probability of rolling a three on a single die). Using a binomial cumulative distribution calculator or a statistical software package, the result is approximately 0.0756.

b. The probability of at most 2990 threes appearing when 12000 dice are tossed can be calculated using the cumulative distribution function of the binomial distribution with parameters n = 12000 and p = 1/4. Using a binomial cumulative distribution calculator or a statistical software package, the result is approximately 0.0544.

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The relative extrema for the following function. f (x )equals 3 x minus x cubed is (1, 2)

Using the first derivative test, you may check for any sign changes of f′ around the critical points of a given function and discover relative extrema, or local maxima and minima.

A critical point must transition from rising, or positive f′, to decreasing, or vice versa, at that point for the function to be said to have local extrema.

So, start by determining the first derivative of f,

f(x) = 3x-x³

f'(x) = 3 - 3x²

To determine the function's critical points,

f' = 0

3 - 3x² = 0

3x² = 3

x = 1

Putting value of 1 in f(x) = 3x-x³ = 3-1 = 2

So the local maxima is (1,2)

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The polynomial in this problem is factored as follows:

x² - 12x + 27 = (x + 9)(x + 3).

The polynomial for this problem is defined as follows:

x² - 12x + 27.

The format to factor the polynomial is given as follows:

x^2+(a+b)x+ab=(x+a)(x+b)

Which means that:

The two numbers that satisfy this are:

-9 and -3.

As a product of it's linear factors, the polynomial is factored as follows:

(x + 9)(x + 3).

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Answer:

Step-by-step explanation:

d

Answer:

C. 6

Step-by-step explanation:

The range is the difference between the highest and lowest numbers so you do 8-2 which gets you 6.

14. A. The total area to be painted is 144 ft²

B. Total cost to paint wall is 12.8$

The shape of the wall to be painted is rectangular and to find out the area to be painted by subtracting the total area of the wall from the area of the door and the area of the window.

The total area of the walls painted is equal to X

Total wall area = T

door area is equal to D

The window area is equal to W

X= T - ( D+W)

X= (12*15) -((8*3) +(4*3))

X= 180-(24+12)

X=180-36

X= 144 ft²

Per 90ft² needs $8, so if you want to paint all 144ft² it will be

144 :90= 1.6

this 1.6 multiply with $8 =12.8$

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Answer:

  0.042563 km

Step-by-step explanation:

You want the distance 4256.3 cm expressed in km.

The prefixes centi- and kilo- refer to 1/100 and 1000, respectively.

  4256.3 cm = 4256.3 (1/100 m) = 42.563 m

Then 1 km = 1000 m, so 1 m = 1/1000 km, and we have ...

  42.563 m = 42.563 (1/1000 km) = 0.042563 km

Answer:

0.042563 km

Step-by-step explanation:

To convert centimeters to kilometers, divide the number of centimeters by 100,000.

So, 4256.3 cm = 4256.3 cm / 100,000 cm/km = 0.042563 km.

Answer:

$0.139/oz

$2.224/lb

Step-by-step explanation:

1 lb = 16 oz

10 oz × (1 lb)/(16 oz) = 0.625 lb

$1.39 for 10 oz of ketchup =

= $1.39 / 10 oz = $0.139/oz

= $1.39 / 0.625 lb = $2.224/lb

The mathematical expression, for the given mathematical phrase, 8 less than y times 28, is 8-28y

Expressions in math are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between.

Given that, a mathematical phrase, 8 less than y times 28,

Converting it into a mathematical expression,

8-28y

Hence, the mathematical expression, for the given mathematical phrase, 8 less than y times 28, is 8-28y

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The amount of money that is due after 4.5 years is equal to $7,754.4.

Mathematically, compound interest can be calculated by using this mathematical expression:

A(t) = P(1 + r/n)^{nt}

Where:

Substituting the given parameters into the compound interest formula, we have;

Future value, A(t) = 6,500(1 + 0.0396/2)^{2 × 4.5}

Future value, A(t) = 6,500(1 + 0.0198)^{9}

Future value, A(t) = 6,500(1.0198)^{9}

Future value, A(t) = 6,500(1.19298523508730964)

Future value, A(t) = $7,754.4.

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There are 97 people who ate pizza last Friday night.

The addition is one of the mathematical operations. The addition of two numbers results in the total amount of the combined value.

We have

184 = watched TV

183 = hung out with friends

37 = watched TV and ate pizza, but did not hang out with friends

31 = watched TV and hung out with friends, but did not eat pizza

40 = hung out with friends and ate pizza, but did not watch TV

29 = watched TV, hung out with friends, and ate pizza

74 = he did not do any of these three activities

We need to find out the number of people eating pizza by deducting the number of people not eating pizza from the total number.

So the people not eating pizza are listed as A, B, D, G

Total = 549

Those who only ate pizza= total - (A+B+D+G)

                                          = 549- (184+183+37+31)

                                          = 549- 415= 97

There are 97 people who ate pizza, last Friday night.

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The area of the lake is 252000 square meters.

In geometry, the dimensions of a rectangle refer to its length and width. The length of a rectangle is its longer side, while the width is its shorter side. The length and width of a rectangle can be thought of as the horizontal and vertical sides of the rectangle, respectively.

A rectangle can be specified by its dimensions in the form (length, width). For example, a rectangle with length equal to 5 units and width equal to 3 units can be written as (5, 3).

It's important to note that the length and width of a rectangle can be interchanged without changing the shape of the rectangle. In other words, a rectangle with dimensions (5, 3) is the same shape as a rectangle with dimensions (3, 5).

To determine the area of the lake, we first need to determine the size of the clay rectangle in terms of meters. We can do this by multiplying the dimensions of the rectangle on the map by 100 (since 1 centimeter on the map represents 100 meters in real life).

So, the dimensions of the clay rectangle in meters would be:

4.5 cm * 100 = 450 meters (width)

5.6 cm * 100 = 560 meters (length)

Next, we can find the area of the rectangle by multiplying its length by its width:

450 meters * 560 meters = 252000 square meters

Therefore, the area of the lake is 252000 square meters.

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[tex] \frac{1}{4} \sqrt{3} \: a {}^{2} [/tex]

Step-by-step explanation:

for equilateral triangle

99% of the time, the student council president knows the names of 598 to 1058 students at that high school. This interval is 1000. So the president's claim is correct.

The interval is the range of numbers between two given numbers of it and includes all real numbers between the two numbers.

A real number is any number you can think of, such as 7.53, 172, 5, -0.157. As an example that can explain the interval itself, if someone says they have at least 5 books but less than 10, those book counts are included in the interval.

An interval is a series of numbers that can be written using inequalities, number lines, or interval notation. However, there is a special way of indicating whether two given numbers are within an interval, called endpoints.

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The area of the trapezoid is 1.56 square units

The area of trapezoid is calculated using the formula

= 1/2 (sum of bases) * height

= 1/2 (a + b) * height

In the problem the dimensions made available are

height =

base (a) = 1

base (b) = 1 3/5

substituting the values into the formula

= 0.5 (1 + 1 3/5) * 1 1/5

= 0.5 * 13/5 * 1 1/5

= 1.56 square units

The area of the trapezoid is 1.56 square units

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Answer:

Step-by-step explanation:

they cause heavy rain fall no problom

Answer:

The limits lim csc(x) cot(x) and lim x→6 (x - 6) cos(x-6) can be solved using direct substitution.

The Squeeze Theorem is a way to solve limits without using direct substitution or other methods from the notes. This theorem states that given three functions f(x) ≤ g(x) ≤ h(x) and lim x→c f(x)=lim x→c h(x)=L, then it follows that lim x→c g(x)=L. This theorem is useful for situations where direct substitution is difficult or impossible to do. For example, the limit x→0 lim x cos(x) cannot be solved using direct substitution because the function cos(x) is undefined at x = 0. In this case, the Squeeze Theorem can be used to solve the limit. By defining a lower bound and an upper bound, we can use the Squeeze Theorem to conclude that lim x→0 lim x cos(x) = 1. The same is true for the limit x→0 lim x’ sin(1/x). By defining a lower bound and an upper bound, we can use the Squeeze Theorem to conclude that lim x→0 lim x’ sin(1/x) = 0.

The limit lim csc(x) cot(x) does not require the Squeeze Theorem and can be solved using direct substitution. By substituting x = 0 into both functions, we can see that the limit is equal to 1. Finally, the limit lim x→6 (x - 6) cos(x-6) does not require the Squeeze Theorem and can be solved using direct substitution. By substituting x = 6 into the function, we can see that the limit is equal to 0.

In conclusion, the limits lim x→0 lim x cos(x) and lim x→0 lim x’ sin(1/x) require the Squeeze Theorem and cannot be solved using direct substitution. The limits lim csc(x) cot(x) and lim x→6 (x - 6) cos(x-6) can be solved using direct substitution.

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Answer:

SAS

Step-by-step explanation:

2 sides if 20in. , the middle one is the same for both, and both have 22° angles

The percentage of data under the shaded region is 68%.

The 68 95 99 rule, commonly referred to as the empirical rule in statistics, asserts that given normal distributions, 68% of observed data points will fall within one standard deviation of the mean, 95% will fall within two standard deviations, and 99.7% will occur within three standard deviations.

From the given table the shaded region is one standard deviation away.

Therefore, By the empirical rule, the percentage of data underneath this normal curve is 68%.

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