What is the rule of proof?

The amount of lavender oil y is proportional to the amount of jojoba oil x

Two quantities are said to be linearly proportionate if change in one quantity produce proportionate amount of change in the other quantity, or the ratio of the two quantities is fixed.

Given,

In the first case, 20 drops of lavender oil and 8 teaspoons of jojoba oil.

In second case, 50 drops of lavender and 20 teaspoons of jojoba oil.

Ratio in first case:

x:y  =  20:8

      = 5:2

Ratio in second case:

x:y  =  50:20

      = 5:2

Since ratio is fixed in both the cases, the amount of lavender oil y is proportional to the amount of jojoba oil x.

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Prices of two ratios in the ratio is 4 : 1.

The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity.

Ratios can be classified into two types. One is part to part ratio and the other is part to whole ratio. The part-to-part ratio denotes how two distinct entities or groups are related.

For example, the ratio of boys to girls in a class is 12: 15, whereas, the part-to-whole ratio denotes the relationship between a specific group to a whole.

For example, out of every 10 people, 5 of them like to read books. Therefore, the part to the whole ratio is 5: 10, which means every 5 people from 10 people like to read books.

Let us assume price of radios are x and y

According to the question

[tex]\frac{x + 20}{y+20} =\frac{5}{2}\\2x+40=5y+100\\2x-5y=60[/tex]-------------------------- eq(1)

and

[tex]\frac{x-5}{y-5} =\frac{5}{1}\\5y-x = 20[/tex]

5y-x = 20

multiplying this equation by 2

we get

10y-2x = 40 ------------------ eq(2)

adding eq 1 and 2

we get

5y=100

y=20

solving further we get x=80

Thus the ratio is 4 : 1.

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The probability of advancing five spaces is 0.2307.

Given that, the odds in favor of advancing five spaces are 3/10.

Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.

The probability of advancing five spaces can be obtained as:

P =Odds/1+Odds

= [tex]\frac{\frac{3}{10} }{1+\frac{3}{10} }[/tex]

= [tex]\frac{\frac{3}{10} }{\frac{13}{10} }[/tex]

= 3/13

= 0.2307

Therefore, the probability of advancing five spaces is 0.2307.

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Answer: Equivalent fractions are two or more fractions that are all equal even though they have different numerators and denominators. For example, the fraction 1/2 is equivalent to (or the same as) 25/50 or 500/1000.

*(This is not my work)*

Step-by-step explanation:

The rate of change of the table is -500

The average rate of change of a function, graph, table or ordered pair is simply the slope of the relation

The function is represented by the table of values

The interval of the function are the x values

So, we make use of  x values

x = 0 and x = 2

Start by calculating f(0) and f(2)

From the table, we have

f(0) = 30000

f(2) = 29000

So, the average rate over these intervals is calculated as:

Average rate = [f(2) - f(0)]/[2 - 0]

Substitute known values

Average rate = [30000 - 29000]/[0 - 2]

Simplify

Average rate = -500

Hence, the rate of change is -500

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[tex]72x^{2}+3x+3[/tex]Answer:

Step-by-step explanation:

[tex]1. (4x^{2}+34x^{2} +34x^{2} )+(x+x+x)+3\\ 2. 72x^{2} +3x+3[/tex]

Answer:

Step-by-step explanation: How much is 1 t shirt

Each side of quadrilateral A changed by a factor of 1/4

When quadrilateral A was reduced the size of each angle remain unchanged

The dimensions of quadrilateral A are 16, 10, 6, 20

The dimensions of quadrilateral B are  4, 2.5, 1.5, 5

comparing the two dimensions

Let the factor be r

4r = 16

r = 16/4

r = 4

2.5r = 10

r = 10/2.5

r = 4

1.5r = 6

r = 6/1.5

r = 4

5r = 20

r = 20/5

r = 4

Therefore we can conclude that quadrilateral B was increased by 4, similarly, quadrilateral A was reduced by a factor of 1/4

16 * 1/4 = 4

10 * 1/4 = 2.5

6 * 1/4 = 1.5

20 * 1/4 = 5

since the reduction are similar quadrilateral A is similar to quadrilateral B

and the size of each angle were not affected

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The number that should be multiplied by 2/3 to get a product 1 is 3/2

What is the Reciprocal of two numbers?

One of two numbers that, when multiplied together, equals to 1 is known as a reciprocal or multiplicative inverse.

Transposing the numerator and denominator will allow you to get the reciprocal if you can reduce the number to a fraction.

Let the number that would result in product 1 be "x"

So, According to the question:

=> x × 2/3 = 1

Solving for x,

Dividing by 3/2 both sides

⇒x = 3/2 which is the reciprocal of 2/3

Hence,

The number that should be multiplied is 3/2

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Answer:

y=11x-5

Step-by-step explanation:

y=mx+b

28=11(3)+b

28=33+b

b=-5

y=11x-5

The lowest temperature with a z score of of -2.49 is 24.0 ˚F

Z scores, measures the amount of standard deviation sample X is from mean. the formula is below

z = (X - μ) / σ

Definition of variables

mean, μ = 4.4˚F

standard deviation, σ = 12.2 ˚F

X = lowest temperature

z = -2.49

substituting into the formula

-2.49 =  (X - μ) / σ

-2.49 = (X - 64.4) / 12.2

-2.49 * 12.2 = X - 64.4

-30.378 + 64.4 = X

X = 34.022

X = 34.0  (to one decimal place)

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The graph of exponential function y = 5.2^x

exponential function in the form y = ab^x + c

Exponential function:

An exponential function is a mathematical function of the following form: f ( x ) = a^x. where x is also a variable, and a is a constant called the base of the function.

where

a is the initial y - intercept.

b is the constant ratio.

c is the horizontal asymptote.

So, in this problem, we have:

constant ratio = 2

y-intercept at y = 5

horizontal asymptote at y = 0

The graph is in the image uploaded.

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Answer: See attached.

Step-by-step explanation:

        This slope-intercept form equation shows us that there is a y-intercept of -7. This means that (0, -7) will be a point on the line. Next, we have a slope of positive 3. This means we will count up three units, right one unit, plot a point. Then we repeat.

Answer:

start at the y-intercept, plot some points using the slope, then connect them using a straight line.

Step-by-step explanation:

Generally you'll have an equation in slope-intercept form, which is denoted as: [tex]y=mx+b[/tex]

This is a really useful form for a couple of reasons, and it has two key features. Firstly, the "m" represents the slope of the linear equation.

This makes sense the definition of slope is: [tex]\frac{\text{change in y}}{\text{change in x}}[/tex]

and we can generally define the points:

[tex]\text{at x = 1, y = }m(1)+b\implies m+b\\\text{at x = 2, y = } m(2)+b\implies m + m + b\\\text{at x = 3, y = }m(3) + b\implies m + m + m +b\\\text{and so on...}[/tex]

as the x value increases by 1, the change in y is the "m" value, we have one more "m" than the previous x-value, so the slope is the "m" value, which is the coefficient of the x variable.

The second key feature of slope-intercept form is the y-intercept, which is represented as "b" in this form. This also makes sense because the y-intercept is simply when the x-value is zero, and if you plug in x=0 into the equation, you're only left with the "b" value

Knowing these two things we can analyze the equation given:

[tex]y = 3x-7[/tex]

since it's given in slope-intercept form, we know that slope = 3, and y-intercept = -7. We immediately know one point, which is the y-intercept, since the x-value is going to be zero at the y-intercept, so one point on the graph is: (0, -7)

now from this point we can draw the graph by going to the right one, and up three, by definition of a slope (as x increases by 1, the y-value increases by 3).

We can do something similar on the left side, except instead of right one and up three, we go left one, and down three. You may be asking why down three? Well as you go right one, you go up three, this means if you compare the previous point to the new point, the previous point is down three compared to the new point (which is to the right one), so if you were to go back, you would be going left one and down three.

Now from these points you can do the same thing to extend the line a bit, and since it's a linear equation you can connect them using a straight line.

I also attached a graph from desmos to illustrate what I was saying a bit and also show the graph.

The hypotenuse leg theorem states that two right triangles are congruent if the hypotenuse and one leg of one right triangle are congruent to the other right triangle's hypotenuse and leg side.

What is the hypotenuse leg theorem?

The hypotenuse leg theorem states that two right triangles are congruent if the hypotenuse and one leg of one right triangle are congruent to the other right triangle's hypotenuse and leg side.

The HL Postulate states that if the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent.

Hypotenuse-Leg (HL) for Right Triangles. There is one case where SSA is valid, and that is when the angles are right angles. In words, if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.

Hence, if the hypotenuse leg theorem states that two right triangles are congruent if the hypotenuse and one leg of one right triangle are congruent to the other right triangle's hypotenuse and leg side.

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Multiply the fraction's bottom numbers with the top numbers in the two fractions.

The two fractions' top numbers should be multiplied, as should the bottom numbers.

Find the fraction of the following two fractions, for instance:

2/3, and 4/5

= 2/3 × 4/5

= (2×4)/(3×5)

= 8/15

Multiplication:

Mathematicians use multiplication to calculate the product of two or more numbers. It is a fundamental operation in mathematics that is frequently utilized in everyday life. When we need to combine groups of similar sizes, we utilize multiplication. The fundamental concept of repeatedly adding the same number is represented by the process of multiplication. The results of multiplying two or more numbers are known as the product of those numbers, and the factors that are multiplied are referred to as the factors. Repeated addition of the same number is made easier by multiplying the numbers.

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The degree of a polynomial in factored form is the number of zeros or roots of that polynomial.

Given:

Degree of a polynomial:

Degree is the highest or the greatest power of a variable in a polynomial equation.

ex : 2x^2 + 2xy^2

Degree = 3

Degree of a polynomial in factored form:

Degree of a polynomial in factored form is the number of zeros or roots of that polynomial.

Example:

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

number of roots = 3

Degree = 3

= (y-5)(y+3/2)(y-4)(y+1)

number of roots = 4

Degree = 4

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Answer:

Step-by-step explanation:

1) Subtract 5 from both sides.

[tex]m-5=-0.5d[/tex]

2) Divide both sides by -0.5.

[tex]-\frac{m-5}{0.5} =d[/tex]

3) Switch sides.

[tex]d=-\frac{m-5}{0.5}[/tex]

Thank you,

Eddie

Answer:

[tex]\textsf{D.} \quad 5^9=x[/tex]

Step-by-step explanation:

Given logarithmic equation:

[tex]\log_5x-\log_525=7[/tex]

[tex]\textsf{Apply the quotient log law}: \quad log_ax - \log_ay=\log_a \left(\dfrac{x}{y}\right)[/tex]

[tex]\implies \log_5\left(\dfrac{x}{25}\right)=7[/tex]

[tex]\textsf{Apply log law}: \quad \log_ab=c \iff a^c=b[/tex]

[tex]\implies 5^7=\dfrac{x}{25}[/tex]

Multiply both sides by 25:

[tex]\implies 25 \cdot 5^7=x[/tex]

Rewrite 5 as 5²:

[tex]\implies 5^2 \cdot 5^7=x[/tex]

[tex]\textsf{Apply exponent rule}: \quad a^b \cdot a^c=a^{b+c}[/tex]

[tex]\implies 5^{2+7}=x[/tex]

[tex]\implies 5^9=x[/tex]

All the inequalities that model the given situation are;

x + y ≥ 25

x > 10

y > 0

We are told that;

x represents the number of apples.

y represent the number of oranges.

Now, Mia needs to bring at least 25 pieces of fruit to the school picnic. Thus, the inequality is expressed as;

x + y ≥ 25

She is bringing both apples and oranges, and was told to be sure to bring more than 10 apples. Thus, the inequality is expressed as;

x > 10

Since we are not told the maximum number of oranges, then we can say it must be more than zero and as such we have the third inequality as; y > 0

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The number of boats needed for 1491 people are 20.

What is the cross-product?

The cross product, also known as the vector product in mathematics, is a binary operation on two vectors in a Euclidean vector space that is oriented in three dimensions. It is represented by the symbol times.

We have,
134 people decided to go scuba diving,
523 people decided to neither scuba dive nor go on a boat tour,

So, people go to boat tour = 2148 -(134 + 523)
= 2148 - 657
= 1491
now, since each boat holds 78 people.

[tex]\frac{1491}{78} = 19\frac{3}{26}[/tex]

= 20 boats.
Hence, the number of boats needed for 1491 people are 20.

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Complete Question
The 2148 guests at a hotel have the. the choice to either go on a boat tour or go scuba diving.  134 people decided to go scuba diving.  523 people decided to neither scuba dive nor go on a boat tour. Each boat holds 78 people. How many boats are needed? Explain your reasoning.

Answer:

According to the SAS ( Side Angle Side ) rule, two triangles are said to be congruent if any two sides and any angle between the sides of one triangle are equal to the equivalent two sides and angle between the sides of the other triangle.  

Step-by-step explanation:

Two figures are said to be "congruent" if they can be placed exactly over one another.

According to the SAS ( Side Angle Side ) rule, two triangles are said to be congruent if any two sides and any angle between the sides of one triangle are equal to the equivalent two sides and angle between the sides of the other triangle.  

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Answer: A

Step-by-step explanation:

The quotient is the result of the division between two numbers.

Answer:

n/3

Step-by-step explanation:

When you divide you get a quotient.

n divided by 3 will equal a quotient.

The function of piecewise graph shown in the figure is,

f(x) = {y = 6   for x < -5

        { [tex]y = -\frac{19}{9}x-\frac{5}{9}[/tex]    for -5 ≤x < 5

        { y = 8/5x -14   for x ≥ 5

What is piecewise function?

A piecewise-defined function in mathematics is a function with several sub functions, each of which has a certain interval in the domain for which it is applicable. Piecewise definition is a method of expressing the function, not a property of the function.

From graph we can see that the first function is a line y = 6

Hence, f(x) = {y = 6    for x < -5}

Now to find the equation of second function.

From graph we get two points (-5, 10) and (4, -9).

First to find the slope

[tex]m = \frac{-9-10}{4+5} = \frac{-19}{9}[/tex]

Now to find the equation of line, using slope point form

[tex]y - 10 = \frac{-19}{9} (x-(-5))\\y-10=-\frac{19}{9}x-\frac{95}{9} \\ y=-\frac{19}{9}x-\frac{95}{9} + 10\\y = -\frac{19}{9}x -\frac{5}{9}[/tex]

So, f(x) = { [tex]y= -\frac{19}{9}x-\frac{9}{5}[/tex],    for -5 ≤ x < 5}

Now to find the equation of third line.

From graph we get two points (5, -6) and (10, 2).

First to find the slope of the line

[tex]m = \frac{2-(-6)}{10-5} = \frac{8}{5}[/tex]

Now to find the  equation of line, using slope point form

[tex]y-(-6)=\frac{8}{5}(x-5)\\ y+6=\frac{8}{5}x-8\\ y = \frac{8}{5}x - 14[/tex]

So, f(x) = {[tex]y = \frac{8}{5}x-14[/tex],   for x ≥ 5}.

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With percent increase or decrease situations, you have to consider the starting point as being 100%.  So if the account increased by 27%, it's now 100% + 27% or 127% of it's original value.

As a decimal, 127% = 1.27.  So 1.27 is your multiplier in this example.

And 1.27 • 7600 = 9652, which is the ending balance.

This gets the same result as finding 27% of 7600 and adding it back on to 7600, but does it more efficiently.

Exponential decay graph looks like see the given image attached.

The process of diminishing an amount by a constant percentage rate over time is known in mathematics as exponential decay. The quantity drops gradually, followed by a dramatic decline in the rate of change and growth over time.

A linear function reduces the original number by the same amount each time, but exponential decay reduces the original amount by a constant percentage over time. Exponential decay differs from linear decay in this way: the decay factor depends on a percentage of the original quantity.

The exponential decay formula is used to determine this decline in growth.

Formula of exponential decay is given by

Its graph is given by

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Answer: Ad Hominum

Step-by-step explanation:

It just attacks the person.

1) The estimated probability of patrick flipping a head with his coin is; 0.45

2) The margin of error for Patrick's estimated probability is; 0.218

3) The confidence interval for the chance of flipping a head with his coin is; CI = (0.232, 0.668)

No, the 95% confidence interval does not contain 50% as a probability, so it will be illogical to think that Patrick's lucky coin has a probability different from 0.5

We are given;

Number of trials; n = 20

Number of successful outcomes; x = 9

Thus; Proportion; p^ = x/n = 0.45

Now, the formula for the confidence interval of proportions is;

CI = p^ ± z√(p^(1 - p^)/n)

where;

z√(p^(1 - p^)/n) is margin of error

1) Probability of Patrick flipping a head with his lucky coin is;

p^ = x/n = 9/20 = 0.45

2) The margin of error is from the formula;

z√(p^(1 - p^)/n)

z for 95% confidence level is 1.96. Thus'

E = 1.96√(0.45(1 - 0.45)/20)

E = 0.218

3) Confidence interval is;

CI = 0.45 ± 0.218

CI = (0.232, 0.668)

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On solving the given question, we got Area of the base prism = bh/2 = (4.819 X 51)/2 = 122.89 cm2

A prism is a polyhedron in geometry that has an n-sided polygonal basis, a second base that is a shifted version of the first base, and n additional faces—necessarily all parallelograms. Connect the matching sides of the two bases.

h = 51 cm

a = 2cm

therefore,

volume of a Triangular Prism = (½) abh cubic cm

245.8 = [tex]\frac{1}{2}[/tex] X 2 X b X 51

[tex]= > b = \frac{245.8 X 2}{2 X 51}[/tex]

[tex]= > b = \frac{491.6}{102}[/tex]

[tex]= > b = 4.819[/tex]

Now Area of base of prism

Area of the base prism = bh/2 = (4.819 X 51)/2 = 122.89 cm2

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yes

Step-by-step explanation:

5