Given: θ = 20° opposite = 45 Find: hypotenuse

Answer:

Parabolas

A quadratic function is a function that can be written in the form f(x)=ax2+bx+c

where a,b, and c are real numbers and a≠0

. This form is called the standard form of a quadratic function.

The graph of the quadratic function is a U-shaped curve is called a parabola.

The graph of the equation y=x2

, shown below, is a parabola. (Note that this is a quadratic function in standard form with a=1 and b=c=0

.)

In the graph, the highest or lowest point of a parabola is the vertex. The vertex of the graph of y=x2

is (0,0)

.

If a>0

in f(x)=ax2+bx+c, the parabola opens upward. In this case the vertex is the minimum, or lowest point, of the parabola. A large positive value of a makes a narrow parabola; a positive value of a which is close to 0

makes the parabola wide.

If a<0

in f(x)=ax2+bx+c, the parabola opens downward. In this case the vertex is the maximum, or highest point, of the parabola. Again, a large negative value of a

makes the parabola narrow; a value close to zero makes it wide.

For an equation in standard form, the value of c

gives the y

-intercept of the graph.

The line that passes through the vertex and divides the parabola into two symmetric parts is called the axis of symmetry.

The equation of the axis of symmetry for the graph of y=ax2+bx+c

, where a≠0, is x=−b2a

In all the above graphs, the axis of symmetry is the y

-axis, x=0. In the graphs below, the axis of symmetry is different (marked in red.) Note that c still gives the y

-intercept.

If you write a quadratic function like x=f(y)=ay2+by+c

, where x is a function of y (instead of a y as a function of x

), you get a parabola where the axis of symmetry is horizontal.

Note that in this case, c

is the x-intercept. If a is positive, the graph opens to the right; if a

is negative, the graph opens to the left.

Example:

Write the equation of the axis of symmetry, and find the coordinates of the vertex of the parabola y=−3x2−6x+4

.

The equation of the axis of symmetry for the graph of y=ax2+bx+c

.

x=−b2a

Substitute −3

for a and −5 for b

in the equation of the axis of symmetry.

x=−−62(−3)    =−1

So, the equation of the axis of symmetry is x=−1

.

Since the equation of the axis of symmetry is x=−1

and the vertex lies on the axis, the x-coordinate of the vertex is −1

.

To find the y

-coordinate of the vertex, first substitute −1 for x

in the given equation.

y=−3(−1)2−6(−1)+4

Simplify.

y=−3+6+4    =7

Therefore, the coordinates of the vertex of the parabola is (−1,7)

.

Step-by-step explanation:

Really hope this works!

Answer:

#4 x=70

Step-by-step explanation:

Because this is an Isosceles triangle, we know that the 2 base angles have to be equivalent. So subtract 40 from 180 which leaves you with 140, then divide 140 by 2 because there are 2 base angles. Then arrive at the answer of x=70.

-Hope this helped & Happy Holidays!

In linear equation, Samuel bought 7 bags of popcorn and 10 bags of candies.

What in mathematics is a linear equation?

A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept.

                              Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.

Let x represent the number of bags of popcorn and y represent the number of candies that Samuel bought.

She bought $43.75 worth of bags of popcorn and candies. Hence:

5x + 4.75y = 43.75  ...........  (1)

She bought a total of 17 bags of popcorn and candies altogether. Hence:

x + y = 9       (2)

Solving equations 1 and 2 simultaneously gives:

x = 4, y = 5

Therefore Samuel bought 4 bags of popcorn and 5 bags of candies.

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The graph that could be used to find the distance between the points

(-3, -13) and (11, 9) is given as follows:

B. graph of a right triangle with hypotenuse beginning at negative 3 comma negative 13 and ending at 11 comma 9.

What is distance between two points?

The distance between two points in a plane is the length of the line segment separating the two points. Typically, the equation

d=√((x2 - x1)^2 + (y2 - y1)^2) is used to calculate the distance between two points.

Suppose we have a pair of points, with the coordinates given as follows:

[tex](x_1, y_1)[/tex]  and  [tex](x_2, y_2)[/tex].

The two points form a right triangle, in which the sides are given as follows:

[tex]y_2-y_1\\x_2-x_1[/tex]

The distance between these two points is given by the hypotenuse of the right triangle.

Hence, applying the Pythagorean Theorem, the distance is obtained as follows:

[tex]d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}[/tex]

In this problem, the coordinates of the points are given as follows:

(−3, −13) and (11, 9).

Hence the correct option is given by option B.

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The values of a such that the linear equation 2 · a - 4 · x = 4 - a² · x have a solution in the interval (- 8, - 2) is a = 2 and - 7 / 4 ≤ a ≤ - 1.

In this problem we find the case of a linear equation whose values of variable a must be determined in terms of an interval of values of variable x. First, write the defined equation:

2 · a - 4 · x = 4 - a² · x

Second, form the standard form of a quadratic equation in terms of a:

x · a² + 2 · a - 4 · (x + 1) = 0

Third, use the quadratic formula:

a = - 2 / (2 · x) ± [1 / (2 · x)] · √[4 - 4 · x · [- 4 · (x + 1)]]

a = - 1 / x ± (1 / x) · √[1 + 4 · x · (x + 1)]

a = - 1 / x ± (1 / x) · √(4 · x² + 4 · x + 1)

There are two cases for the given interval:

Case 1

a = - 1 / x + (1 / x) · √(4 · x² + 4 · x + 1)

x = - 8

a = - 1 / (- 8) + [1 / (- 8)] · √[4 · (- 8)² + 4 · (- 8) + 1]

a = - 7 / 4

a = - 1 / (- 2) + [1 / (- 2)] · √[4 · (- 2)² + 4 · (- 2) + 1]

a = - 1

Case 2

a = - 1 / x - (1 / x) · √(4 · x² + 4 · x + 1)

x = - 8

a = - 1 / (- 8) - [1 / (- 8)] · √[4 · (- 8)² + 4 · (- 8) + 1]

a = 2

a = - 1 / (- 2) - [1 / (- 2)] · √[4 · (- 2)² + 4 · (- 2) + 1]

a = 2

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The exponential function passes through the points (-1,4/5) and (1,20) will be y = 4 × [tex]5^{x}[/tex].

In mathematics, an exponential function is a relationship of the type y = ax, where x is an independent variable that spans the entire real number line and is expressed as the exponent of a positive number.

In other meaning, an exponential function is that function above which a variable x has imposed.

Suppose the exponential function as

y = a[tex]e^{bx}[/tex]

Substitute, (-1,4/5)

4/5 = a[tex]e^{-b}[/tex]                             (1)

Substitute, (1,20)

20 = a[tex]e^{b}[/tex]                                (2)

Multiply (1) with (2)

(4/5)20 = a[tex]e^{-b}[/tex] × a[tex]e^{b}[/tex]

4 × 4 = a²

a = 4

Substitute, a = 4 into (2)

20 = 4[tex]e^{b}[/tex]    

[tex]e^{b}[/tex] = 5

Take log on both sides of the above equation,

ln [tex]e^{b}[/tex]  =  ln 5

b = ln 5

Substitute a = 4 and b = ln 5 into y = a[tex]e^{bx}[/tex]

y = 4 e^(x(ln 5)) = 4 e^(ln5^x)

y = 4 × [tex]5^{x}[/tex]

Hence "The exponential function passes through the points (-1,4/5) and (1,20) will be y = 4 × [tex]5^{x}[/tex]".

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340 ft high

d = 400' distance from building to tower

26° from the window to the top of the tower and 20° from the window to the bottom of the tower

Tan 26 = h2/d  

Tan 20 = h1/d  

h2 = 400Tan26

h1 = 400Tan20

on solving the equations,

h=a+b

Tower = 340 ft high

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The most effectively provide the desired primary data is experimentation

Experimentation is the step in the scientific method that helps people decide between two or more competing explanations—or hypotheses. These hypotheses suggest reasons to explain a phenomenon or predict the results of an action.

With an experiment, you run a test and see what the results are. If you don't get good results, you can try another option, and run another test. Then you can see what the outcomes of the choices are (the info you didn't have when first thinking about the decision), and you can make a better-informed decision now.

According to the question,

A research department wants to test the hypothesis that caffeine from organic coffee more safely stimulates mental alertness than caffeine from nonorganic coffee. Experimentation, one form of primary research, can produce data suggesting cause and effect

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Answer:

a.) The population is years measured after 2000 so 2000 is 0. Any number to the 0 power is 1 so the population is 18.5 (million).

b.) after 8.4 years, so 2008

Step-by-step explanation:

b.) 26.6 = [tex]18.5e^{0.1708t}[/tex]

1.43783784 ≈ [tex]e^{0.1708t}[/tex]

ln 1.43783784= 0.1708t

t=8.4... years

[tex]\textit{internal division of a line segment using ratios} \\\\\\ A(-8,8)\qquad B(-2,-10)\qquad \qquad \stackrel{\textit{ratio from A to B}}{1:2} \\\\\\ \cfrac{A\underline{C}}{\underline{C} B} = \cfrac{1}{2}\implies \cfrac{A}{B} = \cfrac{1}{2}\implies 2A=1B\implies 2(-8,8)=1(-2,-10)[/tex]

[tex](\stackrel{x}{-16}~~,~~ \stackrel{y}{16})=(\stackrel{x}{-2}~~,~~ \stackrel{y}{-10}) \implies C=\underset{\textit{sum of the ratios}}{\left( \cfrac{\stackrel{\textit{sum of x's}}{-16 -2}}{1+2}~~,~~\cfrac{\stackrel{\textit{sum of y's}}{16 -10}}{1+2} \right)} \\\\\\ C=\left( \cfrac{ -18 }{ 3 }~~,~~\cfrac{ 6}{ 3 } \right)\implies {\Large \begin{array}{llll} C=(-6~~,~~2) \end{array}}[/tex]

Answer:

slope = [tex]\frac{5}{4}[/tex]

Step-by-step explanation:

calculate the slope m using the slope formula

m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]

with (x₁, y₁ ) = (0, 5 ) ← coordinates of y- intercept

and (x₂, y₂ ) = (- 4, 0 ) ← coordinates of x- intercept

m = [tex]\frac{0-5}{-4-0}[/tex] = [tex]\frac{-5}{-4}[/tex] = [tex]\frac{5}{4}[/tex]

well, since we know there are 180° in π/2 radians, then, what's 5π/7 radians?

[tex]\begin{array}{ccll} degrees&radians\\ \cline{1-2} 180&\frac{\pi }{2}\\[1em] d&\frac{5\pi }{7} \end{array}\implies \cfrac{180}{d}~~ = ~~\cfrac{\frac{\pi }{2}}{ ~~ \frac{5\pi }{7} ~~ }\implies \cfrac{180}{d}=\cfrac{\pi }{2}\cdot \cfrac{7}{5\pi } \\\\\\ \cfrac{180}{d}=\cfrac{7}{10}\implies 1800=7d\implies \cfrac{1800}{7}=d\implies 257.14^o\approx d[/tex]

Answer:

Step-by-step explanation:

Write it out

[tex](B+6)/4[/tex]

plug in the value

[tex](1.5+6)/4[/tex]

solve

[tex](1.5+6)/4\\7.5/4\\= 1.875[/tex]

The constant of proportionality will be 9.5 when the table displays the proportionate cost for a particular quantity of golf ball buckets at a driving range.

Given that,

The table displays the proportionate cost for a particular quantity of golf ball buckets at a driving range.

We have to find the constant of proportionality.

We know that,

A collection of consecutively ordered numbers a and b expressed as a/b, where b is never equal to zero, is referred to as a ratio. A statement is considered proportionate when there is equality between two items.

The table displays the proportional relationship between the price for a particular quantity of golf ball cans at a driving range.

Buckets                         3               5             7

Price (dollars)            28.50       47.50      66.50

The constant of proportionality is calculated as,

⇒ 25.8 / 3

⇒ 9.5

Therefore, The constant of proportionality will be 9.5 when the table displays the proportionate cost for a particular quantity of golf ball buckets at a driving range.

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Answer:

9.5

Step-by-step explanation:

I took the exam and got it right

Answer:

x = 10

Step-by-step explanation:

Given equation,

→ -24 - x = -34

Now the value of x will be,

→ -24 - x = -34

→ -x = -34 + 24

→ -x = -10

→ [ x = 10 ]

Hence, the value of x is 10.

Answer: x=10

1. Rearrange the terms

-24 - x= -34

-x - 24= -34

2. Add 24 to both sides

-x - 24= -34

-x - 24= -34

+24 +24

3. Simplify(add the numbers)

-x=-10

4. Divide both sides by the same factor

-x=-10

-x/-1 = -10/-1

Simplify (cancel terms, divide numbers)

X=10

Step-by-step explanation:

The distribution is bimodal. -  this is a correct statement.

According to the question, the following distribution is 5, 5, 8, 9, 10, 10, and 11.

        In the above distribution, there are two modes i,e 5 and 10.

                 Median:  Middle value

                In the above distribution, the median is - 9.

        Mean =  [tex]\frac{5 + 5 + 8 + 9 + 10 + 10 + 11}{7}[/tex]  = [tex]\frac{58}{7}[/tex] = 8.28

Hence,

The distribution is bimodal. -  this is a correct statement.

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Each person got 16 fish equally divided among 7 fishermen.

A mathematical statement expressed as a string of numbers and unknowable variables is known as a numerical expression. Statements can be used to create numerical expressions.

Given, One fisherman and 6 his friends were fishing.

So, The total no. of people is (6 + 1) = 7.

one decided to go home so he took 48 fish which was 3/7 of all fish in the box.  

Let, 'x' be the total no. of fish.

Therefore, (3/7)x = 48.

x = 48×(7/3).

x = 112.

Now, by dividing the total no. of fish by total no. of fishermen

we get, (112/7).

= 16.

So, Each person got 16 fish.

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The age of the bones when they were discovered is 13439.66 years

What is half-life of an element?

The half-life of a radioactive isotope is the amount of time it takes for one-half of the radioactive isotope to decay. The half-life of a specific radioactive isotope is constant; it is unaffected by conditions and is independent of the initial amount of that isotope.

Half-life (symbol t½) is the time required for a quantity (of substance) to reduce to half of its initial value

The decay constant λ is = 0.693/t½

where t½ is the half-life of the element

Given data ,

The radioactive element​ carbon-14 has a​ half-life of 5750 years

So , the value of t½ = 5750 years

Now , the decay constant λ = 0.693 / t½

Substituting the values in the equation , we get

Decay constant λ = 0.693 / t½

Decay constant λ = 0.693 / 5750

Decay constant λ = 1.205 x 10⁻⁴

N=N₀e ^ −λt

where N₀ = Mass of radioactive element initially

N = Mass of radioactive element after time t

Apply natural logarithm on both the sides of the equation , we get

ln N = ln N₀ - λt

Let N0 be 100 then N will be 100 - 80.2 = 19.8

Substituting the values in the equation , we get

ln 19.8 = ln 100 -  1.205 x 10⁻⁴ ( t )

ln [ 19.8 / 100 ] = - 1.205 x 10⁻⁴ ( t )

-1.61948 = -1.205 x 10⁻⁴ ( t )

Therefore , the value of t is

t = -1.61948 / -1.205 x 10⁻⁴

t = 1.343966 x 10⁴

t = 13439.66 years

Therefore , the value of t is 13439.66 years

Hence ,

The age of the bones when they were discovered is 13439.66 years

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With the use of regression analysis, you may determine which of the independent factors actually has an effect statistically and learn how the dependent variable changes when one of the independent variables changes.

Regression analysis is used in statistical modeling to determine the associations between two or more variables:

The fundamental element you are attempting to comprehend and anticipate is the dependent variable. The elements that could have an impact on the dependent variable are known as independent variables.

Regression analysis enables you to statistically ascertain which of the independent variables actually has an impact and explains how the dependent variable changes when one of the independent variables changes. There is a difference between a simple and multiple linear regression in statistics. A dependent variable and one independent variable are modeled using a linear function in simple linear regression. Multiple linear regression is used when there are two or more explanatory variables used to predict the dependent variable.

Here,

estimate a simple regression in excel:

On the Data tab, in the Analysis group, click the Data Analysis button.

Select Regression and click OK.

In the Regression dialog box, configure the following settings: Select the Input Y Range, which is your dependent variable.

Click OK and observe the regression analysis output created by Excel.

Regression analysis illustrates how the dependent variable changes when one of the independent factors changes and allows you to statistically determine which of the independent variables actually has an impact.

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The following statements are true based on the frequency table:

Twice as many students prefer green than purple

And

Two more students prefer blue than purple

What is frequency table ?

The information that is gathered is called data. The quantity of occurrences of an event or value is referred to as its frequency. A frequency table is a table that lists items and displays how frequently each item happens.

From the frequency table,

Red= 6

yellow = 2

Blue = 5

Purple = 3

Green = 6

We can see that "Twice as many students prefer green than purple"

And

Two more students prefer blue than purple

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The required average rate of change in the amount of Kendall's account is $50 per month.

Rate is a measurement to determine the change in one quantity with respect to another quantity.

Given that,

The amount of the end of the 8th month in Kendall's account = $450,

And at the end of the 11th month, the saved amount = $600.

To find the rate of change in saving amount

The increased amount = 600 - 450 = 150.

Since, In 3 months the amount increased by 150.

So, the average rate of change in amount

= 150 / 3.

= 50.

The average rate of change in the amount is $50 per month.

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Answer:1400

Step-by-step explanation:

The slope of the line perpendicular to 4x - 6y = - 60 is (-3/2).

The slope is the rate of change of the y-axis with respect to the x-axis.

The equation of a line in slope-intercept form is

y = mx + b, where

slope = m and y-intercept = b.

We know the greater the absolute value of a slope is the more steeper is it's graph or rate of change is large.

We know lines perpendicular to each other have slopes that are negative reciprocals of each other.

Given line is 4x - 6y = - 60 ⇒ - 6y = - 4x - 60 ⇒ y = (2/3)x + 10.

Now the line perpendicular to it will have a slope of (-3/2).

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A confidence interval that will capture the true population mean with 95% confidence. The value is ( 23.6459    , 24.8293 ) .

What is population mean?

A population in statistics is a group of comparable objects or occurrences that are relevant to a particular topic or experiment. A statistical population can be a collection of real things or a hypothetical, possibly limitless collection of objects derived from experience.

Data: 18

for given data

[tex]$$\begin{aligned}\bar{X} & =\frac{\sum X i}{n} \\& =2448 / 101 \\& =24.2376\end{aligned}$$[/tex]

as population variance is unknown so we used sample variance

[tex]$$\begin{aligned}S^2 & =\frac{1}{n-1}\left(\sum X i^2-n \times \bar{X}^2\right) \\S^2 & =\frac{1}{100}\left(60232-101 \times 24.2376^2\right) \\& =8.9830\end{aligned}$$[/tex]

Now

[tex]\frac{\bar{X}-\mu}{S / \sqrt{n}} \sim t_{n-1}$$[/tex]

hence 95 % of population mean is

As n=101 which is large so critical value is 1.98397

[tex]$$\begin{aligned}& =\left(\bar{X}-\sqrt{\frac{S}{n}} \times t_{(n-1, \alpha / 2)}, \bar{X}+\sqrt{\frac{S}{n}} \times t_{(n-1, \alpha / 2)}\right) \\& =\left(24.2376-\sqrt{\frac{8.9830}{101}} \times 1.98397,24.2376+\sqrt{\frac{8.9830}{101}} \times 1.98397\right) \\& =(23.6459,24.8293)\end{aligned}$$[/tex]

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a) The minimum number of dogs that could have a mass of more than 24 kg is 6.

b) The maximum number of dogs that could have a mass of more than 18 kg is 20.

We will first of all have to rearrange the function with the aid of fundamental algebraic concepts that makes us calculate the value of x in the event that the derivative is equal to 0.

Now, the response above will give us the x-coordinate of the function's vertex. This vertex x-coordinate is the point at which  the maximum or minimum will occur. To find the minimum or maximum, we will start by rewriting the solution into the original function.

From the given figures in the question, it is clear that the minimum of the values present in the interval 20 to 40 is 6

From the given figures in the question, it is clear that the maximum of the values present in the interval 20 to 40 is;

(12 + 6) = 18

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The relationship between the cone cylinder and sphere is sphere takes up two-thirds of the volume of the cylinder, the cone takes up one-third of the volume of the cylinder.

Answers of blank :

Blank 1 = 1/3

Blank 2 = Three

Blank 3 = Two

Blank 4 = 2/3

Blank 5 = 1/3

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A small group communication occurs when three or more people work together interdependently for the purpose of accomplishing a task.

Small Group Communication

Small group communication is the term used to describe face-to-face conversation between more than two people. This kind of internal communication exists.

A small group is typically thought of as a collection of three or less people, with a maximum number of twelve to fifteen. A group would not be considered tiny if it had more than fifteen members or if it had just two.

Since almost all businesses and organisations employ people in small squads, there are advantages or even strengths to small group communication. When compared to one individual working alone on a project, it has been found to be more appropriate and produce better results.

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2 * V is equal to 2 v you divide 2 from 8 and your answer is 4

Answer: D

Step-by-step explanation:

B = -6.3

F = -5

M = -4.5

A.) -6.3 is not greater than -4.5

B.) -5 is not less than -6.3

C.) -4.5 is not less than -5

D.) -5 is less than -4.5