if the coefficient a in the general quadratic equation is equal to 9, what is the focal length for
the parabola, c’,equal to?

Answer:

1. 15cm,

2. 22.5ft

3. 24in

4. 24.2m

5. 8.9m

Step-by-step explanation:

1. √(9²+12²) = 15cm

2. √(8²+21²) = 22.47 ft = 22.5ft

3. 7²+x²=25²

x =√(25²-7²) = 24 in

4. √(15²+19²) =24.2 m

5. √(4²+8²) = 8.9m

all the answers related to the pythogorus theorem.

a²+b²=c²

in which a and b are the legs of the right angled triangle where c is the hypotenuse.

note: be careful abt the units.

The central limit theorem's not needed when the original distribution is normal, the distribution of the sample mean is always normal in that case. So, option (a) is correct. If the sample size is reasonably large (for any population) is correct.

Given that,

The central limit theorem is important in statistics because it allows us to use the Normal distribution to find probabilities involving the sample mean,

In layman's words, the theorem asserts that regardless of the form of the initial population distribution, the sampling distribution of the mean tends to resemble a normal distribution as the sample size rises.

Statistics benefits from the Central Limit Theorem because it makes it safe to assume that the sampling distribution of the mean will be typically normal. As we will see in the next section, this means that we can benefit from statistical methods that presume a normal distribution.

The distribution of people's heights within a population is one illustration of how the central limit theorem is put to use. Even though the population's distribution of heights is not normal, when we sample this population, the distribution of the sample averages will be roughly normal.

Therefore,

The central limit theorem's not needed when the original distribution is normal, the distribution of the sample mean is always normal in that case. So, option (a) is correct. If the sample size is reasonably large (for any population) is correct.

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The value of y(0.4) is 1.04384484247985 using  Runge-Kutta method of fourth order .

What is  Runge-Kutta method of fourth order ?

The RK4 method, also known as the fourth-order Runge-Kutta method, is the Runge Kutta technique most frequently employed to get the answer to a differential equation. For a given point x, the Runge-Kutta technique returns an approximation of the value of y. The Runge Kutta RK4 method can only be used to solve first order ODEs.

The step size is h = 0.1

f(x,y) = (x²+y²)/10

Step 1:

x₁ = 0 + 0.1 = 0.1

k₁ =f(0,1) = 0.1

k₂ = f(1/20, 1.005) = 0.1012525

k₃ = f(1/20, 1.005062625) =0.101265088017189

k₄ = f(1/10, 1.01012650880172) = 0.103035556378395

y₁ = 1+ (1/10)/6 +(0.1 + 2⋅0.1012525 + 2⋅0.101265088017189 + 0.103035556378395) = 1.01013451220688

Step 2:

x₂ = x₁ + h = 1/5

k₁ =f(1/10,1.01013451220688)=0.103037173275143

k₂ = f(3/20,1.01528637087064) = 0.105330641487567

k₃ = f(3/20, 1.01540104428126) = 0.105353928072747

k₄ = f(1/5, 1.02066990501415) = 0.10817670550016

y₂ =1.01013451220688 + (1/10)/6 +(0.103037173275143 + 2⋅0.105330641487567 + 2⋅0.105353928072747 + 0.10817670550016) = 1.02067756250514

Step 3:

x₃ = x₂ + h = 3/10

k₁ =f(1/5, 1.02067756250514)=0.108178268660144

k₂ = f(1/4,1.02608647593815)=0.111535345610318

k₃ = f(1/4, 1.02625432978566)=0.111569794940382

k₄ = f(3/10, 1.03183454199918)=0.115468252206266

y₃ =1.02067756250514 + (1/10)/6 (0.108178268660144 + 2⋅0.111535345610318 + 2⋅0.111569794940382 + 0.115468252206266) = 1.03184184253794

Step 4:

x₄ = x₃ + h = 2/5

k₁ =f(3/10, 1.03184184253794) = 0.115469758801209

k₂ = f(7/20,1.037615330478)=0.119914557404297

k₃ = f(7/20, 1.03783757040816)=0.119960682255071

k₄ = f(2/5, 1.04383791076345)=0.1249597583947

y₄ = 1.03184184253794 + (1/10)/6 (0.115469758801209 + 2⋅0.119914557404297 + 2⋅0.119960682255071 + 0.1249597583947) = 1.04384484247985

y(0.4) ≈ 1.04384484247985

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According to Trigonometric Identity , tan(theta) = -5 / 13

Trigonometric Identities are equality statements that hold true for all values of the variables in the equation and that use trigonometry functions. The relationship between a triangle's side length and angle is expressed by a variety of specific trigonometric identities.

Sin, cos, and tan are the three fundamental trigonometric ratios. The reciprocals of sin, cos, and tan are represented by the three additional trigonometric ratios sec, cosec, and cot in trigonometry, respectively.

According to the given information

We know that

tan(theta) = 1/cot(theta)

                = 1 / -2.6

                = -1 / 2.6

                = -10 / 26

                = -5 / 13

According to Trigonometric Identity , tan(theta) = -5 / 13

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We know that the required value of y is -17 using the slope formula.

The slope or gradient of a line in mathematics is a numerical representation of the steepness and direction of the line.

The slope of a line can be used to gauge how steep it is.

The slope is theoretically determined as "rise over run" (change in y divided by change in x).

So, we have the slope 5/2.

The points are: (−6, y) and (4, 8)

The slope formula: m = y2 - y1/x2 - x1

Now, insert the values and calculate y as follows:

m = y2 - y1/x2 - x1

5/2 = 8 - y/4 + 6

5/2 = 8 - y/10

5/2 * 10 = 8 - y

25 = 8 - y

-y = 25 - 8

-y = 17

y = -17

Therefore, we know that the required value of y is -17 using the slope formula.

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Answer:x+19-x+10 is equal to 29.

Answer:

The answer is (x + 10)

Step-by-step explanation:

(9x + 6) - (8x - 4)

9x + 6 - 8x + 4

Rearrange it

9x - 8x + 6 + 4

Combine like terms

x + 10

Thus, The answer is (x + 10)

So, the ordered pair which lies on the graph of the function y = -2x + 1 is (3,-5)

We will use the substitution method and check for all the ordered pair which are given

If the ordered pair lies on the graph, then it should satisfy the function.

So,

Given,

y = -2x+1

First check for (0,-2)

x=0, y=-2

Substitute the values,

y = -2x+1

-2=-2(0)+1

-2=1

It is false.

Next check for (2,-2)

x=2, y=-2

Substitute the values,

y = -2x+1

-2=-2(2)+1

-2=-3

It is false.

Next check for (3,-5)

x=3, y=-5

Substitute the values,

y = -2x+1

-5=-2(3)+1

-5=-5

It is true.

Next check for (5,2)

x=5, y=2

Substitute the values,

y = -2x+1

2=-2(5)+1

2=9

It is false.

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The empirical rule describes a statistical distribution of data within three standard deviations of the mean on a normal distribution.

According to the empirical rule, also known as the three-sigma rule or 68-95-99.7 rule, practically all observed data for a normal distribution will lie within three standard deviations (denoted by σ ) of the mean or average (denoted by μ ).

In specifically, the empirical rule predicts that 68 percent of observations will fall within the first standard deviation, 95 percent will fall within the first two standard deviations, and 99.7 percent will fall within the first three standard deviations.

An approximate test for a distribution's "normality" is the empirical rule. The distribution may not be normal and may be skewed or follow another distribution if a large number of data points are outside the three standard deviation bounds.

The given question is incomplete so I have given the answer in general according to my knowledge.

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9 pounds of the $50 coffee should be used

A word problem is a few sentences describing a 'real-life' scenario where a problem needs to be solved by way of a mathematical calculation.

let's let p = the pounds of coffee we need.

50p + 80(20) = 6(p+ 20)

The total price of the $50 coffee is going to be the pounds x price per pound. The total amount of coffee is the number of pounds of each added together.

From the equation above 50p + 80(20) = 60(p+ 20)

expanding the brackets and collecting like terms

50p + 1600 = 6p + 120

50p - 6p = 1200 - 1600

44p= 400

p= 9

Therefore, 9 pounds of  $50 coffee should be used

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A 99% confidence interval for the true proportion of college students believing in ghosts is Option(c)  (0.4901, 0.6699)

According to the question,

A professor wanted to know what proportion of college students believe in ghosts.

For that he drawn a sample from the college.

Sample size : n = 200

Proportion of students that believed in ghosts in sample = 58%

=> p = 0.58

Proportion of students who does not believe in ghost : q = 1 - p

=> q = 0.42

We have to create a 99% confidence interval for a true proportion of college students believing in the ghosts

Formula of C.I = [tex]p \pm ( z_{\alpha} \sqrt{\frac{pq}{n}})[/tex]

Where p and q is sample proportion , z is value of CI at given α level of significance and n is sample size

C.I at 99% = 0.58 ± 2.576 √0.58×0.42/200

=> 0.58 ± 2.576 √0.001218

=>  0.58 ± 2.576 × 0.035

=> 0.58 ± 0.08958

C.I = ( 0.58 - 0.08958 , 0.58 + 0.08958)
=> C.I = (0.4901, 0.6699)

Hence, Option (c) is correct

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Answer:

41%

Step-by-step explanation:

The matrix resulting from the multiplication of matrices B and C is given as follows:

[-92, -730; 3350, 19648]

The matrices in this problem are defined as follows:

For the product of the matrices, the row of matrix B multiplies the column of matrix C.

Then the element at the first row and the first column of the product matrix is given as follows:

2 x 4 - 10 x 10 = -92

The element at the first row and the second column of the product matrix is given as follows:

2 x -25 - 10 x 68 =  -730.

The element at the second row and the first column of the product matrix is given as follows:

60 x 4 + 311 x 10 = 3350.

The element at the second row and the second column of the product matrix is given as follows:

60 x -25 + 311 x 68 =  19648.

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1. The required amount y after t years is as shown below:

When t=1,  y=74550

When t= 5, y = 88750

When t= 10, y = 106500

When t= 15, y = 124250

When t= 20, y = 142000

2. The required amount y after t years is as shown below:

When t=1, y = 73840

When t= 5, y = 86382.35607  

When t= 10, y = 105097.3442

When t= 15, y = 127866.988

When t= 20, y = 155569.743

What is the percentage?

It's the ratio of two integers stated as a fraction of a hundred parts. It is a metric for comparing two sets of data, and it is expressed as a percentage using the percent symbol.

1. x = 71000

  r =  3550

Use the below formula to find amount y after t years

y = x+( rt )

When t=1,   y= 71000 + ( 3550 * 1) = 74550

Similarly,  

When t= 5, y = 88750

When t= 10, y = 106500

When t= 15, y = 124250

When t= 20, y = 142000

2.  4% = 0.04

Use the below formula to find amount y after t years

y = x *( 1 + r )^ t

=> y = 71000*( 1.04)^t

When t=1,   y= 71000 * ( 1.04)^1 = 73840

Similarly,  

When t= 5, y = 86382.35607  

When t= 10, y = 105097.3442

When t= 15, y = 127866.988

When t= 20, y = 155569.743

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Answer:

See below and attached graph

Step-by-step explanation:

If you are using a graphing calculator it is a breeze. You write down the equation as it is and it should plot it, depending on what calculator you use.

If you are doing this manually here are the steps
We are given y - 2 = 3(x-2)

Expand the brackets on the right hand side
3(x - 2) = 3x - 6

So the original equation becomes
y - 2 = 3x - 6

Adding 2 to both sides
y - 2 + 2 = 3x -6 + 2

y = 3x - 4   [1]

This form of equation is known as a slope-intercept form of the line

We need two points to plot a line. This can be done by choosing a value for x, finding the y value which gives one point. Then repeat for another value of x to get y for that value giving another point. Draw a straight line through both points

Point 1
Let's choose x = 2 and plug this into the equation [1]
y = 3·2 - 4 = 6- 4 = 2.
So (2, 2) is one point on the graph

Point 2

Choose another value for x, say x = 4

y = 3·4 - 4

y = 12-4

y = 8

So (4, 8) is another point

Draw a line through these two points

Answer:

Y=6x-12

Step-by-step explanation:

whenever the X is at zero the number across from it is the Y-intercept. and to find the slope just count how much they are increasing or decreasing by.

Answer:

Step-by-step explanation:

a) 1012⁵ can be written as:

1012⁵ = 15¹² + 05¹¹ + 1*5¹º

= 5¹² + 5¹º

= 3125 + 625

= 3750

(b) 101⁵ can be written as:

101⁵ = 15⁴ + 05³ + 1*5²

= 5⁴ + 5²

= 625 + 25

= 650

(c) 1.1⁵ can be written as:

1.1⁵ = 15¹ + 15º

= 5¹ + 5º

= 25 + 5

= 30

(d) 10.21⁵ can be written as:

10.21⁵ = 15² + 05¹ + 25º + 15⁻¹

= 5² + 2*5º + 5⁻¹

= 25 + 10 + 0.2

= 35.2

(e) 210⁵ can be written as:

210⁵ = 25⁴ + 15²

= 2*625 + 25

= 1250 + 25

= 1275

The interest accrued in the first four years is $1.69 and a half years is $0.21.

What is interest rate?

The amount that the lender charges the borrower above and beyond the principal amount is referred to as the interest rate. A person who deposits money in a bank or other financial institution also earns additional income in terms of the recipient, known as interest, taking into account the time value of money.

Given:

Jeff took out a 15 year unsubsidized loan for $9.800 to attend four years of college.

The rate is 4.3%.

We have to find the interest accrued in the first four-and-a-half years.

First to find the interest for first four years.

p = $9.800,  r  = 4.3% = 0.043,  t = 4

⇒ I = P x r x t

   I = 9.800 x 0.043 x 4

   I = 1.6856

   I = $1.69

Now to find the interest for a half years.

p = $9.800 ,  r = 4.3% = 0.043,  t = 1/2

I = p x r x t

I = 9.800 x 0.043 x 1/2

I = 0.2107

I = $0.21

Hence, the interest accrued in the first four years is $1.69 and a half years is $0.21.

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The equation of the line  5x + y = -4 in slope intercept form is y = -5x - 4.

The equation of a line can be represented in different form such as slope intercept form, standard form, point slope form and general form.

Therefore, equation of a line can be represented in slope intercept form as follows:

y = mx + b

where

Therefore, the equation of the line is 5x + y = -4. Let's model it to the slope intercept form y = mx + b.

Hence,

5x + y = -4

subtract 5x from both side of the equation

5x - 5x + y = -5x - 4

Therefore, the equation of the line is  y = -5x - 4

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The probability that the other side of the card is not green is 60%

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

P(green) = 40%

Using the complement rule of probabilities, we have

P(not green) = 1 - P(green)

Substitute the known values in the above equation, so, we have the following representation

P(not green) = 1 - 40%

Evaluate

P(not green) = 60%

Hence, the probability is 60%

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Complete question

The probability of a green face is 40%. If you're shown a card with a green face showing what is the probability that the other side of the card is not green

Answer:

0.002

Step-by-step explanation:

The probability that the Islanders will win exactly zero out of six games in a series against the Rangers is 0.002. To find this probability, you need to use the binomial probability formula. This formula calculates the probability of a certain number of successes in a series of independent trials, given the probability of success in a single trial. In this case, the probability of success is 0.87, and the number of trials is 6. The number of successes is 0, because the Islanders are not expected to win any of the games. Plugging these values into the formula, we get:

P(exactly 0 successes) = (6 choose 0) * (0.87^0) * (1 - 0.87)^6

= 1 * 1 * (1 - 0.87)^6

= 0.002

Therefore, the probability that the Islanders will win exactly zero out of six games in a series against the Rangers is 0.002.

A: (f ÷ g)(x) = (3.6)⁻²ˣ⁺¹ is the solution of the function.

Functions are relationships that exist between a set of inputs and outputs. A function is an association of inputs, where each input is directly linked to one or more outputs. Range, codomain, and domain are all attributes of each function.

The given functions are:

f(x) = (3.6)ˣ⁺²

And g(x) = (3.6)³ˣ⁺¹

To find (f ÷ g)(x):

We take division of f(x) by g(x).

(f ÷ g)(x) = f(x) ÷ g(x)

(f ÷ g)(x) = (3.6)ˣ⁺² ÷  (3.6)³ˣ⁺¹

(f ÷ g)(x) = (3.6)ˣ⁺² ÷  (3.6)³ˣ⁺¹

(f ÷ g)(x) = (3.6)ˣ⁺² / (3.6)³ˣ⁺¹

Simplifying,

(f ÷ g)(x) = (3.6)ˣ (3.6)² / (3.6)³ˣ (3.6)¹

(f ÷ g)(x) = (3.6)ˣ⁻³ˣ (3.6)

(f ÷ g)(x) = (3.6)⁻²ˣ (3.6)

(f ÷ g)(x) = (3.6)⁻²ˣ⁺¹

Therefore, the answer is A:  (f ÷ g)(x) = (3.6)⁻²ˣ⁺¹

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Observing the nature of the provided equation and the degree of the polynomial , The answer is Cubic.

Sums of terms with the form  kxⁿ, where K is any number and N is a positive integer, are known as polynomials.

The highest degree of a polynomial's monomials (individual terms) with non-zero coefficients is referred to as the polynomial's degree in mathematics.

Expression : [tex]8x^{3}-1[/tex]

Degree of polynomial =3

The expression is a Cubic Expression.

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Observing the nature of the provided equation and the degree of the polynomial , The answer is Cubic.

Sums of terms with the form  kxⁿ, where K is any number and N is a positive integer, are known as polynomials.

The highest degree of a polynomial's monomials (individual terms) with non-zero coefficients is referred to as the polynomial's degree in mathematics.

Expression : 8x³ - 1

Degree of polynomial =3

The expression is a Cubic Expression.

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Step-by-step explanation:

for question 1, the change in y is the highest value takeaway the lowest value.

change in y = 10 - 1 = 9

change in x = 6 - (-3) = 9

9/9 = 1

therefore m = 1, this works for all the questions

It is reasonable to take the variance of the total income to be [tex]\sigma_x^2 + \sigma_y^2[/tex].

When two variables are added, we have that:

The variances for each variable are given as follows:

Then the standard deviation of the total income, which is the combination of the income of the husband and of the wife, is calculated as follows:

[tex]\sigma = \sqrt{\sigma_x^2 + \sigma_y^2}[/tex]

The variance is the square root of the variance, hence it is obtained as follows:

[tex](\sigma)^2 = (\sqrt{\sigma_x^2 + \sigma_y^2})^2 = \sigma_x^2 + \sigma_y^2[/tex]

Which means that it is reasonable to take the variance of the total income to be [tex]\sigma_x^2 + \sigma_y^2[/tex].

The problem asks if it is reasonable to take the variance of the total income to be [tex]\sigma_x^2 + \sigma_y^2[/tex].

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c. A greater proportion of adolescent girls with anorexia nervosa live in the South than live in the other regions when compared to the general US population.

Null Hypothesis, [tex]H_{0}[/tex] : All proportions are the same  

Alternative Hypothesis, [tex]H_{1}[/tex]: At least two of the proportions are not the same

The alternative and null are both population-based assertions. The reason for this is that the purpose of hypothesis testing is to draw conclusions about a population from a sample.

The null and alternative hypotheses are two competing claims that researchers weigh the evidence for and against using a statistical test:

Null hypothesis: There’s no effect on the population.

Alternative hypothesis: There’s an effect on the population.

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I would select company 2 because it has lower cost and tax amount for complete fencing job.

What is a linear equation?

A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. The variables in the preceding equation are y and x, and it is occasionally referred to as a "linear equation of two variables."

Let, f = Total feet of the fence

For company 1 :

Given that 25 $

eq : (f-1)x25 + 2500

 =  (100 -1)25 + 2500

 =  99 x 25 = 2500

 =  4975 $

For company 2 :

Given that 39 $

Now if company 1 charges 7 % taxes then,

4975 x 7%

=  4975 + 348.25

=  5323.25$

and for company 2

39 x 7%

=  39 + 2.73

=  41.93 $

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To determine if the mean reduction in blood pressure is different between the two groups, you can use a statistical test such as a two-sample t-test. This test compares the means of two groups and determines whether the difference between the means is statistically significant.

To conduct the t-test, you will need the following information:

Once you have this information, you can use a statistical software or calculator to perform the t-test and determine the p-value. The p-value is a measure of the probability that the difference between the means is due to chance.

If the p-value is less than the level of significance (5% in this case), then the difference between the means is statistically significant and you can conclude that there is a significant difference in the mean reduction in blood pressure between the two groups.

If the p-value is greater than the level of significance, then the difference between the means is not statistically significant and you cannot conclude that there is a difference in the mean reduction in blood pressure between the two groups.

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Answer:

Hope this helps!

The number of country A forces in country B decreased to approximately 34,000 in 2014 from a high of about 10,000 in 2008. X represents the number of years after January 2007, this means that the amount of funding was approximately $10.5 billion in the year 2.135 years after January 2007, or approximately September 2009.

Generally, To find the year in which the amount of funding was approximately $10.5 billion, we need to solve the equation:

f(x) = 10.5

Substituting the given function for f(x), we get:

-1.384x^{2} + 5.253x + 5.517 = 10.5

We can solve this quadratic equation by using the quadratic formula:

[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]

where a = -1.384, b = 5.253, and c = 5.517.

Substituting these values into the formula, we get:

[tex]x = \frac{5.253 \pm \sqrt{5.253^2 - 4(-1.384)(5.517)}}{2(-1.384)}[/tex]

Simplifying this expression gives us:

[tex]x = \frac{5.253 \pm \sqrt{27.747029}}{-2.768}[/tex]

Since x represents the number of years after January 2007, we are only interested in the positive solution. Therefore, we can ignore the negative solution.

[tex]x = \frac{5.253 + \sqrt{27.747029}}{-2.768}[/tex]

Simplifying this expression further gives us:

x = 2.135

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The equivalent expression for the given expression is 29/3.

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value for the variable.

The given mixed fraction [tex]9\frac{2}{3}[/tex].

There are equivalent improper fractions for all mixed numbers. By multiplying the integer by the denominator and then adding the numerator, they are transformed. The numerator will not change.

Convert mixed fraction to improper fraction as follows

Improper fraction = [(Numerator × Denominator)+Numerator]/Denominator

[(9×3)+2]/3

=[27+2]/3

= 29/3

Therefore, the equivalent fraction for the given mixed fraction is 29/3.

To learn more about an equivalent expression visit:

https://brainly.com/question/28170201.

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