What is a solution to the system of equations that includes quadratic function f(x) and linear function g(x)? f(x) = 5x2 x 3

The equation of the directrix is x = -3

A mirror-symmetrical, roughly U-shaped plane curve known as a parabola. It corresponds to a number of mathematical models that appear to be superficially dissimilar but yet all define the exact same curves. A point and a line are one way to describe a parabola.

Given the general equation of a parabola expressed as

(y-y0)² = 4a(x-x0) ... 1

Equation of the directrix will be expressed as x = x0 - a

From the equation given;

y² = 12x ... 2

Comparing 1 and 2;

x - x0 = x

-x0 = x - x

-x0 = 0

x0 = 0

Similarly;

y-y0 = y

-y0 = y - y

-y0 = 0

y0 = 0

Also;

4a = 12

a = 12/4

a = 3

Substituting x0 =0, y0 = 0 and a = 3 into the equation of the directrix above, we will have;

x = x0 - a

x = 0 - 3

x = -3

Hence,

the equation of the directrix is x = -3

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[tex] {\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]

Here we go ~

[tex]\qquad \sf  \dashrightarrow \: \cfrac{ {z}^{2} - 14z + 48 }{ {z}^{2} + 6x - 27} ÷ \cfrac{z -8}{z-3} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{ {z}^{2} - 8z - 6z+ 48 }{ {z}^{2} + 9z- 3z- 27} ÷ \cfrac{z -8}{z-3} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{ {z(}^{} z- 8) - 6(z - 8) }{ {z}^{} (z+ 9)- 3(z + 9)} ÷ \cfrac{z -8}{z-3} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{ {(}^{} z- 8) (z - 6) }{ {}^{} (z+ 9)(z - 3)} ÷ \cfrac{z -8}{z-3} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \cfrac{ (z - 6) }{ {}^{} (z+ 9)}[/tex]

Break into two parts and simplify

[tex]\\ \rm\dashrightarrow \dfrac{z^2-14z+48}{z^2+6z-27}[/tex]

[tex]\\ \rm\dashrightarrow \dfrac{z^2-8z-6z+48}{z^2+9z-3z-27}[/tex]

[tex]\\ \rm\dashrightarrow \dfrac{(z-8)(z-6)}{(z+9)(z-3)}[/tex]

Now divide by the denominator

[tex]\\ \rm\dashrightarrow \dfrac{(z-8)(z-6)(z-3)}{(z+9)(z-3)(z-8)}[/tex]

[tex]\\ \rm\dashrightarrow \dfrac{z-6}{z+9}[/tex]

2 x 10² kcal are in the serving report your answer to 2 significant figures.

According to the question

A serving of fish contains 50 g of protein and 4. 0 g of fat.

i.e. content of protein = 50 g

Content of fat = 4 g

kcal are in the serving report your answer to 2 significant figures:

As the serving of fish contains 50g of protein that is 4.0kcal/g

Calories in 50 g protein = (Content of protein) × (caloric value in protein)

50g × (4.0kcal/g) = 200kcal

That means the kcal of the serving is 200kcal

In 2 significant figures: 2 x 10² kcal

Hence,

2 x 10² kcal are in the serving report your answer to 2 significant figures.

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Proved that the cofunction identity sec([tex]\frac{\pi }{2}[/tex]) - u = csc(u)

We have to prove that the cofunction identity using the addition and subtraction formulas.

sec([tex]\frac{\pi }{2}[/tex]) - u = csc(u)

We can prove this by using the identities given below:

[tex]sec(u)=\frac{1}{cos(u)}[/tex]

[tex]\frac{1}{sin(u)} =csc(u)[/tex]

cos(a-b) = cos a cos b + sin a sin b

Now the explanation,

[tex]sec(\frac{\pi }{2} -u) = csc(u)[/tex]

By using trignometric identities,

[tex]cos(u)=\frac{1}{sec(u)}[/tex]   ∴[tex]sec(u)=\frac{1}{cos(u)}[/tex]

So,

[tex]\frac{1}{cos(\frac{\pi }{2}-u) } =csc(u)[/tex]

By substituting the given identities we get,

  [tex]\frac{1}{cos(\frac{\pi }{2})cos(u)+sin(\frac{\pi }{2} )sin(u) }[/tex]

= [tex]\frac{1}{0.cos(u)+(1).sin(u)}[/tex]

=[tex]\frac{1}{sin(u)}[/tex]

= csc(u)

csc(u) = csc(u)

Here we proved that the cofunction identity sec([tex]\frac{\pi }{2}[/tex]-u) = csc(u)

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Answer:

The answer is 314

Step-by-step explanation:

You need to do

3.14(pie) times the radius times the height divided by 3

Answer:

C. 314 [tex]in^{3}[/tex]

Step-by-step explanation:

The volume of a cone is: 1/3[tex]\pi[/tex][tex]r^{2}[/tex]h'

1/3 x 3.14 x [tex]5^{2}[/tex] x 12

1/3 x 3.14 x 25 x 12

1/3 x 3.14 x 300

100 x 3.14

V = 314

If the speed of the ball after kicking is 48 feet per second then the ball will return on the ground after 3 seconds.

Given that the speed of the ball after kicking is 48 feet per second and the function that represents the height of the ball is [tex]-16t^{2} +48t[/tex].

We are required to find the time that the ball took to travel before returning the ground.

We know that speed is the distance a thing covers in a particular time period.

The height of the ball after t seconds is as follows:

h(t)=[tex]-16t^{2} +48t[/tex]

It is at ground at the instants of t.

Hence,

[tex]-16t^{2} +48t[/tex]=0

-16t(t-3)=0

We want value of t different of 0, hence :

t-3=0

t=3.

Hence if the speed of the ball after kicking is 48 feet per second then the ball will return on the ground after 3 seconds.

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Answer:

B. t=3

Step-by-step explanation:

i took the test and got it right

The expression for the volume of the pyramid in terms of the radius of the cone is  [tex]V=\pi r^{3}[/tex].

The expression for the volume of the pyramid in terms of the radius of the cone:

B= base area of the cone.

[tex]V=\pi r^{2}[/tex]

The volume of the Pyramid,

[tex]V=\frac{1}{3} *B*height[/tex]

[tex]$\Rightarrow V=\frac{1}{3} \times \pi r^{2} \times 3 \gamma$[/tex]

[tex]V=\pi r^{3}[/tex]

The expression for the volume of the pyramid in terms of the radius of the cone is [tex]V=\pi r^{3}[/tex]

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Answer:

A

Step-by-step explanation:

the means is less than the median

Answer:

$0.03

Step-by-step explanation:

2.31/11 =.21

2.70/15 =.18

.21-.18 = .03

from 2007 to 2012 is only 5 years, so we can see this as a compound interest with a rate of 1.2% per annum for 5 years, so

[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$113505\\ r=rate\to 1.2\%\to \frac{1.2}{100}\dotfill &0.012\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{per year, thus once} \end{array}\dotfill &1\\ t=years\dotfill &5 \end{cases} \\\\\\ A=113505\left(1+\frac{0.012}{1}\right)^{1\cdot 5}\implies A=113505(1.012)^5\implies A\approx 120481[/tex]

The probability that the service time will be more than three minutes is 0.3012.

Given that the average wait time at the Rivertown bank's neighborhood branch is 2.5 minutes and has an exponential distribution,

The time it takes to succeed in a continuous sequence of independent trials is described by the exponential distribution, which is a continuous probability distribution.

At the River Town Bank's neighborhood branch, the random variable X service time has an exponential distribution with a mean value of 2.5 minutes. Therefore,

[tex]f(x)=\left\{\begin{array}{ll}\theta e^{-\theta x},& \theta \geq 0,0\leq x\leq \infty\\0&\text{otherwise}\end{array}[/tex]

E(x)=1/θ

θ=1/2.5

θ=0.4

The probability that the service time will be longer than three minutes:

[tex]\begin{aligned}P(X > 3)&=1-P(X\leq 3)\\ &=1-(1-e^{-0.4\times 3}\\ &=e^{-1.2}\\ &=0.3012\end[/tex]

Since the service duration has an exponential distribution and a mean of 2.5 minutes, the needed chance that it will exceed 3 minutes is 0.3012.

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The probability of drawing a red card from a standard deck of cards is 1/2.

Given that there is a standard deck of cards.

We are required to find the probability of drawing a rd card from a standard deck of cards.

Probability is the calculation of chance of happening an event among all the events possible. It lies between 0 and 1. It cannot be negative.

Probability=Number of items/Total items.

Total number of cards=52

Number of red cards =26

Number of black cards=26

Probability of drawing a red card =Number of red cards/ Total cards.

=26/52

=1/2

Hence the probability of drawing a red card from a standard deck of cards is 1/2.

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Answer:

17, 5

Step-by-step explanation:

Let the width be x

width = x

length = x + 12

length x width = x(x + 12)

length x width = x² + 12x

x² + 12x = 85

x² + 12x - 85 = 0

Now it is just a quadratic:

The 2 numbers we need are 17 and -5

(x + 17)(x - 5) = 0

x = -17 or x = 5.

Since width and length cannot be negative, x must equal 5.

Width = 5

length = 17

The answer is 17, 5.

We are given two things :

We can represent the length and width as x + 12 and x respectively.

Now, the formula for the area of a rectangle is :

Area = Length x width

Now, let's substitute the values for length and width in the formula along with the area.

We now have a quadratic equation, which can either be solved by splitting the middle term, or by using the quadratic formula. For convenience purposes, we'll go with the first one.

Answer:

7/10

Step-by-step explanation:

take 3, 2, 1 from the equation and your left with 7 numbers. Add those together you get 10. 7/10 is the probability that you will draw a number greater than 3

Using the Fundamental Counting Theorem, there are 20 possibilities for their big night out.

It is a theorem that states that if there are n things, each with [tex]n_1, n_2, \cdots, n_n[/tex] ways to be done, each thing independent of the other, the number of ways they can be done is:

[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]

For this problem, we have that:

The number of possibilities is given by:

N = 5 x 4 = 20.

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The correct graph is in the attached image.

Answer:

  B.  A linear graph of solid line intercepts X-axis at the point (2, 0) and Y-axis at the point (0, -4)

Step-by-step explanation:

The boundary line of an inequality is graphed as though the expression were an equality. The nature of the line used will depend on the nature of the inequality.

When the inequality includes the "or equal to" case (≤ or ≥), the boundary line is part of the solution set. It is drawn as a solid line.

When the inequality excludes the "or equal to" case (< or >), the boundary line is not part of the solution set. It is drawn as a dotted or dashed line.

The given inequality

  y ≤ 2x -4

includes the "or equal to" case, so the boundary line is solid.

The equation of the boundary line is written in slope-intercept form:

  y = mx +b . . . . . . . line with slope m and y-intercept b

  y = 2x -4 . . . . . . . boundary line with slope 2 and y-intercept -4.

This tells you that the boundary line intercepts the y-axis at the point (0, -4).

There are 105 numbers in the  the first 1000 positive integers contain only 0s and 1s when written in base 3

The number base is given as:

Base 3

To determine the count of numbers, we make use of the following Python program

count = 0

for n in range(1,1001):

   digits = []

   while n:

       digits.append(int(n % 3))

       n //= 3

   myStr = ''.join(map(str,digits[::-1]))

   if(myStr.count('0')+myStr.count('1') == len(myStr)):

       count+=1

print(count)

The above program counts the numbers in the  the first 1000 positive integers contain only 0s and 1s when written in base 3

The output of the program is 105

Hence, there are 105 numbers in the  the first 1000 positive integers contain only 0s and 1s when written in base 3

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(C) [tex]8x^{3} -7x^{2} -6x+5[/tex] represents a cubic expression.

To determine which expression represents a cubic expression:

Therefore, (C) [tex]8x^{3} -7x^{2} -6x+5[/tex] represents a cubic expression.

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The complete question is given below:

Choose the expression that represents a cubic expression.

(A) −7x + 14

(B) 7x2 − 5x + 6

(C) 8x3 − 7x2 − 6x + 5

(D) 9x4 + 8x3 − 6x2 − 2x + 11

If the vertices of parallelogram ABCD is A(-6,-4),B(0,-4),C(3,4),D(-3,4) ,then the area of parallelogram ABCD will be 48 square units.

Given that the vertices of parallelogram ABCD is A(-6,-4),B(0,-4)

,C(3,4),D(-3,4).

We are required to find the area of parallelogram ABCD.

Let the point of intersection of height of parallelogram from point A is point E.

When we see the graph on which the parallelogram then we will be able to know that point E has coordinates of (-3,-4).

Area of parallelogram=Base*Height

=AB*DE

DE=[tex]\sqrt{(-4-4)^{2} +(-3+3)^{2} }[/tex]

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

=8 units

AB=[tex]\sqrt{(-4+4)^{2} +(0+6)^{2} }[/tex]

=[tex]\sqrt{36}[/tex]

=6 units

Area=6*8

=48 square units.

Hence if the vertices of parallelogram ABCD is A(-6,-4),B(0,-4),C(3,4),D(-3,4) ,then the area of parallelogram ABCD will be 48 square units.

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Answer:

B={1,2,3,4,6and12}

n (B) =6

Step-by-step explanation:

Factor, in mathematics, a number or algebraic expression that divides another number or expression evenly—i.e., with no remainder. For example, 3 and 6 are factors of 12 because 12 ÷ 3 = 4 exactly and 12 ÷ 6 = 2 exactly. The other factors of 12 are 1, 2, 4, and 12.

Hope it helps!

The value of the missing side x is 38.40 units given the triangle is a right angled triangle with angle 38° and side opposite to the angle is 30 units. This can be obtained by using trigonometric ratios.

Trigonometric identities,

If in a right triangle ΔABC where A is the right angle

In the question,

x is the adjacent side of the angle 38° and 30 units is the opposite side.

The ratio with opposite side and adjacent side is tan A

Therefore,

⇒ tan 38° = [tex]\frac{opposite\ side}{adjacent\ side}[/tex]

⇒ tan 38° = 30/x

⇒ x = 30/tan 38°

∵ tan 38° = 0.781285627

⇒ x = 30/0.781285627

⇒ x = 38.398249

⇒ x = 38.40

Hence the value of the missing side x is 38.40 units given the triangle is a right angled triangle with angle 38° and side opposite to the angle is 30 units.

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Answer:

1 hour

Step-by-step explanation:

Answer:

[tex]12.50 \space\ h = 2500.75[/tex]

Step-by-step explanation:

We know from the question that the student earned $12.50 per hour.

Using this information, we can say that if the student worked for h hours, they would make a total of 12.50 × h dollars.

We also know that the total money they earned is $2500.75.

∴ Therefore, we can set up the following equation:

[tex]\boxed {12.50 \times h = 2500.75}[/tex]

From here, if we want to, we can find the number of hours worked by simply making h the subject of the equation and evaluating:

h = [tex]\frac{2500.75}{12.50}[/tex]

  = 200.6 hours

Answer: 12.50h = 2500.75

Step-by-step explanation:

We know from the question that the student earned $12.50 per hour.

Using this information, we can say that if the student worked for h hours, they would make a total of 12.50 × h dollars.

We also know that the total money they earned is $2500.75.

The standard deviation exists as the positive square root of the variance.

So, the standard deviation = 4.47.

Given data set: 13 17  9 21

To calculate the mean of the data.

We know that mean exists as the average of the data values and exists estimated as:

Mean [tex]$=\frac{13+17+9+21}{4} \\[/tex]

Mean [tex]$=\frac{60}{4} \\[/tex]

Mean = 15

To find the difference of each data point from the mean as:

Deviation:

13 - 15 = -2

17 - 15 = 2

9 - 15 = -6

21 - 15 = 6

Now we have to square the above deviations we obtain:

4, 4, 36, 36

To estimate the variance of the above sets:

variance [tex]$=\frac{4+4+36+36}{4}$[/tex]

Variance [tex]$=\frac{80}{4}$[/tex]

Variance = 20

The standard deviation exists as the positive square root of the variance. so, the standard deviation [tex]$=\sqrt{20 }=4.47$[/tex].

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Answer:

4.4

Step-by-step explanation:

The answer above is correct.

Answer:[tex]x^3-x^2-17x+20[/tex]

Step-by-step explanation:

Using proportions, it is found that the value of n is of [tex]n \geq 11[/tex].

A proportion is a fraction of a total amount, and the measures are related using a rule of three. Due to this, relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.

On the first 15 games, the team wins 2 games and losses 13. Then, in each week, the team plays 3 games, winning 2. Then:

The proportion of wins is given as follows:

[tex]\frac{2 + 2n}{15 + 3n}[/tex]

The proportion is of at least 50%, hence:

[tex]\frac{2 + 2n}{15 + 3n} \geq 0.5[/tex]

We solve the inequality for n, similarly to how we would solve an equality, applying cross multiplication. Then:

[tex]2 + 2n \geq 7.5 + 1.5n[/tex]

[tex]0.5n \geq 5.5[/tex]

[tex]n \geq \frac{5.5}{0.5}[/tex]

[tex]n \geq 11[/tex]

The value of n is of [tex]n \geq 11[/tex].

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Answer:

not sure but I think it is this