(40 points awarded for solving) The quadratic function f(x) has roots of 4 and −6, and it passes through the point (1, 21). What is the vertex form of the equation of f(x)?

f(x) = −(x − 1)2 + 25
f(x) = −(x + 1)2 + 25
f(x) = (x − 1)2 + 25
f(x) = (x + 1)2 − 25

For a 0.48 liter solution, Samantha will use 0.16 liters of water.

Given that

she can make a certain solution by mixing 0.04 liters of chemical a with 0.02 liters of water

she wants to make a total of 0.48 liters of this new solution.

a quantity 0.48/0.06 = 8 times greater than that produced by one recipe.

Water in the amount of 0.02 liters is required.

Samantha will require 0.16 liters of water, or 8 times as much water, to create the solution.

For a 0.48 liter solution, Samantha will use 0.16 liters of water.

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The slope of a line parallel to 1 2 is also 1 2. The equation for a line parallel to the line with equation y = mx + b is y = mx + b + c, where c is any constant.

The slope of a line parallel to 1 2 is the same as the slope of 1 2, which is 1 2.

The slope of a line is calculated by dividing the change in the y-values by the change in the x-values. In the case of 1 2, this is (2-1)/(2-1), which equals 1/1, or 1 2. So the slope of any line parallel to 1 2 is also 1 2. Parallel slope intercept form is a form of linear equation that describes a line parallel to a given line. It is written as y = mx + b, where m is the slope of the line and b is the y-intercept of the line. This form is useful for finding equations of lines that are parallel to a given line.

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The lengths of the edges of the considered rectangular box which will give the minimum surface area are all ∛34.

How to find the surface area and volume of a closed rectangular box?

Suppose the dimensions of the rectangular box be 'a', 'b' and 'c' units respectively. Then, we get:

Surface area of the box, S = 2(ab + bc + ca) unit²

Volume of the box, V = abc unit³

For the considered box, let the dimensions be 'a', 'b' and 'c' cm.

Then, as we're provided that the volume of the box is 34 cubic cm, we get:

abc = 34

c = 34/ab

Thus, the surface area of the considered figure would be given as:

S = 2(ab + bc + ca) unit²

S = 2(ab+ c(b+a))

= 2(ab+ (34/ab) (b+a))

=( 2a²b² + 68b + 68a)/ ab cm²

There is symmetry in the equation we obtained for surface area between 'a' and 'b'.

That means, they will be of same measure to minimize or maximize the surface area. It is because if one of them is bigger than other, it is possible to prove that other one, by same tricks, can be proved bigger than first one.

Thus, we have a = b for S to be maximum or minimum.

Thus, we get:

Smax or min = (2a⁴+136a) / a²

= (2a²+ 136) / a

= f(a)

Taking first derivative of f(a), we get:

f'(a) = 4a - 136 / a²

f''(a) = 4 + 272/a³

Putting second rate = 0, we get critical points as:

f'(a) = 0

4a = 136/ a²

a³ = 34

a = ∛34 = b

Thus, the only critical value found is corresponding to minima.

Thus, S is minimum when a = b = ∛34

The value of c is:

c = 34/ab

= 34/ [tex]\sqrt[3]{34^{2} }[/tex]

= (34)¹/³

c = ∛34

Thus, the lengths of the edges of the considered rectangular box which will give the minimum surface area are all ∛34.

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Answer:

25

Step-by-step explanation:

To find the value of a number given that 12% of it is 150, we can set up the following equation:

0.12x = 150

Solving for x, we get:

x = 150 / 0.12 = 1250

To find 2% of that number, we simply multiply 1250 by 0.02:

2% of 1250 = 0.02 * 1250 = 25.

Therefore, 2% of the number for which 12% is 150 is 25.

Yes, One of them must be even and this is because Pigeonhole Principle.

What is Pigeonhole Principle?

According to the pigeonhole principle, at least one container must hold more than one item if n things are placed into m containers, where n > m.  As an illustration, if one has three gloves (none of which are ambidextrous or reversible), at least two of them must be right-handed or left-handed as there are three items but only two categories of handedness to classify them into. It is possible to establish potentially surprising consequences using this seemingly obvious assertion, a type of counting argument.

From series {1,2......2n},  we need to select 'n+1' integers.

Even integers series from above series is {2,4,6..........2(n-1), 2n}

Odd integers series from above series is {1,3,5......2(n-2), 2n-1}

We can choose only n odd integers from the series. Therefore, if we choose n+1 integers, at least one of them must have to be even integer.

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The standard equation of parabola that has focus (5, 0) and directrix x = -5 is y² = 20x.

Any point in a parabola has an identical distance from the fixed point, making it resemble a curve.

Given that,

The focus of the parabola is (5, 0).

And Directrix x = -5

Since the directrix is horizontal here, use the standard equation of parabola,

(y - k)² = 4 p (x  - h)

Vertex = (h, k)

= ((x coordinate of focus + directrix)/2 , 0)

= ((5 - 5)/2, 0)

= (0, 0)

p = distance from focus to the vertex = 5 - 0 = 5

The required equation of parabola,

(y - 0)² = 4 × 5(x - 0)

y² = 20x

The equation of parabola is  y² = 20x.

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The amount of butter that is used this month will be 320 kilograms.

The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100. The phrase % translates to one hundred percent. It is symbolized by the character '%'.

A pastry shop utilized 25% less spread for the current month than a month ago. The pastry shop utilized 240 kilograms of spread for the current month.

Let 'x' be the amount of butter used in the last month. Then the equation is given as,

(1 - 0.25) x = 240

Simplify the equation, then we have

(1 - 0.25) x = 240

0.75x = 240

x = 240 / 0.75

x = 320 kilograms

The amount of butter that is used this month will be 320 kilograms.

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Answer: A total of 30 teachers

Step-by-step explanation:

20% = 6 (science) teachers

20x5= 100

6x5=30 teachers

Hope this helps you!

Answer:

30 teachers

20% is 2/10 of 100

20 x 5 = 100

6 x 5 = 30

Therefore, 30 teachers.

the probability that in a given year more than 5 earthquakes occur is 72.1%.

What is probability?

By simply dividing the favorable number of possibilities by the entire number of possible outcomes, the probability of an occurrence can be determined using the probability formula. Because the favorable number of outcomes can never exceed the entire number of outcomes, the chance of an event occurring might range from 0 to 1.

Let X denotes the number of earthquakes during the next year in this area

X ~ Poisson(2)

The probability mass function of X is

The probability that there will be 1 to 3 (inclusive) earthquakes during the next year in this area P(1 ≤X≤3)

P = 0,721

Hence the probability that in a given year more than 5 earthquakes occur is 72.1%.

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The remaining root of f(x) will be 3 - [tex]\sqrt{11}[/tex]

To answer this question we have to find the conjugate of 3+[tex]\sqrt{11}[/tex],

For example, the conjugate of p+q will be p-q or vice versa.

Let us consider x= 2+[tex]\sqrt{11}[/tex]

To simplify the equation let us subtract 3 on both sides of the equation

x - 3 = 3+[tex]\sqrt{11}[/tex] - 3

x - 3 = [tex]\sqrt{11}[/tex]

Square both,

[tex](x - 3)^{2}[/tex] = 11

[tex](x - 3)^{2}[/tex] - 11 =0

Now we will use the principle of Square difference to factorize both sides-

[tex](d^{2} - e^{2}) = (d - e)(d + e)[/tex]

(x - 3) - 11 × (x - 3) + 11 = 0

on solving the equation we get,

(x - 3) - 1 = 0  let us consider this first equation

or (x - 3) + 11 = 0 let us consider this second equation

Add [tex]\sqrt{11}[/tex] on both sides of the first equation,

[tex]x - 3 = \sqrt{11}[/tex]

Subtract [tex]\sqrt{11}[/tex] on both sides of the second equation,

[tex]x - 3 = -\sqrt{11}[/tex]

Now by adding 3 on both sides of the first and second equation,

[tex]x = 3 + \sqrt{11}[/tex] or [tex]x = 3 - \sqrt{11}[/tex]

Therefore, the remaining root of f(x) will be = [tex]3 - \sqrt{11}[/tex]

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Complete Question - If a polynomial function f(x), with rational coefficients, has roots 0, 4, and 3 + [tex]\sqrt{11}[/tex], what must also be a root of f(x)?.

278.5 if im not wrong

Step-by-step explanation:

75 x 15 1125 ten percent of that is 133.5 take 1125 - 133.5 =1011.5 take that add 210 get 1221.5 -1500 and you have 278.5

Answer: p-3[tex]\leq[/tex]29

Step-by-step explanation:

Answer:

I could be wrong but I think its B

Step-by-step explanation:

Answer: 65

Step-by-step explanation: correct answer is 65 b

The contrapositive of the given conditional statement will  be If I did not buy a car, then I did not win the lottery.

When two statement P and Q are written in the formate of if P then Q,  it means Q must be true whenever P is true, these types of statements that follow some conditions are called conditional statement.

Given that, a conditional statement, "If I won the lottery, then I would buy a car."

Here the P part is If I won the lottery and Q part is I would buy a car.

The rule for statement being contrapositive, for two statement P and Q is,

If not-Q then not-P

Therefore, the contrapositive will be, If I did not buy a car, then I did not win the lottery.

Hence, The asked statement is If I did not buy a car, then I did not win the lottery.

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The [tex]\sqrt{2}[/tex]  is not a rational number, it is an  irrational number

The number is [tex]\sqrt{2}[/tex]

The rational number is the number that can be written in the form of p/q where q ≠ 0

Therefore assume [tex]\sqrt{2}[/tex] is a rational number

[tex]\sqrt{2}[/tex] = p/q

Find the square of the both side

2 = p^2 / q^2

2q^2 = p^2

The value of p^2 will be the multiple of 2

p will be the multiple of 2

Consider the multiple as m

p = 2m

p^2 = 4m^2

Substitute the values in the above equation

2q^2 = 4m^2

q^2 = 2m^2

Therefore the value of the q will be the multiple of 2

The p and q is the multiple of 2, therefore they have coprime

Our assumption is wrong, [tex]\sqrt{2}[/tex] is an irrational number

Therefore, the [tex]\sqrt{2}[/tex] is not a rational number

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The percentage of the guests ordered food is 40%

"Information available from the question"

In the question:

There were 425 guests at a hotel. 170 guests ordered food.

To find the what percentage of the guests ordered food?

Now, According to the question:

Total guests in hotel is 425

and, 170 guests ordered food

=> Divide 170 by 425

= 170/ 425 = 0.4

The percentage of the guests ordered food:

=> 0.4 x 100 = 40%

Hence, The percentage of the guests ordered food is 40%

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Rounded Answer:

0.005 or 0.0049

The measure of angle ABP is 90° and the measure of angle APB is 48°.

In the given figure, the measure of arc AB is 108° and the measure of angle ACB is (x+3)°.

Tangent and radius of a circle meet at 90°. If we draw a radius that meets the circumference at the same point, the angle between the radius and the tangent will always be exactly 90°.

Question 6:

Here, The measure of arc AB = 2×The measure of angle ACB

108 = 2(x+3)

⇒ x+3=54

⇒ x =51°

Question 7:

Given that, ∠PAB=42° and line PB is tangent to circle A

Now, ∠ABP =90° (Angle formed by tangent and radius is 90°)

Here, ∠ABP+∠PAB+∠APB=180°

⇒ 90°+42°+∠APB=180°

⇒ 132°+∠APB=180°

⇒ ∠APB=180°-132°

⇒ ∠APB=48°

Therefore, the measure of angle ABP is 90° and the measure of angle APB is 48°.

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Answer:

6x-9

Step-by-step explanation:

you do

Answer: 6x-9

Step-by-step explanation:

We know that the word than is a switch word. Therefore, the nine will go after the minus sign. X is the variable (unknown number). Times is a multiplication word but you do not put a multiplication symbol when there is just a variable.

Hence, our answer is 6x - 9

Answer:

The slope is 3/8

Step-by-step explanation:

The slope of the line given by the equation 3x - 8y = -16 is 3/8.

To find the slope of the line given by the equation 3x - 8y = -16, we need to rearrange the equation into slope-intercept form, y = mx + b, where m represents the slope.

Starting with the given equation:

3x - 8y = -16

We can isolate y by subtracting 3x from both sides:

-8y = -3x - 16

Next, we divide both sides of the equation by -8 to solve for y:

y = (3/8)x + 2

Now, we can observe that the equation is in slope-intercept form, y = mx + b, with m = 3/8.

This means that for every increase of 1 in the x-coordinate, the y-coordinate increases by 3/8.

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Answer:

n = -8 , n = -12

Step-by-step explanation:

First we factorise the quadratic using the factorisation method :

n²+8n+12n+96 = 0

n(n+8)+12(n+8) = 0

(n+12)(n+8) = 0

Now we equate each bracket to 0 :

Either

n + 12 = 0  

n = -12

OR

n + 8 = 0

n = -8

So the real solutions for that equation is

n = -8 , n = -12

Answer: The answer is A) True.

Step-by-step explanation: It is possible, although the only thing that wouldn't be possible is an equilateral triangle, since there is one side length that is different.

I hope this helps!!

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

Given two points, the slope is given by the change in y divided by the change in x, hence the slope of segment RS is given as follows:

m = (16 - 4)/(9 - 3) = 2.

When two lines are perpendicular, the multiplication of their slopes is of -1, hence the slope of the perpendicular bisector is obtained as follows:

2m = -1

m = -1/2.

The midpoint of a segment is given by the mean of the coordinates, hence:

The perpendicular bisector is a linear function with slope of -1/2 going through the point (6,10), hence the intercept b is calculated as follows:

10 = -0.5(6) + b

b = 13.

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To find the the measures of missing angles, it's important to know the total sum in the angles and then subtract by the angles given.

Let's take for instance, a triangle has a total angle of 180°. It should be noted that a triangle has three sides and angles.

Let's say that two of the angles are 50° and 70°, therefore, the value of the third angle be be represented thus:

(50 + 70 + x) = 180

120 + x = 180

Collect like term

x = 180 - 120

x = 60°

The third angle is 60°

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The area that park will need to cover 67.95 square feet of the ramp.

The trigonometric ratios are defined for a right angled triangle.

The example of these ratios are sin, cos, tan, cosec, sec and cot.

The trigonometric ratios are useful in determining the heights and distances of the large objects.

Suppose the length of the wood be x.

The length of wood forms the right triangle as shown in the figure.

Apply trigonometric ratio to the triangle to get,

Cos (28°) = 12/x

⇒ x = 12/Cos (28°) = 13.59

Now, the area of the wood is given as,

Length × width

⇒ 13.59 × 5 = 67.95

Hence, the area of wood required is 67.95 square feet.

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The 90% confidence interval is given by (3.9911; 7.9911)

It is 90% confident that the true average speed of a vehicle on a highway is between 3.9911 and 7.9911 miles per hour.

1) Former concept

A confidence interval is "a range of values ​​that is likely to encompass the values ​​of a population with some degree of confidence. Often expressed as a percentage.

The error bar is the range of values ​​above and below the sample statistic in the confidence interval.

A probability distribution that is symmetric about the mean and indicates that data close to the mean are more common than data far from the mean.

X = 2, represents the sample mean of the sample

μ = population mean (variable of interest)

s = 1 represents the standard deviation of the sample

n = 11 represents the sample size

2 ) Confidence Interval:

Confidence Interval for mean is given below:

                  X ± t (α/2) s/√n

To compute the critical value, we first need to find the degrees of freedom given by

df = n -1

   = 11 -1 = 10

Confidence is 0.90 or 90%, so the value of and is α = 0.1 , and α/2 = 0.05 you can use Excel, a calculator, or a spreadsheet to find the critical values. The Excel command would be "=-T.INV(0.05,10)".

2 - 1.81 (1/√11) = 3.9911

2 + 1.81 (1/√11) = 7.9911

So in this case the 90% confidence interval is (3.9911 , 7.9911)

We are 90% confident that the true mean for the speeds of vehicles traveling on a highway is between 3.9911 and 7.9911 miles per hour.

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The slope of the tangent line to the given polar curve at the point specified by the value of theta is m= 1/√3.

A straight line that just touches a plane curve at a particular location is referred to as the tangent line (or tangent) to the curve in geometry. It was described by Leibniz as the line connecting two points on the curve that are endlessly near to one another. A line passes through the point (c, f(c)) on the curve and has a slope of f'(c), where f' is the derivative of f. This is more properly referred to as being a tangent of the curve y = f(x) at a point x = c. Curves in n-dimensional Euclidean space and curves in space have a comparable definition.

The tangent line is "passing" through the point of tangency—the intersection of the tangent line and the curve as it travels.

Given,

r=6cosθ

f'(θ)=-6sinθ

θ=π/3

dy/dx=(f'(θ)sinθ+f(θ)cosθ)/(f'(θ)cosθ-f(θ)sinθ)

dy/dx=( -6sinθsinθ+6cosθcosθ)/(-6sinθcosθ-6cosθsinθ)

=(6cos²θ-6sin²θ)/-12 sinθcosθ

=6(cos²θ-sin²θ)/-6(2sinθcosθ)

=6cos2θ/-6sin2θ

=-cos2θ/sin2θ

=-cot2θ

Slope of the given polar curve at θ=π/3 is

m=dy/dx

m= -cot(2(π/3))

m = cot(-2π/3)

m= 1/√3

Therefore, the slope of the tangent line to the given polar curve at the point specified by the value of theta is

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The exponential growth model below that fits the points (0, 2720) and (8, 10,225) as  [tex]y = 2720 \times0.039^t.[/tex]

To find the exponential growth model that fits the points (0, 2720) and (8, 10,225), we need to find the values of the constants k and y0 in the equation [tex]y = y_0e^(kx)[/tex]. We can do this by substituting the coordinates of the two points into the equation and solving for the constants.

First, we substitute the coordinates of the first point into the equation:

[tex]2720 = y_0e^(k\times0)[/tex]

We can simplify this expression by substituting 1 for e⁰: 2720 = y⁰ˣ¹

This equation tells us that y0 = 2720.

Next, we substitute the coordinates of the second point into the equation: [tex]10,225 = y_0e^(k\times8)[/tex]

Substituting the value of y0 that we found above, we get: [tex]10,225 = 2720e^(k\times8)[/tex]

To solve for k, we divide both sides of the equation by 2720: [tex]\dfrac {10,225}{2720} = e^(k\times8)[/tex]

We can take the natural logarithm of both sides to get: ln(10,225/2720) = k8

Dividing both sides of the equation by 8, we get: k = ln(10,225/2720)/8

We can approximate the value of k by using a calculator to evaluate the logarithm: k ≈ 0.039

We now have the values of the constants k and y0, so we can write the exponential growth model that fits the given points: [tex]y = 2720 \times0.039^t.[/tex]

This is the exponential growth model that fits the points (0, 2720) and (8, 10,225).

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