Find all of the zeros of the function f(x) = x3 – 23x2 + 161x – 303.

Answer:

Step-by-step explanation:

The dot product is the sum of products of the corresponding coordinate values:

[tex]\text{\bf{a}$\cdot$\bf{b}}=a_{x}b_{x}+a_{y}b_{y}[/tex]

The value of this can also be written in terms of the magnitudes of the vectors and the angle θ between them:

  |a|·|b|·cos(θ)

But the angle between the vectors is the same as the difference of their individual angles, θ = α - β, so this can also be written as ...

  |a|·|b|·cos(α-β)

And the trig identity for the cosine of the difference of angles lets us write the above as ...

  = |a|·|b|·(cos(α)cos(β) +sin(α)sin(β))

Answer with explanation:

For a Lopsided coin ,probability of getting Head is equal to [tex]\frac{1}{3}[/tex] For a Lopsided coin ,probability of getting Tail  is equal to [tex]\frac{1}{3}[/tex].

→Probability of getting Tail > Probability of getting Head

→Coin is heavier from tail side and lighter from Head side.

→→We have to Calculate number of flips of coin that is needed to have both heads and tails appear at least once.

[tex]\rightarrow P(\text{Head})=\frac{1}{3}\\\\ \frac{1}{3} \times x=1\\\\x=3\\\\\rightarrow P(\text{Tail})=\frac{2}{3}\\\\ \frac{2}{3} \times y=1\\\\y=\frac{3}{2}[/tex]

→We need to find common multiple of 3 and [tex]\frac{3}{2}[/tex].

Least common multiple of 3 and [tex]\frac{3}{2}[/tex] is 6.

→So,on Average number of flips of this coin are needed to have both heads and tails appear at least once=6 tosses

Answer:

For [tex]3,604\ books[/tex] the cost for both methods will be the same

Step-by-step explanation:

Let

y ------> the production cost

x ------> the number of books

we know that

First production Method

[tex]y=19.25x+22,427[/tex] -------> equation A

Second production Method

[tex]y=10.50x+53,962[/tex] -------> equation B

Solve the system of equations

Equate equation A and equation B and solve for x

[tex]19.25x+22,427=10.50x+53,962[/tex]

[tex]19.25x-10.50x=53,962-22,427[/tex]

[tex]8.75x=31,535[/tex]

[tex]x=3,604\ books[/tex]

Answer:

264 transistors.

Step-by-step explanation:

By proportion the total number made if  16 are faulty is (35/2) * 16

=  280.

The number of good ones is 280 - 16

= 264 transistors.

Answer:

264 good ones

Step-by-step explanation:

2 in 35 are faulty = in a batch of 35, 2 will be faulty = there are 33 good for every 2 faulty ones there are

16 faulty have been made - to figure out how many batches there have been do 16/2=8

8 batches of 35, 8 × 35 = 280 total

280 total - 16 faulty = 264 good

OR do 8 batches × 33 good per batch = 264 good

at least thats how i interpreted it!

There are 511 positive integers whose digits strictly increase from left to right.

To determine how many positive integers have digits that strictly increase from left to right, we need to consider combinations of digits. Since the digits must strictly increase, we cannot repeat any digit, and we can only use digits from 1 to 9.

Each number can be considered a unique combination of these digits, where the order of digits matters. For example, the digits {1, 2, 3} can only form the number 123. The total number of such combinations is given by the binomial coefficient C(9, k), where k is the number of digits.

Answer:

  5y = 5

Step-by-step explanation:

When the second equation is multiplied by -2, it becomes ...

  -2(x -2y) = -2(-1)

  -2x +4y = 2

Adding this to the first equation gives ...

  (2x +y) +(-2x +4y) = (3) +(2)

  2x +y -2x +4y = 5 . . . . . . . . . eliminate parentheses

  5y = 5 . . . . . . . . . . . . . . . . . . .collect terms

Answer:

c

Step-by-step explanation:

Answer:

answer would be The mean must have a mean of 0 and a standard deviation of 1. hope this helps

Step-by-step explanation:

Answer:

428.7 mm²

Step-by-step explanation:

The area (A) of the sector is calculated as

A = area of circle × fraction of circle

   = πr² × [tex]\frac{170}{360}[/tex]

   = π × 17² × [tex]\frac{17}{36}[/tex]

   = 289π × [tex]\frac{17}{36}[/tex]

   = [tex]\frac{289(17)\pi }{36}[/tex] ≈ 428.7 mm²

The area of a sector with a central angle of 170° and a radius of 17 mm is calculated using the sector area formula, resulting in approximately 428.7 mm², rounded to the nearest tenth.

To find the area of the sector, we use the formula [tex]\( \text{Area} = \frac{\text{Central Angle}}{360\°} \times \pi \times \text{Radius}^2 \)[/tex]. Substituting the given values, we get [tex]\( \text{Area} = \frac{170\°}{360\°} \times \pi \times 17^2 \)[/tex]. Simplifying, we have [tex]\( \text{Area} = \frac{17^2}{2} \times \pi \)[/tex]. Evaluating this expression, we find [tex]\( \text{Area} \approx 428.7 \)[/tex] mm². Therefore, the area of the sector, rounded to the nearest tenth, is approximately 428.7 mm². This calculation represents the portion of the circle enclosed by the given central angle and radius, providing the area of the sector.

Answer:

  A ≈ 32°

Step-by-step explanation:

You are given two sides and the angle between them, so the law of cosines applies. The measure of side b can be found to be ...

  b² = a² + c² -2ac·cos(B)

  b² = 2² +3² -2·2·3·cos(95°) ≈ 14.0459

  b ≈ 3.74778

Then the law of sines can help you find angle A, the angle opposite the shortest side.

  sin(A)/a = sin(B)/b

  A = arcsin(a/b·sin(B)) = arcsin(2/3.74778·sin(95°)) ≈ 32.11°

  A ≈ 32°

The smallest angle is about 32°.

Answer:

The probability that the card drawn is a club is 0.25.

The probability that card drawn is a club, when it is given that the card is black is 0.5.

Step-by-step explanation:

In a standard​ deck of cards:

Total number of cards = 52

Total number of cards of each suit (club, spade,heart, diamond) = 13

The probability that the card drawn is a club is

[tex]P=\frac{\text{Favorable outcomes}}{\text{Total outcomes}}[/tex]

[tex]P=\frac{^{13}C_1}{^{52}C_1}=\frac{13}{52}=0.25[/tex]

Therefore the probability that the card drawn is a club is 0.25.

Let A and B represents the following events:

A : Card is black

B : Card is a club

Total number of black cards = 26

[tex]P(A)=\frac{26}{52}=\frac{1}{2}=0.5[/tex]

From the above parts

[tex]P(B)=0.25[/tex]

Total number of black club cards = 13

[tex]P(A\cap B)=\frac{13}{52}=\frac{1}{4}=0.25[/tex]

We need to find the probability that card drawn is a club, when it is given that the card is black.

[tex]P(\frac{B}{A})=\frac{P(A\cap B)}{P(A)}[/tex]

[tex]P(\frac{B}{A})=\frac{0.25}{0.5}=0.5[/tex]

Therefore the probability that card drawn is a club, when it is given that the card is black is 0.5.

Finding the slope using two points:

The formula for slope is

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

In this case...

[tex]y_{2} =7\\y_{1} =6\\x_{2} =8\\x_{1} =4[/tex]

^^^Plug these numbers into the formula for slope...

[tex]\frac{7 - 6}{8 - 4}[/tex]

C) [tex]\frac{1}{4}[/tex]

^^^This is your slope

Hope this helped!

~Just a girl in love with Shawn Mendes

Answer:

Explanation:

1) Set the algebraigic expression that represents the combinations of sofa and pillow orders:

2) Predict the graph of the inequality x + 2y ≥ 4

        x = 0 ⇒ 2y = 4 ⇒ y =4/2 ⇒ y = 2 ⇒ point (0,2)

        y = 0 ⇒ x = 4 ⇒ point (4,0)

        Then the lines goes through (0,2) and (4,0) ... [the four graphs meet this]

3) Conclusion:

That means that the correct graph is the last one: solid line, shaded area over the line y = 2 - x/2.

Note: a more detailed graph would include the fact that the items cannot be negative, i.e. x ≥ 0 and y ≥ 0, which would result in that the shaded area would be on the first quadrant.

Answer: D

Step-by-step explanation:

The expression that shows the total attendance is 7 x 40 + 7 x 2.

An algebraic expression is consists of variables, numbers with various mathematical operations,

Given that,

Total number of rows = 8.

Number of seats in each row = 42.

Total number of seats = 42 x 8 = 336.

Since, an act needs to keep first row empty, but rest of the seats sold.

So the remaining seats = 336 - 42 = 294.

So, the expression for the total attendance can be given as,

7 x 40 + 7 x 2 = 280 + 14 = 294.

The required expression is 7 x 40 + 7 x 2.

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Answer:

It would take Tom 8 hours alone

Step-by-step explanation:

We are looking at this basic equation:

Tom + Shirley = 8 hours

Tom takes x hours to cut the grass; this means that 1/x of the job gets done in 1 hour

It takes Shirley 3 times as long as Tom, so it takes her 3x hours to cut the grass; this means that 1/3x of the job gets done in 1 hour

If the total number of hours it takes them to do the job is 6 hours; this means that 1/6  of the job gets done in 1 hour

Looking back at the original basic equation, we will fill in our info:

[tex]\frac{1}{x}+\frac{1}{3x}=\frac{1}{6}[/tex]

We can solve for x by first finding the LCD and eliminating the denominators.  The LCD is 6x, since all the denominators go into 6x evenly.

We will multiply the rational equation through by the LCD:

[tex]6x[\frac{1}{x}+\frac{1}{3x}=\frac{1}{6}][/tex]

Dividing each denominator into the LCD gives us:

6 + 2 = x so

x = 8

This means that it takes Tom 8 hours to do the job alone.  It would take Shirley 24 hours alone.  Ugh.

The time it would take Tom to mow the lawn alone is 3 hours

From the information given;

Let Tom's working hours be x.

Let Shirley's working hours be y.

We have  that;

Shirley and Tom work 6 hours together, this is expressed as;

x + y = 6

Given that Shirley work 3 hours alone, we have;

y = 3

Substitute the value of y as 3 in the equation

x + 3 = 6

x = 6 - 3

x = 3

Thus, the time it would take Tom  to mow the lawn alone is 3 hours

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Answer:

  remaining zeros: negative i comma 5 minus i

Step-by-step explanation:

The remaining two zeros are the conjugates of the two zeros given. That brings the total number to 4 zeros, consistent with the number of zeros expected for a 4th-degree polynomial.

The conjugate of a complex number has the same real part and the opposite imaginary part.

Answer:

-i, 5-i

Step-by-step explanation:

Given that a function f(x) has only real coefficients and also of degree 4.

Since any polynomial with real roots have imaginary roots only with conjugate pairs, we can find other two roots easily

Degree of polynomial = 4

No of roots = 4

GIven roots are i, 5+i

Conjugate of the given roots are -i, 5-i

Hence remaining zeroes of f are -i, 5-i

Answer:

r = +2√3

Step-by-step explanation:

The equation of a circle with center at (0, 0) and radius r is

x^2 + y^2 = r^2.

Here we have

x^2 + y^2 = 12,

and so we can deduce that r^2 = 12.  Then r = +2√3.

Answer:

The radius is: [tex]2\sqrt{3}[/tex]

Step-by-step explanation:

The equation of a circle in center-radius form is:

[tex](x - h)^2 + (y - k)^2 = r^2[/tex]

Where the center is at the point (h, k) and the radius is "r".

So, given the equation of the circle:

[tex]x^2+y^2=12[/tex]

You can identify that:

[tex]r^2=12[/tex]

Then, solving for "r", you get that the radius of this circle is:

[tex]r=\sqrt{12}\\\\r=2\sqrt{3}[/tex]

Answer:

B

Step-by-step explanation:

We are given f(x) is height in cm while x is days old.

We are also given f(60)=210.

If you compare f(60) to f(x) you should see that x is 60 so we have the sunflower is 60 days old.  Since f(60)=210, then you have the height of the sunflower is 210 cm tall.

Answer:

B.) The height of the sunflower plant is 210 cm when it is 60 days old.

Step-by-step explanation:

Answer:

I think its 21.6 or 22 (Mostly 21.6 though) Sorry if its not right but i'm mostly sure of it like this!

Answer:

[tex] x=2 \pm \sqrt{7} [/tex]

Step-by-step explanation:

Given this form ax^2+bx=k, here are my steps for completing the square while answer your question:

First step: Divide both sides by what is in front of x^2.  You want the coefficient of x^2 to be 1.  To do this for your question, divided both sides by 3.

This gives us x^2-4x  = 3.

Second step:  We are ready to begin the completing the square process at this step.  We are going to add (b/2)^2 on both sides.  For this question b=-4.

So we will be adding (-4/2)^2 on both sides.

This gives us x^2-4x+(-4/2)^2=3+(-4/2)^2.

Third step:  I like to simplified the things inside the square and I do not actually apply the square at this step.  It makes a later step easier in my opinion.

So this step gives us  x^2-4x+(-2)^2=3+(-2)^2.

Fourth step:  I'm actually going to write the left hand side as a square.  Just drag the things that are inside the squares down into ( )^2.

This is what I mean x^2-4x+(-2)^2=(x-2)^2.

So at the end of this step we have (x-2)^2=3+(-2)^2.

Fifth step: I'm going to simplify the right hand side.

This step gives us (x-2)^2=7

Sixth step:  We are ready to square root both sides.  

This gives us [tex] x-2=\pm \sqrt{7} [/tex]

Seveth step:  Get x by itself like you normally would with a linear equation.  My step here is just to add 2 on both sides.

Final answer:  [tex] x=2 \pm \sqrt{7} [/tex]

[tex]3x^2-12x=9\\x^2-4x=3\\x^2-4x+4=7\\(x-2)^2=7\\x-2=\sqrt7 \vee x-2=-\sqrt7\\x=2+\sqrt7\vee x=2-\sqrt7[/tex]

Answer:

Area=14.22π

Step-by-step explanation:

Area=πr²(C/360)

C  is the central angle in degrees

r  is the radius of the circle of which the sector is part.

Area = π(8)²(80/360)

Area=π(64)(0.2222)

Area=π(14.22)

Area=14.22π....

[tex]\bf ~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$1000\\ r=rate\to 8\%\to \frac{8}{100}\dotfill &0.08\\ t=years\dotfill &10 \end{cases} \\\\\\ A=1000e^{0.08\cdot 10}\implies A=1000e^{0.8}\implies A=2225.54~\hfill \stackrel{interest = A - P}{1225.54}[/tex]

btw, we could have used the compound formula just the same, by simply using a compounding cycle of 365, namely daily, assuming a year has 365 days.

Answer:

Step-by-step explanation:

For principal amount P, the future value of the simple interest account will be ...

  P(1 + rt) = P(1 + .026·20) = 1.52P

The future value of the compound interest account will be ...

  P(1 +r)^t = P(1.026^20) ≈ 1.6708875P

The value of the compound interest account is about 10% greater after 20 years.

I would choose the compound interest account because it has a higher rate of return.

Answer:

  A = √29

Step-by-step explanation:

The short of it is that ...

  A² = 2² + 5² = 29

  A = √29

_____

Amplitude

If you expand the second form using the sum-of-angles formula, you get ...

  Asin(ωt +φ) = Asin(ωt)cos(φ) +Acos(ωt)sin(φ)

Comparing this to the first form, you find ...

  c₂ = 2 = Acos(φ)

  c₁ = 5 = Asin(φ)

The Pythagorean identity can be invoked to simplify the sum of squares:

  (Asin(φ))² + (Acos(φ))² = A²(sin(φ)² +cos(φ)²) = A²·1 = A²

In terms of c₁ and c₂, this is ...

  (c₁)² +(c₂)² = A²

  A = √((c₁)² +(c₂)²) . . . . . . . formula for amplitude

_____

Phase Shift

We know that tan(φ) = sin(φ)/cos(φ) = (Asin(φ))/(Acos(φ)) = 5/2, so ...

  φ = arctan(c₁/c₂) . . . . . . . formula for phase shift*

  φ = arctan(5/2) ≈ 1.19029 radians

___

* remember that c₁ is the coefficient of the cosine term, and c₂ is the coefficient of the sine term.

In this mathematics problem, to rewrite in the appropriate form, the amplitude, A, has to be first determined using the given values and the Pythagorean identity. This gives the formula as A=sqrt( + ), where A is the calculated amplitude.

To rewrite in the form, we first need to find the amplitude A. Given values and, if we use the Pythagorean identity, we can solve for A. According to the Pythagorean identity, the sum of the squares of the values equals the square of the amplitude. In mathematical terms, A=sqrt( + ). The result will give you the correct amplitude. Therefore, the given value can be rewritten in the form Acos(ωt+ϕ).

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Answer:

Radius of the Earth is 3978.8 Miles

Circumference of the Earth is 25000 Miles

Step-by-step explanation:

The angle of the sun shone at an angle of 7.2° to the zenith

This means that the angle of the sector of the circle is 7.2° (θ)

S = Length of the sector of the circle = 500 miles

r = radius of earth

Converting 7.2° to radians

[tex]\theta =7.2\frac{\pi}{180}[/tex]

[tex]S=r\theta\\\Rightarrow r=\frac{S}{\theta}\\\Rightarrow r=\frac{500}{7.2\frac{\pi}{180}}\\\Rightarrow r=3978.8\ Miles[/tex]

∴ Radius of the Earth is 3978.8 Miles

[tex]C=2\pi r\\\Rightarrow C=2\times \pi \frac{500}{7.2\frac{\pi}{180}}\\\Rightarrow C=25000\ Miles[/tex]

∴ Circumference of the Earth is 25000 Miles

The radius and circumference are respectively; r = 3978.86 miles and C = 25000 miles

We are told that the angle of the sun shone at an angle of 7.2° to the zenith. Thus, we can liken this to the angle of a sector and so;

Angle of the sector of the circle; θ = 7.2° = 0.125664 rad

Length of the sector of the circle; S = 500 miles

Formula for length of arc is;

S = rθ

where;

S is length of arc

r is radius

θ is angle of sector in radians

Thus;

r = S/θ = 500/0.125664

r = 3978.86 miles

Formula for circumference is;

C = 2πr

C = 2 * π * 3978.86

C = 25000 miles

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6x + y = 28.  The standard form of the equation that is parallel to y = -6x + 5 and goes through point (4, 4) is 6x + y = 28.

The equation is written in the slope-intercept form y = mx +b. So:

y = -6x + 5

The slope m = -6

Since the slopes of parallel lines are the same, we are looking for a slope line m = -6 and goes through point (4, 4).

With the slope-intercept form:

y = mx + b

Introducing the slope m = -6:

y = -6x + b

Introducing the point (4, 4):

4 = -6(4) + b

b = 4 + 6(4)

b= 24 + 4

b = 28

Then

y = -6x + 28

write the equation in standard form ax + by = c:

y = -6x + 28

6x + y = 28

Step-by-step explanation:

the e standard form of parabola with vertex (h,k) is

y=a(x-h)²+k

here (h,k)=(-3,6)

so the answer to your question is

y=-3(x-(-3))²+6

y=-3(x+3)²+6

Answer:

Option D is correct.

Step-by-step explanation:

The vertex is (-3,6)

We will check which equation satisfies the given vertex.

A) y = -3(x-3)^2 - 6

if x = -3 then value of y should be 6

Checking:

y = -3(-3-3)^2 - 6

y = -3(-6)^2 - 6

y = -3(36) -6

y = -114

if x= -3, y ≠ 6

B) y = -3(x+3)^2 - 6

if x = -3 then value of y should be 6

Checking:

y = -3(-3+3)^2 - 6

y = -3(0)-6

y = -6

if x= -3, y ≠ 6

C) y = -3(x-3)^2 + 6

if x = -3 then value of y should be 6

Checking:

y = -3(-3-3)^2 + 6

y = -3(-6)^2 + 6

y = -3(36) + 6

y = -102

if x= -3, y ≠ 6

D) y = -3(x+3)^2 + 6

if x = -3 then value of y should be 6

Checking:

y = -3(-3+3)^2 + 6

y = -3(0)^2 + 6

y = 6

So, if x= -3, y =6 so, if the vertex of parabola is at (-3,6) the equation will be

y = -3(x+3)^2 + 6

So. Option D is correct.

Answer:

quarters = 6+x

dimes = x

Total value = 325

Dime value = 10

Quarter value = 25

Step-by-step explanation:

25x+150+10x=325 (simplify)

35x=175

x=5  

6+5 = 11  

Hence Zoe has 5 dimes and 11 quarters.

Hope this helps!!❤

Answer:

Option 2: (x-4)^2 + (x-5)^2 = 4

Step-by-step explanation:

The given circle's center is: (4,5)

And the radius is 4 units.

We are told in the question that the center will remain same only the radius will be changed.

The center is denoted by (h,k) = (4,5) and radius is 2

Standard equation of circle is:

(x-h)^2 + (y-k)^2 = r^2

So the equation of another circle with same center and radius of 2 units will be:

(x-4)^2 + (x-5)^2 = 2^2

(x-4)^2 + (x-5)^2 = 4  

Hence option 2 is correct ..

Answer:

  3

Step-by-step explanation:

Segment BC corresponds to segment DF. The length of BC is the distance between coordinates (0, 2) and (3, 2). These points are on the same horizontal line (y=2), so the distance between them is the difference of their x-coordinates: 3 - 0 = 3.

Answer:

DF = 3

Step-by-step explanation:

If ABC is equivalent to EDF, then DF is equivalent to BC, which form the following ordered pairs:

D = (0,2)

F = (3,2)

It can be seen that both pairs have the same value of "y" or second value, that is 2.

As a rule, when the points are located on the y-axis (of the ordinates) or on a line parallel to this axis, the distance between the points corresponds to the absolute value of the difference of their ordinates.

So,

DF = D(x) + F(x) = 0 + 3 = 3

If we apply the equation of the distance between two points we get the same result,

[tex]DF=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}   }=\sqrt{(3-0)^{2}+(2-2)^{2}   }=\sqrt{(3)^{2}+(0)^{2}   }=\sqrt{9+0   }=\sqrt{9}=3[/tex]

Hope this helps!

Answer:

400,000,000 tons

Step-by-step explanation:

You could reword this sentence to ask the question, "100,000,000 is 25% of how much?"

The word "is" means equals, the percent will be .25 in decimal form, the word "of" means to multiply, and "how much" is x, our unknown.  In equation form, then, this looks like

100,000,000 = .25 × x

We will divide both sides by .25 to get that

x = 400,000,000

That's how much grain was grown in 2008

Answer: 400 million tons

Step-by-step explanation:

100 millions is 1/4 of 400 million