There are 100 students each enrolled in at least one of three science classes. Of those students, 60 are enrolled in chemistry, 45 in physics, and 30 in biology. Some students are enrolled in two science classes, and 10 students are enrolled in all three. (a) How many students are enrolled in exactly two science classes? (b) There are 9 students taking both chemistry and physics (but not biology), and 4 students taking both physics and biology (but not chemistry). How many are taking both chemistry and biology (but not physics)? Lewis, Harry. Essential Discrete Mathematics for Computer Science (p. 57). Princeton University Press. Kindle Edition.

Answer:

Descartes' rule states that the possible number of the positive roots of a polynomial is equal to the number of sign changes in the coefficients of the terms or less than the sign changes by a multiple of 2.

The Fundamental Theorem of Algebra states that every polynomial equation over the field of complex numbers of degree higher than 1 has a complex solution, furthermore any polynomial of degree n has n roots.  

Remember that the complex numbers include the real numbers.

Suppose we are given the polynomial x^3+3x^2-x-x^4-2, we arrange the terms of the polynomial in the descending order of exponents:

-x^4+x^3+3x^2-x-2, count the number of sign changes, there are 2 sign changes in the polynomial, so the possible number of positive roots of the polynomial is 2 or 0.

returning to our polynomial above, -x^4+x^3+3x^2-x-2, it has degree 4 and so has n roots. Note that complex roots always come in pairs, so here is what can be said from these two rules:  

degree 1 has 1 real root

degree 2 has 2 real roots or 2 complex roots

degree 3 has 3 real roots or 1 real root and 2 complex roots

degree 4 has 4 real roots or 2 real roots and 2 complex roots  

note that if the degree is odd, there will be at least 1 real root

Step-by-step explanation:

The number of complex roots and the possible positive and negative real roots of a polynomial can be predicted using the Fundamental Theorem of Algebra and Descartes' Rule of Signs. The Fundamental Theorem of Algebra ensures at least one complex root for every polynomial, while Descartes' Rule predicts possible real roots based on the number of sign changes in the polynomial equation.

To predict the number of complex roots and find the number of possible positive and negative real roots to a polynomial, you can use both Descartes' Rule and the Fundamental Theorem of Algebra. These two concepts in mathematics can give us interesting insights into the roots of polynomial equations.

First, the Fundamental Theorem of Algebra states that every non-constant polynomial equation has at least one complex root. This theorem can ensure us that we have a starting point, knowing that every polynomial equation will have at least one solution, even if it's complex.

Second, you can use Descartes' Rule of Signs to predict possible positive and negative roots. This rule uses the number of sign changes in the polynomial to give possible number of positive real roots. You can get the possible number of negative real roots by replacing x with -x and count the sign changes again.

For example, for the equation f(x) = x5 – 4x4 + 2x3 + 8x2 -12x + 6, there are two sign changes in the original equation, so there can be two or zero positive real roots. If we replace x with -x, we get three sign changes, suggesting three or one negative real roots.

An important aspect to remember is that Descartes' Rule of Signs gives us possible quantities of roots, but not the exact amount or their values, and it can't predict complex roots. However, by utilizing the Fundamental Theorem of Algebra in conjunction with Descartes' Rule, we can get a fuller picture of the roots of a polynomial equation.

Answer:

5 or B

Step-by-step explanation:

Answer:

5

Step-by-step explanation:

The amplitude is the distance from the highest to lowest points divided that by 2.

Or the simpliest way when given the function

f(x) = A sin(x)

Where A is the amplitude

To determine the amplitude of the function \( f(x) = 5 \sin x \), let's review the concept of amplitude in the context of sinusoidal functions like sine and cosine.
The general form of a sine function is:
\[ f(x) = A \sin(Bx + C) + D \]
- \( A \) is the amplitude of the function, which determines the height of the wave's peak or the depth of its trough, relative to the center line of the wave.
- \( B \) affects the period of the function, which is the distance over which the wave pattern repeats.
- \( C \) is the phase shift, which determines where the function starts on the x-axis.
- \( D \) is the vertical shift, which moves the wave up or down on the y-axis.
The amplitude \( A \) is always a non-negative number. It represents the maximum value that the function reaches from its middle position (equilibrium). In other words, it's the distance from the middle of the wave to its peak or trough.
In the function you've provided, \( f(x) = 5 \sin x \), there's no phase shift (\( C \)) or vertical shift (\( D \)), and since there's no coefficient multiplying \( x \) inside the sine function, the period is not affected (\( B = 1 \)). The coefficient of \( \sin x \), here \( 5 \), is the amplitude of the function.
So, the amplitude of the function \( f(x) = 5 \sin x \) is simply the coefficient in front of the sine term, which in this case is \( 5 \). Therefore, the amplitude of \( f(x) \) is \( 5 \).

Answer:

Step-by-step explanation:

After the 440 pennies are added, the bank is 3/5 full.  It started out 1/10 full, so

3/5 - 1/10 is the amount of space in the bank that the 440 pennies took up, or

1/2.  Use proportions to solve this, with number of pennies on the top and the fraction of the bank that is filled on the bottom:

[tex]\frac{440}{\frac{1}{2} }=\frac{x}{1}[/tex]

where x is the number of pennies (our unknown) that it will take to fill the bank (1).  Cross multiply to get

[tex]\frac{1}{2}x=440[/tex]

so x = 880

y = k/x

We want to find k first.

4 = k/8

4(8) = (k/8)(8)

32 = k

We now need to find y knowing that k = 32 and x = 2.

y = k/x

y = 32/2

y = 16

Did you follow?

Answer:

16

Step-by-step explanation:

Inverse variation follows this equation:

y = k/x

We know one point, so we input x and y, and solve for k.

4 = k/8

k = 32

The inverse variation equation is

y = 32/x

Now we let x = 2, and use our equation to find y.

y = 32/2

y = 16

Answer:

(d):  0.7734

Step-by-step explanation:

Using the normcdf( statistical function built into my old TI 83-Plus calculator, I find that the area under the standard normal curve from the extreme left to the value 69 is 0.2266.  Thus, about 0.2266 of students would fail and

1.000-0.2266, or 0.7734 would pass.  This corresponds to Answer (d).

Answer:

D

Step-by-step explanation:

The midpoint formula is

[tex]M=(\frac{x_{1}+x_{2}  }{2},\frac{y_{1}+y_{2}  }{2})[/tex]

Filling in our coordinates where they go gives us:

[tex]M=(\frac{1+4}{2},\frac{1-16}{2})[/tex] so

[tex]M=(\frac{5}{2},\frac{-15}{2})[/tex]

-- average price of gas ... continuous.  An average can come out to be any number, with a huge string of decimal places.  There are no numbers it CAN'T be.

-- car's speed ... continuous.  Between zero and the car's maximum top speed, there are no numbers it CAN'T be.

-- number of cars ... discrete.  It has to be a whole number.  There can't be a half a car or 0.746 of a car passing through.  

-- number of phone calls ... discrete.  It has to be a whole number.  There can't be a half of a call or 0.318 of a call made.

-- salaries ... I'm a little fuzzy on this one.  The employer can set a person's salary to be anything he wants it to be.  If they want it to be a whole number, or ANY fraction, they can do it ... there's no number it CAN'T be.  BUT ... when it comes time to actually pay him, THAT has to be a whole number of pennies.  There are actually a lot of numbers that they CAN'T pay, because they can't give him half of a penny, or 0.617 of a penny.

So I'm going to say that salary is a discrete variable.  

Answer:

  C) All integers where n ≥ 1

Step-by-step explanation:

N is the term number, so is a "counting number," or "natural number." It is a positive integer.

Answer:

216.4 mm^2

Step-by-step explanation:

The polygon has 9 sides.

Divide the polygon into 9 congruent triangles. Each triangle has 2 sides of length 8.65 mm, so each triangle is isosceles. The measure of each internal angle of the polygon is (9 - 2)(180)/9 = 140 degrees. The base angles of an isosceles triangle measures 70 deg. The vertex angle measures 40 deg. Draw a segment from the center of the polygon to the midpoint of a side. This segment is the altitude of the triangle. Now the triangle has been split into two right triangles. The angles of the right triangle are 70, 90, and 20. 3.65 mm is the length of the hypotenuse. The length of the altitude is found with trig.

sin A = opp/hyp

sin 70 = h/8.65

h = 8.65 sin 70

h = 8.1283 mm

Now with the altitude, we can find the length of half of a side of the polygon.

a^2 + b^2 = c^2

x^2 + h^2 = 8.65^2

x^2 + 8.1283^2 = 8.65^2

x = 2.9585

Half a side measures 2.9585 mm.

The side of the polygon measures 5.9169 mm.

The area of the polygon is 9 times the area of one triangle.

area = 9 * base * height/2

area = 9 * 5.9169 mm * 8.1283 mm / 2

area of polygon = 216.4 mm^2

NOTE: (gºf)(6) means g(f(6)).

First find f(6).

f(6) = 2(6) - 6

f(6) = 12 - 6 or 6.

We now find g(6).

g(6) = (6)^3

g(6) = 216

ANSWER:

(gºf)(6) = 216

Answer:

A) open up if [tex]a[/tex] is positive .

B) open down if [tex]a[/tex] is negative.

Step-by-step explanation:

If you are given a quadratic in vextex form, [tex]y=a(x-h)^2+k[/tex], then the parabola is:

A) open up if [tex]a[/tex] is positive .

B) open down if [tex]a[/tex] is negative.

Answer:

See below.

Step-by-step explanation:

The coefficient of x^2 will tell you this. If it is positive the graph opens upwards , negative it will open downwards.

For example

-(x - 4)^2 + 1  will open downwards and (x - 4)^2 + 1 will open upwards.

Answer: 6 hours

Step-by-step explanation:

Given : Time taken by Type JQ pump to fill the tank = 72 hours

Then , Time taken by Type JQ pump to fill half of the tank = 36 hours

Time taken by Type JT pump to fill the tank = 18 hours

Then, Time taken by Type JT pump to fill the tank = 9 hours

Now, if he tank starts at half full then the time taken by two Type JQ pumps and a Type JT pump all three pumps working together to fill the tank :-

[tex]\dfrac{1}{t}=\dfrac{1}{36}+\dfrac{1}{36}+\dfrac{1}{9}\\\\\Rightarrow\dfrac{1}{t}=\dfrac{1+1+4}{36}=\dfrac{1}{6}\\\\\Rightarrow t=6[/tex]

Hence, it will take 6 hours to fill the tank.

Explanation:

An absolute value function in the form ...

  f(x) = a|x -h| +k

will have its vertex at (x, y) = (h, k). The sign on scale factor "a" will tell you whether it opens upward (a > 0) or downward (a < 0).

If a is positive, the vertex is a minimum, and the range is [k, ∞).

If a is negative, the vertex is a maximum, and the range is (-∞, k].

The range of an absolute value function is determined by its vertex and the value of 'a'. If 'a' is positive, the function opens upwards and the minimum of the range is the y-coordinate of the vertex. If 'a' is negative, the function opens downwards, the maximum of the range is the y-value of the vertex.

The range of an absolute value function can be determined using the vertex and the value of 'a' in the function’s equation. In an absolute value function, the vertex is the lowest or highest point on the graph, depending on whether the function opens upwards or downwards. The value 'a' influences the direction of the opening: if 'a' > 0, the graph opens upwards, and if 'a' < 0, it opens downwards.

For example, consider the function |a(x-h)|+k, where (h,k) is the vertex. If 'a' is positive, then the minimum range of the function will be 'k', and the function will extend to positive infinity, making the range [k, ∞). If 'a' is negative, the function will extend towards negative infinity, making its maximum value 'k', and thereby setting the range to (-∞, k].

This means, for instance, if we have a function like y = 3|x - 2| + 1, the value of 'a' is 3, which is positive, thus the function opens upwards, and the vertex is (2,1), which indicates that the range of this function is [1,∞).

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Answer:

y

Step-by-step explanation:

Flip b/x to the proportion x is now y.

Answer:

b/x = (x + y)/b

where c = x + y then

b/x = c/b

Step-by-step explanation:

Considering the larger of the three triangles the image represents,

b and a in light of the angle A form by both sides (the angle opposite the side a) would have a relationship

Cos A = x/b

hence b/x = 1/Cos A

Considering the largest triangle,

Cos A = b/(x +y)

hence,

(x + y)/b = 1/Cos A

as such,

b/x = (x + y)/b

Answer:

  1

Step-by-step explanation:

There are about 110,000 days in 300 years, so the expected number of failures is about 10/11 ≈ 1.

_____

This assumes the launch conditions are identical for each of the launches, or that whatever variation there might be has no effect on the probability. These are bad assumptions.

A failure rate of 1 in 100,000 with daily launches over 300 years would statistically result in approximately 1 failure.

If the chance of a space shuttle failure is 1 in 100,000, we can calculate the expected number of failures over 300 years with the assumption that a shuttle is launched every day. There are 365 days in a year, so over 300 years, there would be 300 × 365 = 109,500 launches. Given the failure rate of 1 in 100,000, we would then expect approximately 109,500 / 100,000 = 1.095 failures, which rounded to the nearest integer is 1 failure.

In other words, if the failure rate estimated by the shuttle program management were accurate and a shuttle was launched every day for 300 years, one would expect about 1 failure during that time span.

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Answer:

∠A ≅ ∠A; reflexive property

Step-by-step explanation:

Statement and reason are missing in the proof is "∠A ≅ ∠A reflexive property". Option 3 is the right choice.

The image shows a proof of the congruence of two isosceles triangles, ΔABC and ΔAYX. The proof uses the reflexive property of congruence to show that ∠A ≅ ∠A, and then the SAS similarity theorem to show that ΔABC ≅ ΔAYX.

The reflexive property of congruence is a fundamental property of congruence, and it is used in many proofs in geometry. It is important to understand the reflexive property of congruence and how it can be used in proofs.

Option 3 is the right choice.

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Answer:

$165,000

Step-by-step explanation:

$50,000 + $100,000 + $15,000 = $165,000

Hope this helps C:

Everett McCook's annual payment for liability insurance can be calculated by dividing his total liability coverage by the annual distance he drives to college.

To determine Everett McCook's annual payment for liability insurance, we need to calculate the total premium he pays. Since he drives 5 miles each way to the college, his daily distance is 10 miles. To calculate his annual distance, we multiply this by the number of days he drives to college in a year. Assuming he works on all weekdays, he drives 5 days a week for 52 weeks, resulting in 260 days a year. Therefore, his annual distance is 260 * 10 = 2600 miles.

Now, let's calculate his annual payment. Since his liability insurance includes $50,000 for single bodily injury, $100,000 for total bodily injury, and $15,000 for property damage, we can add these amounts together. Therefore, his liability coverage is $50,000 + $100,000 + $15,000 = $165,000.

Lastly, we divide the total liability coverage by the annual distance to find the cost per mile. So, the annual payment is $165,000 / 2600 = $63.46 per mile.

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Answer:

Step-by-step explanation:

The given parallelogram is attached.

First, we have to define some terms:

So, if we observe points D, G and J, from given options, they are non-collinear and non-coplanar, because they are not on the same line, nor plane.

A because it has a set between the whole numbers 1 and 4 (and slightly beyond those in decimals)

Answer:

40,000 workers

Step-by-step explanation:

This question deals with simple rates proportion.

If it takes 20 years for X workers to complete a job,

if we double the amount of workers, we will take half the time.

hence to halve the time from 20 years to 10 years, we will need to double the amount of workers.

In this case we will need 20,000 workers x 2 = 40,000 workers

Answer: 40,000

Step-by-step explanation:

(20÷10)×20,000

Answer:

Option 1:(x-4)^2+y^2=100

Step-by-step explanation:

Given center = (h,k) = (4,0)

The point (-2,8) lies on circle which means the distance between the point and center will be equal to the radius.

So,

The distance formula will be used:

[tex]d = \sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}} \\=\sqrt{(4+2)^{2}+(0-8)^{2}}\\=\sqrt{(6)^{2}+(-8)^{2}}\\=\sqrt{36+64}\\ =\sqrt{100}\\ =10\ units[/tex]

Hence radius is 10.

The standard form of equation of circle is:

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

Putting the values

(x-4)^2+(y-0)^2=10^2

(x-4)^2+y^2=100

Hence option 1 is correct ..

Answer:

A

Step-by-step explanation:

Answer:

  B.  a circle inscribed inside the triangle

Step-by-step explanation:

The incenter is the center of the inscribed circle. The incenter is the intersection of angle bisectors.

___

The center of mass and balance is the centroid, the intersection of medians.

Answer:

B.

Step-by-step explanation:

That is obtained when you split the angle of each vertex into two equal parts and trace the bisector. The intersection point is the incenter or the center of the circle inscribed inside the triangle.

Answer:

Radians help make calculations easier. Also, They measure arc-length on the circle, giving you angle in the sense which you can actually represent on the number line without any conversion.

Answer:

[tex]2.6\times 10^{19}m^{3}[/tex]

Step-by-step explanation:

we have given thickness of atmosphere =50 km

radius of earth =6400 km

average temperature of atmosphere=15°C

Average pressure of atmosphere= 0.20 atm

we have to calculate the volume of atmosphere

so the we have to calculate the volume of atmosphere = ( volume of earth +atmosphere) - volume of earth

volume of atmosphere =[tex]\frac{4}{3}\times\pi \times \left ( 6400+50 \right )^{3}-\frac{4}{3}\times \pi\times 6400^3[/tex]

=[tex]2.6\times 10^{19}m^{3}[/tex]

To calculate the total volume of the Earth's atmosphere, one has to use the principles of geometry for spheres and subtract the volume of the Earth from the volume of the Earth and its atmosphere.

The volume of the Earth's atmosphere can be calculated by using the geometry of spheres and the characteristics provided: the average thickness of the Earth's atmosphere (50 km) and Earth's radius (6,400 km). We model Earth and its atmosphere as a larger sphere encapsulating a smaller one and define the larger sphere's radius as the Earth's radius plus the thickness of the atmosphere.

To get the most accurate results, use the accurate value of π and proper bracket organization for your calculations.

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Explanation:

Tommy calculated the area of the base by squaring the diameter, not the radius. If he's going to use the diameter in the area formula, he needs to divide the result by 4.

  A = πr² = πd²/4

Answer:

Sample response: Tommy used the diameter instead of the radius in the formula to find the area of the base. The diameter needs to be divided by 2 to find the radius. The correct area of the base is 1 pi square inches. The correct total volume is 3 pi cubic inches.Step-by-step explanation:

Answer:

C!!!!!! :She made a mistake when finding the additive inverse of (7n2 – 5n + 6).

Step-by-step explanation:

i just did the assignment on edge

She made a mistake when finding the additive inverse of (7n2 – 5n + 6).

Additive inverse absolutely means converting the signal of the wide variety and adding it to the original range to get a solution identical to zero. The properties of additive inverse are given underneath, based totally on the negation of the original wide variety. as an instance, if x is the authentic quantity, then its additive inverse is -x.

Additive inverse is what you upload to a number to make the sum zero. for instance, the additive inverse of 4 is -four due to the fact their sum is 0. while numbers are delivered together to get 0, then we are saying both the numbers are additive inverses of every different.

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Answer:

y=4x-4

Step-by-step explanation:

The equation of a line is slope-intercept form is: y=mx+b where m is the slope and b is the y-intercept. This is the required form I think. Your document says write in slope... can't read the rest because it is cut off.

I'm actually going to use point-slope form which is: y-y1=m(x-x1) where m is the slope and (x1,y1) is a point we know that is on the line.

We have m=4.

We can actually find a point on the line. Both the line and the curve y=x^2 cross at x=2.

So we find the corresponding y-coordinate on our line to x=2 by plugging into x^2.

x^2 evaluated at x=2 gives us 2^2=4.

So we have the slope m=4 and a point (x1,y1)=(2,4) on the line.

Let's plug it into the point-slope form:

y-4=4(x-2)

Now the goal was y=mx+b form so let's solve our for y.

y-4=4(x-2)

Distribute 4 to terms in ( ):

y-4=4x-8

Add 4 on both sidea:

y=4x-4

Answer:

8.4 gallons

Step-by-step explanation:

We have a boater traveled 532 miles and we want to know how much gas he used for this trip.

We are also given if he travels 63 miles then has used 1 gallon.

532 miles ->x gallons

63 miles   ->1  gallon

The information here is lined up for you already.

We have from the line, this proportion:

[tex]\frac{532}{63}=\frac{x}{1}[/tex]

x/1=x so we have:

[tex]\frac{532}{63}=x[/tex]

[tex]8.\overline{4}=x[/tex]

So approximately 8.4 gallons was used on this trip of 532 miles.

Answer:

8.4 gallons

Step-by-step explanation:

A boater travels 532 miles. Assuming the boat averages 63 miles per gallon, there are 8.4 gallons of gasoline that were used.