Nancy and Harry are sliding a stone statue and moving it to a new location in their garden. Nancy is pushing the statue with a force of 120N at a 60° angle to the horizontal and Harry is pulling the statue with a force of 180N at a 40° angle with the horizontal. What is the magnitude of the horizontal force exerted on the statue?

Answer:

Option D (12).

Step-by-step explanation:

The law of outcomes states that if there are m ways to do Event 1 and n ways to do Event 2, then if both Event 1 and Event 2 are combined, then the possible outcomes will be m*n. Similarly, in this case, there are 2 types of products, 2 types of materials, and 3 types of colours. So according to the law of outcomes, simply multiply the numbers to gain the total possible outcomes:

Possible outcomes = 2 * 2 * 3 = 4 * 3 = 12.

So Option D is the correct answer!!!

Answer:

12 is correct.

Step-by-step explanation:

Answer:

  D.  (x, y) → (-x, -y)

Step-by-step explanation:

A. (x,y) → (y,x) . . . . reflects across the line y=x

B. (x,y) → (y,-x) . . . . rotates 90° CCW

C. (x,y) → (-y,-x) . . . . reflects across the line y=-x

D. (x,y) → (-x,-y) . . . . rotates 180° about the origin

Answer:

The correct option is D.

Step-by-step explanation:

If a point rotating 180° counterclockwise about the origin, then the sign of each coordinate is changed.

Consider the coordinates of a point are (x,y).

If a (x,y) rotating 180° counterclockwise about the origin, then the rule of rotation is defined as

[tex](x,y)\rightarrow (-x,-y)[/tex]

In which (x,y) is the coordinate pair of preimage and (-x,-y) is the coordinate pair of image.

Therefore the correct option is D.

If a point reflects across the line y=x , then

[tex](x,y)\rightarrow (y,x)[/tex]

If a point rotated 90° clockwise, then

[tex](x,y)\rightarrow (y,-x)[/tex]

If a point reflects across the line y=-x, then

[tex](x,y)\rightarrow (-y,-x)[/tex]

Answer:

Step-by-step explanation:

f(10) = 120 tells you that after x = 10 minutes, the coffee is 120 degrees

The statement f(10)=120 indicates that after 10 minutes, the coffee's temperature is 120 degrees Fahrenheit.

When we see an equation such as f(10)=120, it tells us that after 10 minutes, the temperature of the coffee has cooled down to 120 degrees Fahrenheit. The function f(x) describes the temperature of the coffee after x minutes, so the specific point f(10)=120 provides us with a snapshot of the temperature at that particular time.

Answer:

2 gallons

Step-by-step explanation:

At this speed, the car uses 1 gallon of fuel for a distance of 25 miles.

We need the number of miles the car travels in 75 minutes to find the amount of fuel it uses.

75 minutes * (1 hour)/(60 minutes) = 1.25 hours

speed = distance/time

distance = speed * time

distance = 40 miles/hour * 1.25 hours = 50 miles

In 75 minutes, at 40 mph, the car travels 50 miles.

(1 gal)/(25 miles) = x/(50 miles)

x = 2 gal

Answer: 2 gallons

The car will use 2 gallons of fuel when traveling at a constant speed of 40 miles per hour for 75 minutes.

To find the number of gallons of fuel used when the car travels at a constant speed of 40 miles per hour for 75 minutes, we can use the formula:

Gallons of fuel used = (Distance traveled in miles) / (Miles per gallon)

Since the car travels at a constant speed of 40 miles per hour, it covers a distance of 40 miles in 1 hour. Therefore, in 75 minutes it will travel 40 miles * (75 minutes / 60 minutes per hour) = 50 miles.

Now, we can calculate the number of gallons of fuel used: Gallons of fuel used = 50 miles / 25 miles per gallon = 2 gallons.

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Answer:

(3,4,5)

(6,8,10)

(5,12,13)

(8,15,17)

(12,16,20)

(7,24,25)

(10,24,26)

(20,21,29)

(16,30,34)

(9,40,41)

Just choose 2 numbers from {1,2,3,4,5,6,7,8,...} and make sure the one you input for x is larger.

Post the three in the comments and I will check them for you.

Step-by-step explanation:

We need to choose 2 positive integers for x and y where x>y.

Positive integers are {1,2,3,4,5,6,7,.....}.

I'm going to start with (x,y)=(2,1).

x=2 and y=1.

[tex](2^2+1^2)^2=(2^2-1^2)^2+(2\cdot2\cdot1)^2[/tex]

[tex](4+1)^2=(4-1)^2+(4)^2[/tex]

[tex](5)^2=(3)^2+(4)^2[/tex]

So one Pythagorean Triple is (3,4,5).

I'm going to choose (x,y)=(3,1).

x=3 and y=1.

[tex](3^2+1^2)^2=(3^2-1^2)^2+(2\cdot3\cdot1)^2[/tex]

[tex](9+1)^2=(9-1)^2+(6)^2[/tex]

[tex](10)^2=(8)^2+(6)^2[/tex]

So another Pythagorean Triple is (6,8,10).

I'm going to choose (x,y)=(3,2).

x=3 and y=2.

[tex](3^2+2^2)^2=(3^2-2^2)^2+(2\cdot3\cdot2)^2[/tex]

[tex](9+4)^2=(9-4)^2+(12)^2[/tex]

[tex](13)^2=(5)^2+(12)^2[/tex]

So another is (5,12,13).

I'm going to choose (x,y)=(4,1).

[tex](4^2+1^2)^2=(4^2-1^2)^2+(2\cdot4\cdot1)^2[/tex]

[tex](16+1)^2=(16-1)^2+(8)^2[/tex]

[tex](17)^2=(15)^2+(8)^2[/tex]

Another is (8,15,17).

I'm going to choose (x,y)=(4,2).

[tex](4^2+2^2)^2=(4^2-2^2)^2+(2\cdot4\cdot2)^2[/tex]

[tex](16+4)^2=(16-4)^2+(16)^2[/tex]

[tex](20)^2=(12)^2+(16)^2[/tex]

We have another which is (12,16,20).

I'm going to choose (x,y)=(4,3).

[tex](4^2+3^2)^2=(4^2-3^2)^2+(2\cdot4\cdot3)^2[/tex]

[tex](16+9)^2=(16-9)^2+(24)^2[/tex]

[tex](25)^2=(7)^2+(24)^2[/tex]

We have another is (7,24,25).

You are just choosing numbers from the positive integer set {1,2,3,4,... } and making sure the number you plug in for x is higher than the number for y.

I will do one more.

Let's choose (x,y)=(5,1).

[tex](5^2+1^2)^2=(5^2-1^2)^2+(2\cdot5\cdot1)^2[/tex]

[tex](25+1)^2=(25-1)^2+(10)^2[/tex]

[tex](26)^2=(24)^2+(10)^2[/tex]

So (10,24,26) is another.

Let (x,y)=(5,2).

[tex](5^2+2^2)^2=(5^2-2^2)^2+(2\cdot5\cdot2)^2[/tex]

[tex](25+4)^2=(25-4)^2+(20)^2[/tex]

[tex](29)^2=(21)^2+(20)^2[/tex]

So another Pythagorean Triple is (20,21,29).

Choose (x,y)=(5,3).

[tex](5^2+3^2)^2=(5^2-3^2)^2+(2\cdot5\cdot3)^2[/tex]

[tex](25+9)^2=(25-9)^2+(30)^2[/tex]

[tex](34)^2=(16)^2+(30)^2[/tex]

Another Pythagorean Triple is (16,30,34).

Let (x,y)=(5,4)

[tex](5^2+4^2)^2=(5^2-4^2)^2+(2\cdot5\cdot4)^2[/tex]

[tex](25+16)^2=(25-16)^2+(40)^2[/tex]

[tex](41)^2=(9)^2+(40)^2[/tex]

Another is (9,40,41).

Answer:

Option D

Step-by-step explanation:

We have the following variable definitions:

sofas: x

pillows: y

Pillows come in pairs so we have 2y pillows

The total order for all the possible combinations is:

[tex]x+2y[/tex]

The wholesaler requires a minimum of 4 items in each order from its retail customers. This means the retailers can order 4 or more.

Therefore the inequality is:

[tex]x+2y\ge4[/tex]

To graph this inequality, we graph the corresponding linear equation, [tex]x+2y=4[/tex]  with a solid line and shade above.

The correct choice is D

See attachment

The first step is to add 7 to both sides, applying the addition property of equality:

[tex]-2(4x-1)-7+7=5+7 \iff -2(4x-1)=12[/tex]

Answer:

The equation can become −2(4x − 1) = 12 by applying the commutative property of addition

Step-by-step explanation:

Answer:

2x^3+8x^2-54x-84

Step-by-step explanation:

Answer:

2(x + 7)(x² - 3x - 6) = 2x³ + 8x² - 54x - 84

Step-by-step explanation:

Simplification is a method used to reduce the complexity or the component parts of an algebraic equation which makes it simpler and easier to understand.

The given equation is: 2(x + 7)(x² - 3x - 6).

Simplifying the given algebraic equation:

⇒ 2 (x + 7) (x² - 3x - 6)

⇒ (2x + 14) (x² - 3x - 6)

⇒ 2x³ + 14x² - 6x² - 42x - 12x - 84

⇒ 2x³ + 8x² - 54x - 84

Answer:

Question 9: False

Question 10: False

Step-by-step explanation:

The third side is always greater than the other two sides.

Question 9

a = 6, b = 6, c = 5

Since the third side is the smallest, it would not create a triangle.

Question 10

a = 7, b = 2, c = 5

Since the third side is the smallest, it would not create a triangle.

Answer:

Question 9: True

Question 10: False

Step-by-step explanation:

The Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than the last side.

To test if the three lengths create a triangle you would have to test the three combinations if the two numbers are greater than the last number.

The three lengths 6, 6, 5 create a triangle.

First check the first two numbers.

Next check the first and last number.

Last check the second and last number.

The answer for question 9 is TRUE.

The three lengths 7, 2, 5 create a triangle.

Check the first two numbers.

Check the first and last number.

Finally, check the second and last number.

The answer for question 10 is FALSE.

Answer:

Step-by-step explanation:

Part A

xf = xo + vo* t + 1/2 a*t^2                    Subtract xo

xf - xo = 0*t + 1/2 a*t^2                        multiply by 2

2(xf - xo) = at^2                                    divide by t^2

2(xf - xo ) / t^2 = a

Part B

Givens

xo =0

vo = 0

a = 10  m/s^2

xf = 120 m

Solution

xf = xo + vo* t + 1/2 a*t^2              Substitute the givens

120 = 0 + 0 + 1/2 * 10 * t^2            Multiply by 2

120*2 = 10* t^2                  

240 = 10*t^2                                  Divide by 10

240/10  = t^2

24 = t^2                                          take the square root of both sides.

√24 = √t^2

t = √24

t = √(2 * 2 * 2 * 3)

t = 2√6

The maximum difference in the lengths of the radii is 1/2.

To solve this problem, let's denote the radius of the smaller circle as ( r ) and the radius of the larger circle as ( R ). We're given that [tex]\( r + R = 10 \)[/tex].

The area of a circle is given by the formula [tex]\( A = \pi r^2 \)[/tex], where ( r ) is the radius.

We want the absolute difference in the areas of the two circles to be less than or equal to [tex]\( 5\pi \)[/tex]. So, we can set up the following inequality:

[tex]\[ |(\pi R^2) - (\pi r^2)| \leq 5\pi \][/tex]

[tex]\[ |(\pi (10-r)^2) - (\pi r^2)| \leq 5\pi \][/tex]

Expanding and simplifying:

[tex]\[ |(100\pi - 20\pi r + \pi r^2) - (\pi r^2)| \leq 5\pi \][/tex]

[tex]\[ |100\pi - 20\pi r| \leq 5\pi \][/tex]

[tex]\[ 100 - 20r \leq 5 \][/tex]

[tex]\[ 100 - 5 \leq 20r \][/tex]

[tex]\[ 95 \leq 20r \][/tex]

[tex]\[ \frac{95}{20} \leq r \][/tex]

[tex]\[ r \geq \frac{19}{4} \][/tex]

So, the maximum difference in the lengths of the radii is when [tex]\( r = \frac{19}{4} \)[/tex] and [tex]\( R = 10 - r = 10 - \frac{19}{4} = \frac{21}{4} \)[/tex].

The maximum difference in the lengths of the radii is [tex]\( \frac{21}{4} - \frac{19}{4} = \frac{2}{4} = \frac{1}{2} \)[/tex].

Answer:

[tex]AC=4.3\ in[/tex]

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

The triangle AOC is an isosceles triangle

OA=OC=5/2=2.5 in -----> the radius of the circle

∠AOC=180°-60°=120°

∠CAO=∠ACO=120°/2=60°

Applying the law of cosines find the length of the chord AC

[tex]AC^{2}=OA^{2}+OC^{2}-2(OA)(OC)cos(120\°)[/tex]

substitute

[tex]AC^{2}=2.5^{2}+2.5^{2}-2(2.5)(2.5)cos(120\°)[/tex]

[tex]AC^{2}=18.75[/tex]

[tex]AC=4.3\ in[/tex]

4(x+2)=3(x-2)

Multiply the first bracket by 4

Multiply the second bracket by 3

4x+8=3x-6

Move 3x to the left hand side, whenever moving a number with a letter the sign changes ( positive 3x to negative 3x)

4x-3x+8=3x-3x-6

x+8=-6

Move positive 8 to the right hand side

x+8-8=-6-8

x=-14

Check answer by using substitution method

Use x=-14 into both of the equations

4(-14+2)=3(-14-2)

-56+8=-42-6

-48=-48

Answer is -14- D)

The algebraic equation 4(x + 2) = 3(x − 2) is solved by distribution, combining like terms, and isolating the variable x, which results in x = -14.

This is a simple algebraic equation problem. We solve 4(x + 2) = 3(x − 2) by following these steps:

So, x = -14 is the solution.

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Answer:

0

Step-by-step explanation:

100 - 10 = 90

90 - 30 = 60

60 - 10 = 40

40 - 10 = 30

Answer:

a. 0.17

Step-by-step explanation:

Total number of light bulbs = 8

The probability that each individual light bulb works = 0.8

The working of light bulbs is independent of each other, this means one light bulb does not influence the other light bulbs.

We need to calculate the probability that all eight of the light bulbs work. Since the light bulbs work independently, the overall probability of independent events occurring together is the product of their individual probabilities. Therefore,

Probability that all eight of the light bulbs work = 0.8 x 0.8 x 0.8 x 0.8 x 0.8 x 0.8 x 0.8 x 0.8

= [tex](0.8)^{8}[/tex]

= 0.16777216

≈ 0.17

Thus, option a gives the correct probability that all eight of the light bulbs work

You can use binomial distribution, and thus, its probability function to find the needed probability.

The probability that all eight of the light bulbs work is 0.167

Binomial distributions consists of n independent Bernoulli trials.

Bernoulli trials are those trials which end up randomly either on success (with probability p) or on failures( with probability 1- p = q (say))

Suppose we have random variable X pertaining binomial distribution with parameters n and p, then it is written as

[tex]X \sim B(n,p)[/tex]

The probability that out of n trials, there'd be x successes is given by

[tex]P(X =x) = \: ^nC_xp^x(1-p)^{n-x}[/tex]

Since all the light bulbs' working is independent, and each bulb's chance of working is 0.8 and there are 8 bulbs, thus,

n = 8

p = 0.8

and Let X be a random variable tracking how many out of 8 bulbs are working, then we have:

[tex]X \sim B(8, 0.8)[/tex]

Then, the needed probability is P(X = 8) (since we need to know probability that all 8 bulbs will work)

By using the probability mass function of binomial distribution, we get:

[tex]P(X =x) = \: ^nC_xp^x(1-p)^{n-x}\\P(X = 8) = \:^8C_8(0.8)^8(1-0.8)^{8-8} = 1 \times (0.8)^8 \times 1 \approx 0.167[/tex]

Thus,

The probability that all eight of the light bulbs work is 0.167

Learn more about binomial distribution here:

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Answer:

Answer would be 9y+36

Step-by-step explanation:

Because if you distribute the 9 inside the parenthesis, you'd get

9*y=9y and 9*4=36

so 9y+36

Hope my answer was helpful to you!

Final answer:

The Distributive Property is used to rewrite the expression 9(y + 4) as 9y + 36 by multiplying 9 by each term inside the parentheses.

Explanation:

To use the Distributive Property to rewrite the expression 9(y + 4), we would distribute the number 9 to both y and 4 inside the parentheses. This means we multiply 9 by y and then multiply 9 by 4, combining the results with the addition operations between them.

Using the distributive property, we get:

9 times y = 9y

9 times 4 = 36

So, the expression will be rewritten as:

9y + 36

Therefore, by distributing the 9, we have turned the original expression into a sum of two terms, which are a number, variable, or a product/quotient of numbers and/or variables separated by + or - signs. In this case, the terms are 9y and 36.

Answer: 19.2 inches would be the most reasonable answer, since the first two is too small, and the last answer would be too tall.

if there was 2 feet of water, it would be 24 inches full. taking 1 gallon out, wouldn't make the difference to make it go up or down much.

Answer:

c) 19.2 inches

Step-by-step explanation:

Height of water when full = 2 feet = 24 inches

Radius of cylinder = 4 inches

Volume of tank = 5 gallon

Gallon per inch height of tank = [tex]\frac{5}{24}[/tex]

Inch per gallon of height = [tex]\frac{24}{5}[/tex]

So, when 1 gallon is removed

[tex]24-1\times \frac{24}{5}=\frac{96}{5}=19.2\ inches[/tex]

∴ Height of the water in the tank after Alex filled the 1 gallon jug is 19.2 inches.

Volume of cylinder after 1 gallon was removed

[tex]\pi r^2h=4\times 231\\\Rightarrow h=\frac{4\times 231}{\pi 4^2}\\\Rightarrow h=18.38\ inches[/tex]

∴Height of the water in the tank after Alex filled the 1 gallon jug is 18.38 inches

The different height arises due to the thickness of the rank which is not given.

The first method is more accurate

Answer:

  B.  46,368

Step-by-step explanation:

Each Fibonacci number is the sum of the previous two. The next one is the sum of the two that are given.

  F(24) = F(22) +F(23) = 17,711 +28,657 = 46,368

Answer:

Length of arc AB is,

= 2πr (angle between AB) /360

=2×3.14×90/360

=1.57 cm

For this case we have that by definition, the arc length of a circle is given by:

[tex]AL = \frac {x * 2 \pi * r} {360}[/tex]

Where:

x: Represents the angle between AB. According to the figure we have that x = 90 degrees.

[tex]r = 7.9 \ cm[/tex]

So:

[tex]AL = \frac {90 * 2 \pi * 7.9} {360}\\AL = \frac {90 * 2 * 3.14 * 7.9} {360}\\AL = \frac {4465,08} {360}\\AL = 12.403[/tex]

Answer:

[tex]12.4\ cm[/tex]

Answer:

24 kilometers.

Step-by-step explanation:

The shortest path between two points is a straight segment that connects the two points.

Refer to the diagram attached. The 6-km segment and the 8-km segment are normal to each other. Together with the segment that joins the library and the house, the three segments now form a right triangle.

The length of the hypotenuse can be found with the Pythagorean Theorem.

[tex]\begin{aligned}\text{Hypotenuse} &= \sqrt{(\text{Leg 1})^{2} + (\text{Leg 2})^{2}}\\&= \sqrt{6^{2} + 8^{2}}\\&= \sqrt{36 + 64} \\&= \sqrt{100}\\&= \rm 10\;km\end{aligned}[/tex].

The length of the round-trip will equal to the sum of the length of the three segments: [tex]\rm 6\;km + 8\;km + 10\;km = 24\;km[/tex].

Answer:

20,845 square meters

Step-by-step explanation:

We can use the formula for area of a triangle to figure this out easily.

Area = [tex]\frac{1}{2}abSinC[/tex]

Where

a and b are the two side lengths of the triangle given, and

C is the ANGLE BETWEEN the two sides

Clearly, we see that one side is 218.5 and other is 224.5 and the angle between them is given by 58.2 degrees. Now we simply substitute these values into the formula and get the area:

[tex]A=\frac{1}{2}abSinC\\A=\frac{1}{2}(218.5)(224.5)Sin(58.2)\\A=20,844.99[/tex]

Rounding, we get the area to be 20,845 square meters

Answer:

20,845 m2

Step-by-step explanation:

I got it correct on founders edtell

Answer:

0.889

Step-by-step explanation:

I think it is multiplied.

10 *9 = 90

which is the total number of ways you can draw 2 cards without replacement.

I think it is easier to figure out how many possibilities there are over 62 and go from there.

10*9

10*8

10*7

===========

9*8

9*7

So there are 5 combinations that are over 62. There are 5 more possibilities because you could draw them in the reverse order

9 * 10

8 * 10

7 * 10  

8* 9

7 * 9

In all there are 10 ways of drawing numbers that are over 62

So what is the probability of drawing 2 cards above 62?

10/90 = 1/9 = 0.111

Therefore, there must be a probability of 1 - 0.111 for under 62 = 0.889

Answer:

x = 3 or x = 2 or x = 0 thus: 5

Step-by-step explanation:

Solve for x over the real numbers:

x^2 = (6 x)/(5 - x)

Cross multiply:

x^2 (5 - x) = 6 x

Expand out terms of the left hand side:

5 x^2 - x^3 = 6 x

Subtract 6 x from both sides:

-x^3 + 5 x^2 - 6 x = 0

The left hand side factors into a product with four terms:

-x (x - 3) (x - 2) = 0

Multiply both sides by -1:

x (x - 3) (x - 2) = 0

Split into three equations:

x - 3 = 0 or x - 2 = 0 or x = 0

Add 3 to both sides:

x = 3 or x - 2 = 0 or x = 0

Add 2 to both sides:

Answer:  x = 3 or x = 2 or x = 0

Answer:

 55

Step-by-step explanation:

Let x represent the middle integer. Then the smallest is x-2 and the largest is x+2. Your requirement is that ...

  (x-2)/(x+2) > 2/3

  3x -6 > 2x +4 . . . . cross multiply

  x > 10 . . . . . . . . . . .add 6-2x

The smallest integer satisfying this requirement is x=11. The sum of the 5 integers is 5x = 55.

The smallest sum is 55.

Answer:

55

Step-by-step explanation:

Answer:

[tex]9x^2y-5xy[/tex]

Step-by-step explanation:

Split it up like this to make it easier to work with:

[tex]\frac{27x^2y}{3}-\frac{15xy}{3}[/tex]

Since the only thing in the denominator of those fractions is a 3, we can only divide the 27 by 3, not the x or y terms.  Same thing with the second fraction.  27 divided by 3 is 9 and 15 divided by 3 is 5, so

[tex]9x^2y-5xy[/tex]

is the solution.  It is not completely simplified, but that isn't what you asked for, so this should suffice as the answer.

Answer:

Option 4: (x+1)^2+(y-1)^2 = 16

Step-by-step explanation:

The radius of the given circle in attached picture is: 4 units

The center is denoted by (h,k) = (-1,1)

So,

The standard form of equation with center at (h,k) and radius r

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

Putting the values

(x-(-1))^2 + (y-1)^2 = 4^2

(x+1)^2+(y-1)^2 = 16

Hence option number 4 is correct ..

Answer:

[tex]\frac{8}{17}[/tex] of fish in this aquarium are guppies.

Step-by-step explanation:

Let x be the number of guppies and y be the number of swordtails in the aquarium,

According to the question,

[tex]\frac{3}{4}\text{ of } x=\frac{2}{3}\text{ of }y[/tex]

[tex]\frac{3x}{4}=\frac{2y}{3}[/tex]

By cross multiplication,

[tex]9x=8y[/tex]

[tex]\implies \frac{x}{y}=\frac{8}{9}[/tex]

Thus, the ratio of guppies and swordtail fishes is 8 : 9

Let guppies = 8x, swordtail = 9x

Where, x is any number,

Since, the aquarium contains only two kinds of fish, guppies and swordtails,

So, the total fishes = 8x + 9x = 17x

Hence, the fraction of fish in the aquarium are guppies = [tex]\frac{\text{Guppies}}{\text{Total fishes}}[/tex]

[tex]=\frac{8x}{17x}[/tex]

[tex]=\frac{8}{17}[/tex]

To find what fraction of fish in the aquarium are guppies, you express the given relationship between the number of guppies and swordtails algebraically and solve for the number of guppies relative to the total number of fish, concluding that 8/17 of the fish in the aquarium are guppies.

If 3/4 of the number of guppies is equal to 2/3 of the number of swordtails, we can express this relationship using variables. Let G represent the number of guppies and S represent the number of swordtails in the aquarium. The given relationship can be written as (3/4)G = (2/3)S.

To find the fraction of fish that are guppies, we need to express G in terms of S first. By manipulating the equation, we multiply both sides by (4/3) to get G = (4/3)*(2/3)S = (8/9)S. This equation shows that the number of guppies is (8/9) times the number of swordtails.

Now, to find the total number of fish (T), we add the number of guppies and swordtails: T = G + S. Substituting the value of G from the equation above, we get T = (8/9)S + S = (17/9)S. The fraction of the total that are guppies is then G/T = [(8/9)S]/[(17/9)S] which simplifies to 8/17. Therefore, 8/17 of the fish in the aquarium are guppies.

Answer:

The probability that Andrew will be among the 4 volunteers selected and Karen will not is 2/7.

Step-by-step explanation:

From the given information it is clear that

The total number of volunteers, including Andrew and Karen = 8

The total number of volunteers, excluding Andrew and Karen = 8-2 = 6

We need to find the probability that Andrew will be among the 4 volunteers selected and Karen will not.

Total number of ways of selecting r volunteers from n volunteers is

[tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]

Total number of ways of selecting 4 volunteers from 8 volunteers is

[tex]\text{Total outcomes}=^8C_4=70[/tex]

Total number of ways of selecting 4 volunteers from 8 volunteers, so that Andrew will be among the 4 volunteers selected and Karen will not is

[tex]\text{Favorable outcomes}=^1C_1\times ^6C_3=1\times 20=20[/tex]

The probability that Andrew will be among the 4 volunteers selected and Karen will not is

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

[tex]P=\frac{20}{70}[/tex]

[tex]P=\frac{2}{7}[/tex]

Therefore the probability that Andrew will be among the 4 volunteers selected and Karen will not is 2/7.

The probability that Andrew is selected and Karen is not from a group of 8 volunteers for a 4-person task is 2/7.

The question is asking about the probability of a specific event happening when a group of volunteers is randomly selected. The key to solving this problem is knowing how to calculate combinations.

There are 8 volunteers in total and we know that 4 people are to be selected. The total number of ways 4 people can be selected from 8 is given by the combination formula C(n, r) = n! / (r!(n-r)!), where n is the total number of elements, r is the number of elements to choose, and ! represents the factorial operator.

So, total combinations = C(8, 4) = 8! / (4!(8-4)!) = 70.

Now, we need to find the combinations in which Andrew is chosen and Karen is not. This situation is equivalent to selecting 3 people from the remaining 6 people (excluding Andrew and Karen). Therefore, these combinations = C(6, 3) = 6! / (3!(6-3)!) = 20.

The probability that Andrew will be among the 4 volunteers selected and Karen will not is therefore 20/70 = 2/7.

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Answer:

[tex]50xy[/tex] and [tex]40x^2[/tex].

Step-by-step explanation:

The given polynomial is [tex]36x^3-22x^2-144x[/tex].

The prime factorization of each term are;

[tex]36x^2=2^2\times 3^2\times x^3[/tex]

[tex]-22x^2=-2\times 11\times x^2[/tex]

[tex]-144x=-2^4\times 3^2\times x[/tex]

The greatest common factor of these three terms is [tex]2x[/tex].

Now observe that:

The GCF of [tex]2x[/tex] and 11 is 1

The GCF of [tex]2x[/tex] and 50xy is 2x

The GCF of [tex]2x[/tex] and [tex]40x^2[/tex] is 2x

The GCF of [tex]2x[/tex] and 24 is 2

The GCF of [tex]2x[/tex] and 10y is 2

The correct options are [tex]50xy[/tex] and [tex]40x^2[/tex].