expand (1+2)^4 using the Pascal Triangle​

Answer:

No, a, h, and k behave differently in the square root function than they did in the quadratic function and/or cubic functions from the previous course. There is no variable (a) in front of the square root word in the square root function, and the shifts of the function are represented by h and k, respectively. Unlike in quadratic or cubic functions, where h stands for the vertex's horizontal shift and k for its vertical shift. The square root function's scope and range also vary from those of cubic and quadratic functions.

After answering the presented question, we can conclude that Therefore, the solution to the system of equations is x = 2 and y = 5.

An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "[tex]2x + 3 = 9[/tex]" asserts that the phrase "[tex]2x + 3[/tex]" equals the value "9." The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. The variable x is raised to the second power in the equation "[tex]x2 + 2x - 3 = 0[/tex]." Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.

solve the equation

[tex]y=5x-5[/tex]

[tex]-x+3(5x-5)[/tex][tex]=3[/tex]

[tex]x+15x-15=13[/tex]

[tex]14x=28[/tex]

[tex]x=2[/tex]

[tex]5(2)-y=5[/tex]

[tex]10-y=5\\[/tex]

[tex]y=5[/tex]

Therefore, the solution to the system of equations is x = 2 and y = 5.

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To be in the top 20% of scores on the test, a student must achieve a minimum score of 125.5 using the z-score corresponding to the 80th percentile, which was then converted back to the original scale using the mean and standard deviation of the distribution.

The z-score corresponding to the top 20% of the scores is the 80th percentile, which can be found using a standard normal distribution table or calculator. In terms of z-scores, the 80th percentile is at 0.84.

To find the minimum score needed to be in the top 20%, we can use the formula:

minimum score = (z-score x standard deviation) + mean

When we substitute the values specified in the problem, we obtain:

minimum score = (0.84 x 18) + 110

minimum score = 125.52

Therefore, the minimum score needed to be in the top 20% of the scores on the test is 125.5.

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The equation in x is 2[tex]x^{2}[/tex]-3x-$27=0, which can be solved using the quadratic formula to find the value(s) of x that satisfy the equation. The expressions derived in parts (a) and (b) are used to set up an equation involving x.

To solve this problem, we need to use the information given in parts (a) and (b) to set up an equation involving x.

(a) Let the number of large sacks Paul buys be y. The cost of each large sack is given as x, so the total amount Paul spends is:

y * x = $27

Solving for y, we get:

y = 27/x

(b) Let the number of small sacks Rula buys be z. We are told that each small sack costs $2 less than a large sack, so the cost of each small sack is:

x - $2

The total amount Rula spends is $25, so we can set up an equation:

z * (x - $2) = $25

Simplifying this equation, we get:

z = 25/(x - $2)

(c) We are told that Rula buys 4 more sacks than Paul. Using the expressions we derived in parts (a) and (b), we can set up an equation:

z = y + 4

Substituting the expressions we derived in parts (a) and (b), we get:

25/(x - $2) = 27/x + 4

Multiplying both sides by x(x - $2), we get:

25x = 27(x - $2) + 4x(x - $2)

Simplifying this equation, we get:

25x = 27x - $54 + 4x² - 8x

Rearranging terms and simplifying, we get:

2x² - 3x - $27 = 0

So the equation in x is 2x²-3x-$27=0, which we can solve using the quadratic formula to find the value(s) of x that satisfy the equation.

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With a basic interest rate of 5%, a deposit earns $102 after 36 months. The duration and overall cost are 3 years and $782, respectively.

To resolve this issue, we can utilize the simple interest formula:

I = Prt

where I stands for interest, P for the principal (original deposit), r for interest rate, and t for the time in years.

We know that the interest earned is $102, the interest rate is 5%, and we want to find the time and total value. Let's call the principal "P" and the time "t". Then we have:

102 = P x 0.05 x 3

Simplifying this equation, we get:

102 = 0.15P

Dividing both sides by 0.15, we get:

P = 680

So the initial deposit was $680.

Now, we can calculate the time using the same formula:

102 = 680 x 0.05 x t

Simplifying this equation, we get:

102 = 34t

Dividing both sides by 34, we get:

t = 3

So the time was 3 years (or 36 months).

To find the total value, we simply add the interest earned to the initial deposit:

Total value = P + I = $680 + $102 = $782

Therefore, the time was 3 years and the total value after 36 months was $782.

The complete question is:-

A deposit earns $102 after 36 months at a simple interest rate of 5%

What is the time and total value

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Answer:

Step-by-step explanation:

first to solve this you should make the numbers on one side after the equal sign with reversing the sign of each moved number  like this :

- note that the number 3 was in a negative sign ( -3 ) before it's moved to the other side , and also the number 60 was in a positive sign ( + 60 ) before it's moved to the other side.

Answer:

We can start by using conservation of momentum and conservation of energy to solve this problem.

Conservation of momentum:

Before the explosion, the momentum of the projectile is:

p = mv = (228 kg)(146 m/s) = 33312 kg·m/s

After the explosion, the momentum is still conserved, so the momentum of each piece must be equal to 1/3 of the initial momentum:

p = (1/3) mv1 + (1/3) mv2 + (1/3) mv3

where v1, v2, and v3 are the velocities of the three pieces after the explosion.

Conservation of energy:

At the highest point of the arc, the projectile has no kinetic energy and all of its energy is in the form of potential energy. Therefore, we can use conservation of energy to find the potential energy at this point and use that to find the initial kinetic energy of the projectile.

mgh = (1/2) mv²

where h is the maximum height of the projectile and v is the initial velocity of the projectile.

Solving for v, we get:

v = sqrt(2gh)

where g is the acceleration due to gravity (9.81 m/s²).

Substituting the given values, we get:

v = sqrt(2(9.81 m/s²)(h)) = 146 m/s

Now we can use the conservation of momentum equation to solve for the velocity of the third fragment:

p = (1/3) mv1 + (1/3) mv2 + (1/3) mv3

33312 kg·m/s = (1/3)(228 kg)(146 m/s) + (1/3)(228 kg)(146 m/s) + (1/3)(228 kg)(v3)

Simplifying and solving for v3, we get:

v3 = 306.23 m/s

Therefore, the magnitude of the velocity of the third fragment immediately after the explosion is 306.23 m/s.

To determine the direction of the velocity of the third fragment immediately after the explosion, we can use the fact that two of the fragments move with the same speed right after the explosion as the entire projectile had just before the explosion. This means that the horizontal component of the velocity of the third fragment must be equal in magnitude to the horizontal component of the velocity of the original projectile (146 m/s), and the vertical component must be opposite in sign to the vertical component of the velocity of the fragment that moves vertically downward.

Since the fragment that moves vertically downward has a velocity that is purely vertical, its vertical component is equal in magnitude to its total velocity. Therefore, the vertical component of the velocity of the third fragment must also be equal in magnitude to 306.23 m/s.

Using the Pythagorean theorem, we can find the magnitude of the horizontal component of the velocity of the third fragment:

sqrt((146 m/s)² - (306.23 m/s)²) = 259.31 m/s

Therefore, the velocity of the third fragment immediately after the explosion has a magnitude of 306.23 m/s and is directed at an angle of arctan(-306.23 m/s / 259.31 m/s) = -51.4° below the horizontal.

Answer:

Step-by-step explanation:

To solve this question, we need to use conservation of momentum.

1. The problem tells us that is breaks into 3 pieces of equal mass. So each piece weighs: [tex]\frac{228}{3}=76kg[/tex]

2. This problem also tells us that the projectile breaks up into 3 parts at the highest point of its arc. This tells us that the initial momentum of the system in the y-direction is 0.

3. Before we solve, we need to clear something up. The velocity of each fragment is NOT [tex]146\frac{m}{s}[/tex]. Since we launch at an angle, the velocity of the fragments will be [tex]146cos(70)=49.93[/tex], because each fragment move with the same speed after the explosion, and our speed after the explosion only has an x-component to it, as the y-component is 0.

4. To solve we need the equations:

[tex]m_{1}v_{1_{x}}+m_{2}v_{2_{x}}+m_{3}v_{3_{x}}=m_{1}v'_{1_{x}}+m_{2}v'_{2_{x}}+m_{3}v'_{3_{x}}[/tex]

[tex]m_{1}v_{1_{y}}+m_{2}v_{2_{y}}+m_{3}v_{3_{y}}=m_{1}v'_{1_{y}}+m_{2}v'_{2_{y}}+m_{3}v'_{3_{y}}[/tex]

NOTE: fragment 1 is going to be the one that travels horizontally. Fragment 2 will be the one traveling vertically downward. 3 will be our unknown

5. Solving for x:

[tex]m_{1}v_{1_{x}}+m_{2}v_{2_{x}}+m_{3}v_{3_{x}}=m_{1}v'_{1_{x}}+m_{2}v'_{2_{x}}+m_{3}v'_{3_{x}}[/tex]

[tex]228(146)cos(70)=76(49.93)cos(0)+76(50)cos(270)+76v'_{3_{x}} = > v'_{3_{x}} \approx 100[/tex]

6. Solving for y:

[tex]m_{1}v_{1_{y}}+m_{2}v_{2_{y}}+m_{3}v_{3_{y}}=m_{1}v'_{1_{y}}+m_{2}v'_{2_{y}}+m_{3}v'_{3_{y}}[/tex]

[tex]0=76(49.93)sin(0)-76(50)sin(270)+76v'_{3_{y}} = > v'_{3_{y}} \approx 50[/tex]

7. Solve for v:

[tex]v'_{3}=\sqrt{100^2+50^2}=111.8\frac{m}{s}[/tex]

8. Solve for theta:

[tex]\theta=tan^{-1}(\frac{50}{100} )=26.57[/tex]

Hope this helped. :)

Answer:

B.a low mean and a high MAD

Hence, the range [(-, 13/8] is the domain of g(s) as the expression must be positive to be under the square root sign.

The set of all potential input values (commonly referred to as the "two factors") for which a function is defined is known as the domain of the function in mathematics. The set of real numbers for which the function yields a valid output value is the set of all real numbers (often referred to as the "dependent variable").

given

As the expression must be positive to be under the square root sign, we have:

13 - 8s ≥ 0

To solve for s, we obtain:

s ≤ 13/8

Hence, the range [(-, 13/8] is the domain of g(s) as the expression must be positive to be under the square root sign.

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The probability that a randomly chosen student who speaks French is female is 33/110 and a female student speaks French is 15/41.

What is probability?

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.

a) Let's use Bayes' theorem to calculate the probability that a randomly chosen female student speaks French. Let F represent the event that a student speaks French, and let F' represent the event that a student does not speak French. Similarly, let Fm and Ff represent the events that the student is male and female, respectively. Then we have:

P(F|Ff) = P(Ff|F)P(F) / P(Ff)

where P(Ff|F) = 3/10 is the probability that a female student speaks French, P(F) = 5/22 is the overall probability that a student speaks French, and P(Ff) = (3/10)(5/22) + (2/12)(17/22) = 41/220 is the probability that a randomly chosen student is female. Thus, we have:

P(F|Ff) = (3/10)(5/22) / (41/220) = 15/41

So the probability that a randomly chosen female student speaks French is 15/41.

b) Now we want to find the probability that a randomly chosen student who speaks French is female. Let's again use Bayes' theorem, with F and F' representing the events of speaking French and not speaking French, and M and F representing the events of being male and female, respectively. Then we have:

P(Ff|F) = P(F|Ff)P(Ff) / P(F)

where P(F|Ff) = 3/10 is the probability that a female student speaks French, P(Ff) = 41/220 is the probability that a randomly chosen student is female, and P(F) = (3/10)(5/22) + (2/12)(17/22) = 5/22 is the overall probability that a student speaks French. Thus, we have:

P(Ff|F) = (3/10)(41/220) / (5/22) = 33/110

So the probability that a randomly chosen student who speaks French is female is 33/110.

Hence,  the probability that a randomly chosen student who speaks French is female is 33/110 and a female student speaks French is 15/41.

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Answer:

Substituting x = 3 and y = -3 in the given expression, we get: 5x^2 - 2xy + y^2 = 5(3)^2 - 2(3)(-3) + (-3)^2 = 5(9) + 18 + 9 = 45 + 18 + 9 = 72 Therefore, when x = 3 and y = -3, the value of the expression 5x^2 - 2xy + y^2 is 72

Probability is the possibility that something will happen. With the help of standard deviation, we can say that SD(x) = 1.918.

Probability refers to the likelihood that an occurrence will occur.

In actual life, we frequently have to make predictions about the future. We might or might not be conscious of how an event will turn out.

When this occurs, we proclaim that there is a possibility that the event will occur. In conclusion, probability has a broad range of amazing uses in both business and this rapidly developing area of artificial intelligence.

Simply dividing the favorable number of possibilities by the total number of possible outcomes using the probability formula will yield the chance of an event.

According to the given question,

Standard deviation = [tex]\sqrt{(variance)}[/tex]

Variance = Var(x)

Var(X) = Σx²*p(x) - E(x)

∑[tex][(0^2 * 0.09) + (1^2 * 0.32) + (2^2 * 0.27) + (3^2 * 0.17) + (4^2 * 0.11) + (5^2 * 0.04)] - 2.01[/tex]

= 5.69 - 2.01

= 3.68

Hence, With the help of standard deviation, we can say that SD(x) = 1.918.

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Answer:

242.50 [tex]cm^{2}[/tex]

Step-by-step explanation:

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

4850 = [tex]\pi r^{2}[/tex] (20)  Divide both sides by 20

242.5 = [tex]\pi r^{2}[/tex] The area of the base is [tex]\pi r^{2}[/tex]

Helping in the name of Jesus.

Answer:

D

Step-by-step explanation:

Answer:

576 divided by 32 = 18

Step-by-step explanation:

18 times 32 goes into 576 and gives the remainder as 0. (hope this helps!)

According to the question the distance of the railway track is 200 km and the area representing the country map is 8 km² in real life.

The surface area of an object is the sum of all the shapes that make up its surface. This kind of rectangle's area is calculated by multiplying its length and breadth.

(a)If the scale of the map is 1 : 500,000, this means that 1 unit on the map represents 500,000 units in real life. The real length of a railway track, which is shown on the map as a 40 cm track, needs to be determined.

Let x be the actual distance in km. Then, we can establish the ratio shown below:

1 cm on the map represents 500,000 cm in real life, or

1 cm on the map represents (500,000/100,000) km in real life (since there are 100,000 cm in 1 km).

Thus, 1 cm on a map corresponds to 5 km real life.

Therefore, 40 cm on the map represents 40 x 5 = 200 km in real life.

So, the actual distance of the railway track is 200 km.

(b) If the area of the country is 40,000 km², we need to find the area on the map that represents this area in real life.

Since the scale of the map is 1 : 500,000, this means that 1 cm on the map represents 500,000 cm² in real life.

To find the area on the map that represents 40,000 km² in real life, we can set up the following proportion:

1 cm on the map represents 500,000 cm² in real life, or

1 cm² on the map represents (500,000/100,000,000) km² in real life (since there are 100,000 cm² in 1 km²).

So, 1 cm² on the map represents 0.005 km² in real life.

Therefore, the area representing the country on the map is 40,000/0.005 = 8,000,000 cm² or 8 km² in real life.

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Answer: True, the quadrilateral below is a rhombus. Why? Because If all sides of a quadrilateral are congruent, then it’s a rhombus, which in this case, all sides are congruent. So, that's how I know the figure below is a rhombus. :)

Hope this helps!!

Using probability,

Samuel having the game's initial move and winning has a 1/16 chance of happening (0.0625). The best choice is B.

Probability describes the likelihood that an event will occur. In real life, there are many situations where we might need to make predictions about how something will turn out. It's possible for us to know or not know how an event will turn out. This is when we use the phrase "there is a probability that the event will happen or not."

In this case,

Due to the fact that there are four players, and each has an equal chance of going first, the probability that Samuel goes first is 1/4.

Given that there are four players and that each has an equal chance of winning, the likelihood that Samuel will win the game is also 1/4.

We must multiply these two probabilities in order to get the likelihood that Samuel will start the game and win:

P (Samuel goes first and wins)

= P (Samuel goes first) x P (Samuel wins | Samuel is playing)

= (1/4) x (1/4)

= 1/16

= 0.0625

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Answer: 210 square meters

Step-by-step explanation:

20 × 16 = 320

1/2 (20 × 11) = 110

320-110=210

Answer:

670, 630, 590, 550, 510, ....

Step-by-step explanation:

The difference between each of the numbers is -40. This is the constant pattern.

The equation of the parabola is y = 5x² - 6x + 1 .

What is parabola?

A parabola is a U-shaped plane curve. Any location on a parabola is at an equal distance from both the focus, a fixed point, and the directrix, a set straight line.

First, we figure out the value of c or the y intercept, we use the second point (0, 1) and substitute to the equation of the parabola. When x = 0, y = 1. So, c should be equal to 1. The parabola is y = ax^2 + bx + 1

Now, we can substitute the point (1,0) into the equation,

0 = a(1)² + b(1) + 1

0 = a + b + 1

a + b = -1

The slope at x = 1 is equal to 4 which is equal to the first derivative of the equation.

We take the derivative of the  equation ,

y = ax² + bx + 1

y' = 2ax + b

x = 1, y' = 2

4 = 2a(1) + b

4 = 2a + b

So, we have two equations and two unknowns,

2a + b = 4

a + b = -1

Solving simultaneously,

a = 5

b = -6

Therefore, the equation of the parabola is y = 5x² - 6x + 1 .

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The volume of the cylinder with a height of 8 units and a radius of 8 units is approximately 1606.88 cubic units.

The formula for the volume of a cylinder is:

V = πr²h , Where:

V is the volume of the cylinder

π is a mathematical constant approximately equal to 3.14159

r is the radius of the base of the cylinder

h is the height of the cylinder

To find the volume of a cylinder, you need to know the radius and height of the cylinder. Once you have those values, you can substitute them into the formula and solve for V.

Assuming this quesrtion  referring to a cylinder, the dimensions you provided correspond to a cylinder with a height of 8 units and a radius of 8 units.

To find the volume of this cylinder, you can use the formula:

V = πr²h

where V is the volume, r is the radius, and h is the height. Plugging in the given values, we get:

V = π(8²)(8)

V = 1606.88 cubic units (rounded to two decimal places)

Therefore, the volume of the cylinder with a height of 8 units and a radius of 8 units is approximately 1606.88 cubic units.

complete question - Two cones with same base radius 8 cm and height 15 cm are joined together along their bases. Find the surface area of the shape so formed (answer to the nearest whole number).

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Complete qeustion:

What is volume of a cylinder that comes with the height of 8 and radius of 8.

The relative change in the number of text messages from 2010 to 2012 was a 35.04% increase.

A number is a fundamental building block of mathematics. Numbers are used for indexing, counting, measuring, and a variety of other tasks. According to their characteristics, there are various sorts of numbers, including natural numbers, whole numbers, rational and irrational numbers, integers, real numbers, complex numbers, even and odd numbers, etc.

The relative change in the number of text messages from 2010 to 2012 was:

(3,213 - 2,380) / 2,380 x 100% = 35.04%

Therefore, the relative change in the number of text messages from 2010 to 2012 was a 35.04% increase.

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Answer: 500 strips

Step-by-step explanation:

20 x 500 = 10000 / 20 = 500

Answer:

Step-by-step explanation:

Answer: The solution to the system of equations by substitution is (x, y) = (-3, -1).

Answer:

7.5

Step-by-step explanation:

Since the number of numbers is an even number, you're going to have to add 7 and 8 since they are both in the middle. After you added 7 and 8 your answer would be 15, you would have to divide 15 by 2 and your answer will be 7.5. Sorry if it's difficult to understand.

Answer:

7.5

(7+8)/2 = 7.5

Step-by-step explanation:

Hope this helps! =D

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The requried initial loan amount is approximately $128,000.00.

We can use the formula for the present value of an annuity to find the initial loan amount:

[tex]PV = PMT * ((1 - (1 + r/n)^{(-nt))} / (r/n))[/tex]

where PV is the present value (initial loan amount), PMT is the monthly payment, r is the annual interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the total number of years.

Substituting the given values, we get:

[tex]PV = 770.82 * ((1 - (1 + 0.053/12)^{(-12*25))} / (0.053/12))[/tex]
≈ $128,000.00

So the initial loan amount is approximately $128,000.00.

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The prοbability that exactly 4 students attend Tet festivities is apprοximately 0.182.

Prοbability is the study οf the chances οf οccurrence οf a result, which are οbtained by the ratiο between favοrable cases and pοssible cases.

Let X be the number οf students whο attend Tet festivities.

Then, X ~ B(19, 0.24).

P(X = k) = (n chοοse k)[tex]*p^k*(1-p)^{(n-k)}[/tex]

where (n chοοse k) is the binοmial cοefficient, given by:

(n chοοse k) = n! / (k! * (n - k)!)

where n! is the factοrial οf n.

Fοr example, tο find the prοbability that exactly 4 students attend Tet festivities, we can calculate:

P(X = 4) = (19 chοοse 4) * 0.24^4 * (1 - 0.24)^(19 - 4)

P(X = 4) ≈ 0.182

Therefοre, the prοbability that exactly 4 students attend Tet festivities is apprοximately 0.182.

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Answer:

75%

Step-by-step explanation:

Formula:
P = (x/y) * 100

where x is 6 dollars and y is 8 dollars.

P = (6/8) * 100
P = 0.75 * 100
P = 75%