5
14
6
3
1
2
Two friends, Andrew and Liz, are playing a game using this spinner. If the
spinner lands on region 2 or 3, Andrew wins. If it lands on region 4, Liz wins. If
it lands on any other region, neither wins. Is this a fair game?
A. Yes. Each friend has a
probability of winning.
B. Yes. Each friend has a
probability of winning.
C. No. Andrew has a probability of winning and Liz has a
probability of winning.
D. No. Andrew has a probability of winning and Liz has a
probability of winning.

In consecuence, the conjugate complex number x₂ = (- 5 + i √3)/2 is one of the three roots of the cubic equation f(x) = x³ + 6 · x² + 12 · x + 7. (Correct choice: D)

In this case we have a cubic equation, that is, a third order polynomial. There are two strategies to determine the roots of cubic equations: (i) Cardano's formula, (ii) Numerical methods.

The quickest though most effective way consists in determining the roots by numerical methods (i.e. Newton-Raphson method). By using numerical methods we conclude that the cubic equation f(x) = x³ + 6 · x² + 12 · x + 7

x₁ = - 1, x₂ = (- 5 + i √3)/2, x₃ = (- 5 - i √3)/2

In consecuence, the conjugate complex number x₂ = (- 5 + i √3)/2 is one of the three roots of the cubic equation f(x) = x³ + 6 · x² + 12 · x + 7. (Correct choice: D)

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Answer:

  145 square inches

Step-by-step explanation:

The surface area of a square pyramid can be found using the formula ...

  A = s(s +2h) . . . . . where s is the base side length, and h is the slant height

__

Using this formula with s=5 inches and h=12 inches, we find the area to be ...

  A = (5 in)(5 in +2(12 in)) = (5 in)(29 in) = 145 in²

The area of the pyramid is 145 square inches.

_____

Additional comment

This formula comes from the sum of the areas of the square base (s²) and the four triangular faces (4 × 1/2sh). That sum is ...

  A = s² +2sh = s(s +2h)

The set of inequalities do not have a solution.

The relationship between two values that are not equal is defined by inequalities. Inequality means not equal. Generally, if two values are not equal. But to compare the values, whether it is less than or greater than, different inequalities are used.

Given:

f(x)= x²-2

g(x) = -x²+5

To derive, y ≤ x² + 5 simply change the equality sign in the function g(x).

To derive  y > x² - 2 ,  following transformation on the function f(x)

The inequalities of the graphs become

y < x² - 2 and y ≤ x² - 5

From the graph of the above inequalities it can be seen that curves of the inequalities do not intersect.

Hence, the set of inequalities do not have a solution

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Using proportions, it is found that the club needs to sell at least 832 tickets to make their desired profit.

A proportion is a fraction of a total amount, and the measures are related using a rule of three.

The prize is of €58, with the event making a profit of €150, hence the amount of money they have to earn from tickets is given by:
A = 150 + 58 = €208.

The tickets are sold at €0.25, hence the number of tickets they have to sell is:

n = 208/0.25 = 208 x 4 = 832.

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Answer:

x = 7/2

Step-by-step explanation:

1/5 (10x+30)-14= - 1/3 (6x-18)

10x + 30/5 - 14 = -6x + 18/3

(2*5)x+2*3*5/5-14 = -(2*3)x-2*3^2/3

x = 7/2

Answer:

200 Bricks

Step-by-step explanation:

1/3 of the bricks were not used(which was 50 bricks), meaning that 3/3 is 150 bricks of the pile. Since 150 is 3/4 of the original pile, that means that there was 200 bricks at the beginning.

∠2 and ∠4 are vertical opposite angles and they are congruent.

Vertically opposite angles are congruent.

Therefore, let proof ∠2 and ∠4 are vertical angles

Hence,

m∠2 + m∠3 = 180

∠2 and ∠3  are linear pair.

m∠3 + m∠4 = 180

∠3 and ∠4  are linear pair.

∠2 and ∠4 are vertical angles

m∠2 + m∠3 = m∠3 + m∠4

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Answer: View picture below!

Step-by-step explanation: Just did it!

Answer:

200 sqin.

Step-by-step explanation:

10 x 10 + 10 x 10 = 100 + 100
                          = 200

If each of the letters is ordered randomly, the fractional probability of M being the last letter is 1/6.

Probability refers to a possibility that deals with the occurrence of random events.

The probability of all the events occurring need to be 1.

We have to make a tree diagram to show all possible arrangements of the letters in the word MAT.

M     ⇒ A  ⇒ T    

       ⇒ T  ⇒ A

A  ⇒ M ⇒ T

    ⇒ T ⇒ A

T ⇒ M ⇒ A

  ⇒ A  ⇒ M

If each of the letters is ordered randomly, the fractional probability of M being the last letter is 1/6.

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Question 1

We want to choose a white marble, then a black marble.

Multiplying these probabilities, we get (9/24)(8/24) = 1/8.

Question 2

Thus, the probability is 5/10 = 1/2.

Question 3

Multiplying these probabilities, we get (7/24)(9/23) = 21/184.

A slope also known as the gradient of a line is a number. The correct option is A, C, and D.

A slope is also known as the gradient of a line is a number that helps to know both the direction and the steepness of the line.

For the given question,

This can be observed in the graph, as the slope of the graph before 4 pm is positive, it can be concluded that the temperature is increased until 4 pm.

This can be observed in the graph, as the slope of the graph after 4 pm is negative, it can be concluded that the temperature is decreasing after 4 pm.

This can be probed by calculating the slope of the line between the two points. Therefore, the slope between 8 am to 12 pm will be 1, while the slope from 12 pm to 4 pm will be equal to 4/3.

Hence, the correct option is A, C, and D.

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Answer:

ACD

Step-by-step explanation:

Answer:

5x - 10

Step-by-step explanation:

→ Find the sum of the expression

3x - 4 + 2x - 6

→ Collect all the x terms

3x + 2x - 4 - 6

→ Simplify

5x - 10

Part 1

[tex](g\circ h)(x)=g(x-4)=(x-4)^2 + 1=x^2 - 8x+16+1=\boxed{x^2 - 8x+17}[/tex]

Part 2

[tex](-\infty, \infty)[/tex]

The answers in the box are:

Given that the functions are

[tex]g(x)=x^2+1[/tex]

[tex]h(x)=x-4[/tex]

Since clearly g(x) and h(x) exist for all real values of x so the domain of g(x) and h(x) both are all real numbers.

Now the composition of function is given by,

[tex](g\circ h)(x)=g(h(x))[/tex]

               [tex]=g(x-4)[/tex]

               [tex]=(x-4)^2+1[/tex]

               [tex]=x^2-8x+16+1[/tex], since we know that [tex](a-b)^2=a^2-2ab+b^2[/tex]

               [tex]=x^2-8x+17[/tex]

So, [tex](g\circ h)(x)=x^2-8x+17[/tex]

It is a polynomial function so it is defined for all real values of x.

Hence the domain of composition is all real numbers, that is [tex](-\infty,\infty)[/tex]

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Answer:

[tex]y=\dfrac{5}{4}x^3+10[/tex]

Step-by-step explanation:

Given information:

Create two equations by substituting the given points into the given equation:

Equation 1:  point (0, 10)

[tex]\implies a(0)^3+d=10[/tex]

[tex]\implies 0+d=10[/tex]

[tex]\implies d=10[/tex]

Equation 2:  point (2, 20)

[tex]\implies a(2)^3+d=20[/tex]

[tex]\implies 8a+d=20[/tex]

Substitute Equation 1 into Equation 2 and solve for a:

[tex]\implies 8a+d=20[/tex]

[tex]\implies 8a+10=20[/tex]

[tex]\implies 8a+10-10=20-10[/tex]

[tex]\implies 8a=10[/tex]

[tex]\implies \dfrac{8a}{8}=\dfrac{10}{8}[/tex]

[tex]\implies a=\dfrac{10}{8}[/tex]

[tex]\implies a=\dfrac{5}{4}[/tex]

Finally, substitute the found values of a and d into the original formula:

[tex]\implies y=\dfrac{5}{4}x^3+10[/tex]

Check by substituting the x-values of the two given points into the found equation:

[tex]x=0 \implies y=\dfrac{5}{4}(0)^3+10=10 \leftarrow \textsf{correct}[/tex]

[tex]x=2 \implies y=\dfrac{5}{4}(2)^3+10=20 \leftarrow \textsf{correct}[/tex]

Put(0,10)

Now

Put again (2,20) this time

u=-54

you have -54 because

-54(-2)=108/9=12

Answer:

u =-54

Step-by-step explanation:

-2/9u=12

To solve for u multiply each side by -9/2 to isolate u

-9/2 * -2/9 u= 12 * -9/2

u =-54

The value of x equals quantity of (3 ± √23)/2.

A quadratic equation is an algebraic equation of the second degree in x. The quadratic equation in its standard form is ax² + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term.

Here, given equation;

  x² - 3x = -8

    x² - 3x + 8 = 0

 x² - 2 X 3/2 x + 8 = 0

 x² - 2.3/2 x + (3/2)² - (3/2)² + 8 = 0

 (x - 3/2)² - 9/4 + 8 = 0

 (x - 3/2)² + 23/4 = 0

  (x - 1.5)² = -5.75

  x - 1.5 = ±√(-5.75)

 x = 1.5 ± i√(5.75)

  x = (3 ± i√21)/2

Thus, x equals quantity of 3 plus or minus I square root of 23 all over 2.

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Answer:

x=12

Step-by-step explanation:

Given that there is only one expression given in each triangle, if we only use the information in a single triangle, there is no way to set up an equation that we could solve to find the expression.  (For this situation, at best we could use the triangle sum theorem to say that the three angles in one triangle added to 180°, but since we don't know the other two angles, that's 3 unknowns with only 1 equation... that's not solvable)

The only way to solve this problem is to make a relationship of some sort between the two triangles.

Proving a relationship between triangles - option 1

You may have a theorem about the Third angles of two triangles... "Given two arbitrary triangles, [tex]\triangle ABC[/tex] and [tex]\triangle D E F[/tex], if [tex]\angle A \cong \angle D[/tex] and [tex]\angle B \cong \angle E[/tex],  then [tex]\angle C \cong \angle F[/tex] "

If you have this, then the third angles are congruent.

In a more general situation of a similar problem (one in which you don't know two of the angles to begin with), it might be easier to prove triangle congruence, or triangle similarity.

Other Relationships between triangles

There are two main relationships between triangles: similarity and congruence.

Most people learn about congruence first.

In the situation for this problem, the two triangles happen to be congruent (we'll prove it shortly), which implies that all corresponding angles between shapes are congruent (and all corresponding sides between shapes are congruent).

For the purpose of solving for things related to angles, proving that the two triangles are similar is enough to know that angles between triangles are congruent (sides wouldn't necessarily be equal, but would be proportional, and since we're not solving for anything related to side lengths, proving that the triangles are similar would be enough).

Proving a relationship between triangles - option 2 - congruence

Notice that the triangle on top has two angles with markings (a single mark, and a double mark), and one side with a marking (a single tick).  These three pieces are in a configuration of ASA (the side is between the two angles).

Looking at the bottom triangle, it also has angles and sides with corresponding markings, and they are also in an ASA configuration.

Thus, by ASA congruence, these triangles are congruent triangles.

Knowing the triangles are congruent (even though we only used 3 parts), the rest of the corresponding parts (including the third angles) are also congruent.

Proving a relationship between triangles - option 3 - Similarity

For similarity, the process is similar to proving congruence, however the theorems we have for proving similarity are different.

SSS, SAS, or AA similarity.

Since the triangles in our problem do have two angles that are congruent, by AA similarity, the triangles are "similar".

Knowing the triangles are similar (even though we only used 2 parts), the last set of corresponding angles are also congruent.

Building our equation

Since the third angle of each triangle is congruent, by definition of congruent angles, the measures of each of those two angles are equal, and so we can build an equation knowing that the two expressions are equal to each other:

[tex]3x-7=2x+5[/tex]

Solving for x

From there, we need some algebraic properties equality to solve for x.

Subtract 2x from both sides...

[tex](3x-7)-2x=(2x+5)-2x\\x-7=5[/tex]

add 7 to both sides...

[tex](x-7)+7=(5)+7\\x=12[/tex]

Extension

Knowing the value for x, we could solve for the measure of the angles, if that had been requested.  Simply substitute 12 back into the expressions for the angle measure (the results should be the same for both angles, since they were congruent angles)

[tex]3(12)-7\\36-7\\29[/tex]         [tex]2(12)+5\\24+5\\29[/tex]

So, if we had been asked, the measure of the angle is 29°

Answer:

3.036, 3.36, 3.3661 , 3.5

Step-by-step explanation:

Look at the first 2 numbers

3.3661, 3.5, 3.36, 3.036

We can order them as 3.0 , 3.3 , 3.3 . 3.5

So 3.036 is first one

Now we have 2 of the 3.3's

One is 3.3661 and the other one is 3.3600

Because if you add a number to the end it will always be a zero and it wont change the answer

So  3.36 is second and 3.3661 is third one

3.5  is bigger than 3.3

So 3.5 is last

Answer: y = -3/2x + 6, y intercept is (0, 6)

Step-by-step explanation: You are dividing 2 from both -3x and 12. -3/2x cannot be simplified any further unless it's a decimal, so we leave it as -3/2x. But, since the equation was y = -3x+12 originally, we divide 12 from 2 as well and not just from 3x, and we get 6.

Hope this helped.

Considering a line of best-fit, the approximate number of hours in 72 days on Saturn is given by 768.

To find the equation, we need to insert the points (x,y) in the calculator.

In this problem, the points are:

(266.67, 25), (373.3, 35), (426.67, 40), (586.67, 55).

Using a calculator, the equation is:

y = 0.09375x + 0.00003455.

For 72 days, we have that y = 72, hence the number of hours is x.

y = 0.09375x + 0.00003455.

72 = 0.09375x + 0.00003455.

x = (72 - 0.00003455)/0.09375

x = 768 hours.

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If the volume of a cylinder is multiplied by 4/3, it would become the volume of a sphere. Thus, you're a sphere.

Mathematically, the volume of a cone can be calculated by using this formula:

V = 1/3 × πr²h

Where:

Mathematically, the volume of a sphere can be calculated by using this formula:

V = 4/3 × πr³

Similarly, the volume of a cylinder can be calculated by using this formula:

V = πr²h

In this context, we can infer and logically deduce that if the volume of a cylinder is multiplied by 4/3, it would become the volume of a sphere. Also, multiplying the volume of a cylinder by 1/3 would produce the volume of a sphere.

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If x∆y = 3x - y², then

5∆1 = 3×5 - 1² = 15 - 1 = 14

and

14∆6 = 3×14 - 6² = 42 - 36 = 6

So (5∆1)∆6 = 6.

Answer:

B.

Step-by-step explanation:

Answer:

2 cos (2x)

Step-by-step explanation:

[tex]\frac{4 -2 sec^2x}{sec^2x} \\=\frac{4-\frac{2}{cos^2x} }{\frac{1}{cos^2 x} } \\=\frac{\frac{4 cos^2x-2}{cos^2 x} }{\frac{1}{cos^2x} } \\=\frac{2(2 cos^2x-1)}{cos^2x} \times \frac{cos^2x}{1} \\=2(2cos^2x-1)\\=2cos(2x)[/tex]

Part 1

[tex](f\circ g)(x)=f(g(x))=\frac{\frac{1}{x}}{\frac{1}{x}-3}[/tex]

Multiplying the numerator and denominator by x,

[tex](f\circ g)(x)=\boxed{\frac{1}{1-3x}}[/tex]

Part 2

The domain for this function is all values of x except for when the denominator equals 0 (because then, it would be undefined).

The denominator is equal to 0 when x=1/3, so the domain is [tex]\boxed{x \neq \frac{1}{3}}[/tex]

The correct answer is the last choice .

The parent quadratic function is f(x) = x² it's just like the image above .

if you look at the graph horizontal translation has happened two units to right , it means we have -2 inside a bracket next to x , then vertical translation has happened 4 units to up it means we have +4 outside the bracket and after x

there's also a reflection on the graph which means there's a negative before the bracket .

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The solutions to the quadratic equation 50 - x² = 0 are x = ±5√2.

A quadratic equation is a second-order polynomial equation in a single variable x , ax2+bx+c=0. with a ≠ 0 .

The solution(s) to the quadratic equation 50 - x² = 0 can be found by setting the equation equal to zero and solving for x.

50 - x² = 0

To solve this equation, we can rearrange it as follows:

x² = 50

Taking the square root of both sides, we have:

x = ±√50

Simplifying the square root of 50, we get:

x = ±√(25 × 2)

x = ±√25 × √2

x = ±5√2

Therefore, the solutions to the quadratic equation 50 - x² = 0 are x = ±5√2.

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