Select the correct answer. A firework cracker weighing 1.5 kilograms, is launched at 15(m)/(s) by propelling chemicals from combustion at a speed of 55(m)/(s). What is the mass of the chemicals being propelled?

Answer: 2

5x - 7 = 3x + 3

2x - 7 = 3

2 is the new value of the coefficent

Answer:

The missing variable "x" = 20

And the Angle measure = 98°

Step-by-step explanation:

Explaination is given in the picture...

Thank you!

Answer:

the angles are 82 degrees and 98 degrees (5(20) -2), and the missing variable (x) is 20.

Step-by-step explanation:

Let us first look at SL. SL is a straight line and has an angle measure of 180 degrees. Angle RTL is 82 degrees and splits SL into 2. The angle right next to RTL is RTS, which is (5x-2) degrees. Since all of SL adds to 180 degrees, this means that RTL and RTS will add up to 180 degrees, since they are in the middle of it.

82 + 5x-2 = 180

80 +5x = 180

5x = 100

x = 20

Therefore, the angles are 82 degrees and 98 degrees (5(20) -2), and the missing variable (x) is 20.

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By answering the above question, we may infer that So the perimeter of the regular decagon is approximately 38.2 units (rounded to the nearest tenth).

In geometry, a decagon is either a decagon or not. There are 144° of inner angles total in a simple decagon. A regular decagon that self-intersects is known as a decagram. A polygon with 10 sides, ten internal angles, and ten vertices is called a decagon. Geometry may contain the form known as a decagon. It also has ten horns and ten horns. A dodecagon is a polygon with twelve sides. Some unusual types of dodecagons are shown in the photographs above. Particularly, a regular dodecagon has angles that are equally placed around a circle and sides that are of the same length.

Each interior angle of a regular decagon measures:

[tex]$$(n-2)\times180^\circ/n = (10-2)\times180^\circ/10 = 144^\circ$$\\$$\cos(72^\circ) = \frac{x}{2y}$$[/tex]

Solving for x, we get:

[tex]$$x = 2y\cos(72^\circ)$$[/tex]

We can use the fact that[tex]$\cos(72^\circ) = \frac{1+\sqrt{5}}{4}$[/tex](which can be derived using the golden ratio) to get:

[tex]$$x = 2y\cos(72^\circ) = 2y\cdot\frac{1+\sqrt{5}}{4} = \frac{y}{2}(1+\sqrt{5})$$\\$$R = \frac{x}{2\sin(180^\circ/10)} = \frac{x}{2\sin(36^\circ)}$$\\[/tex]

We can use this formula to find[tex]$y$:[/tex]

[tex]$$y = R = \frac{x}{2\sin(36^\circ)} = \frac{x}{2\sin(\frac{1}{2}\times72^\circ)} = \frac{x}{2\cos(72^\circ/2)}$$[/tex]

We can use the half-angle identity [tex]$\cos(\theta/2) = \sqrt{\frac{1+\cos(\theta)}{2}}$ to simplify this expression:[/tex]

[tex]$$y = \frac{x}{2\cos(72^\circ/2)} = \frac{x}{2\sqrt{\frac{1+\cos(72^\circ)}{2}}} = \frac{x}{2\sqrt{\frac{1+\frac{1+\sqrt{5}}{4}}{2}}} = \frac{x}{2\sqrt{\frac{3+\sqrt{5}}{4}}} = \frac{x}{\sqrt{3+\sqrt{5}}}$$[/tex]

Putting it all together, we have:

[tex]$$\text{Perimeter} = 10x = 10\cdot\frac{y}{2}(1+\sqrt{5}) = 5\sqrt{10+2\sqrt{5}}\approx 38.2$$[/tex]

So the perimeter of the regular decagon is approximately 38.2 units (rounded to the nearest tenth).

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i. The joint distribution for the coin and dice events is given by P(C,D) = P(C|D)P(D) = P(D|C)P(C), where C is the event that the coin lands heads and D is the event that the dice lands six.

ii. The prior probabilities for the coin and dice events are P(C) = 0.5 and P(D) = 1/6, since they are both fair.

iii. The full conditional distribution for the event that the coin is heads or the dice is six is given by P(C∪D) = P(C) + P(D) - P(C∩D) = 0.5 + 1/6 - (0.5)(1/6) = 0.5833.

iv. The full conditional distribution for the event that the coin is heads and the dice is six is given by P(C∩D) = P(C|D)P(D) = P(D|C)P(C) = (0.5)(1/6) = 0.0833.

v. The distribution for the coin and dice events given the event that the coin is heads or the dice is six is given by P(C,D|C∪D) = P(C∩D|C∪D)P(C∪D) = (0.0833)(0.5833) = 0.0486.

vi. The probability of observing that the coin is heads or the dice is six is given by P(C∪D) = 0.5833.

vii. The posterior distribution for the joint probability of the coin and dice events given the event that the coin is heads or the dice is six is given by P(C,D|C∪D) = P(C∩D|C∪D)P(C∪D) = (0.0833)(0.5833) = 0.0486.

viii. The marginal distribution for the event that the coin is heads and the dice is six given we know the event heads or six has occurred is given by P(C∩D|C∪D) = P(C,D|C∪D)/P(C∪D) = 0.0486/0.5833 = 0.0833.

ix. The marginal probability that the coin is heads and the dice is six given we know the event heads or six has occurred is given by P(C∩D|C∪D) = 0.0833.

x. The probability of you winning the prize is the same as the probability that the coin is heads and the dice is six, which is given by P(C∩D) = 0.0833.

Therefore, the probability of you winning the prize is 0.0833.

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Answer: To find the nth term of the sequence -5, -2, 3, 10, 19, we need to identify the pattern in the sequence.

Starting with the first term, we can see that we are adding 3 to get to the second term, adding 5 to get to the third term, adding 7 to get to the fourth term, and adding 9 to get to the fifth term.

So, the pattern is that each term is obtained by adding (n-1) x 2, where n is the position of the term in the sequence (starting with n=1 for the first term).

Therefore, the nth term of the sequence is:

-5 + (n-1) x 2

Simplifying this expression, we get:

-5 + 2n - 2 = 2n - 7

So, the nth term of the sequence is 2n - 7.

Step-by-step explanation:

[tex]\huge\begin{array}{ccc}A=75ft^2\end{array}[/tex]

The formula:

[tex]\huge\boxed{A=l\cdot w}[/tex]

[tex]l[/tex] - length of a rectangle

[tex]w[/tex] - width of a rectangle

[tex]l=25ft,\ w=30ft[/tex]

substitute:

[tex]A=25\cdot30=750ft^2[/tex]

There are 34 pupils in the class who do not wear spectacles.

What is percentage?

Percentage is a way to express a number as a fraction of 100. It is commonly used to show the relationship between two numbers, to compare two quantities, or to express a part of a whole. It is expressed as a fraction, decimal, or ratio.

1. To find the number of pupils who wear spectacles, we can multiply the total number of pupils by the percentage of pupils who wear spectacles:

40 pupils x 15% = 6 pupils

2. To find the number of pupils who do not wear spectacles, we can subtract the number of pupils who wear spectacles from the total number of pupils:

40 pupils - 6 pupils = 34 pupils

Therefore, there are 34 pupils in the class who do not wear spectacles.

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a) The null and alternative hypotheses are:
H0: µ1 = µ2
Ha: µ1 ≠ µ2

b) The test statistic is:
t = 0.117

c) The P-value is:
P-value = 0.9072

To test the null hypothesis at α= 0.05 using the pooled t-test, we need to follow these steps:

Step 1: Choose the null and alternative hypotheses. The null hypothesis is that the mean page views per visit for website 1 are equal to the mean page views per visit for website 2. The alternative hypothesis is that the mean page views per visit for website 1 are not equal to the mean page views per visit for website 2.

H0: µ1 = µ2
Ha: µ1 ≠ µ2

Step 2: Calculate the test statistic. The test statistic for the pooled t-test is given by:

t = (y1 - y2) / (sp * √(1/n1 + 1/n2))

where sp is the pooled standard deviation, given by:

sp = √(((n1 - 1) * s1^2 + (n2 - 1) * s2^2) / (n1 + n2 - 2))

sp = √(((70 - 1) * 4.9^2 + (90 - 1) * 5.4^2) / (70 + 90 - 2)) = 5.178

t = (7.5 - 7.4) / (5.178 * √(1/70 + 1/90)) = 0.117

Step 3: Calculate the P-value. The P-value is the probability of observing a test statistic as extreme or more extreme than the one we calculated, assuming the null hypothesis is true. We can use a t-distribution table or a calculator to find the P-value. The degrees of freedom for the pooled t-test are n1 + n2 - 2 = 70 + 90 - 2 = 158.

Using a t-distribution table or a calculator, we find that the P-value is 0.9072.

Step 4: Since the P-value is greater than the significance level of 0.05, we fail to reject the null hypothesis. There is not enough evidence to suggest that the mean page views per visit for website 1 are different from the mean page views per visit for website 2.

Thus:

a) The null and alternative hypotheses are:
H0: µ1 = µ2
Ha: µ1 ≠ µ2

b) The test statistic t = 0.117

c) The P-value = 0.9072

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Check the picture below.

[tex]\begin{cases} x = -1 &\implies x +1=0\\ x = 3 &\implies x -3=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x +1 )( x -3 ) = \stackrel{0}{y}}\implies a(x^2-2x-3)=y\hspace{5em}\stackrel{\textit{\LARGE product}}{x^2-2x-3}[/tex]

now, that's not the equation of the parabola, unless the value of "a" is 1, but in this case, doesn't matter, we just need that product part.

The zeros and their multiplicities of the function f(x)=2x^(5)+11x^(4)-2x^(3)-74x^(2)-60x+48 are:

x = -4, multiplicity 1

x = (-5 + √(41))/(2), multiplicity 1

x = (-5 - √(41))/(2), multiplicity 1

The zeros of a function f(x) are the values of x that make f(x) equal to 0. The multiplicity of a zero is the number of times that zero appears as a solution. To find the zeros and their multiplicities of the given function f(x)=2x^(5)+11x^(4)-2x^(3)-74x^(2)-60x+48, we can use synthetic division or factoring.

First, let's use synthetic division to find one of the zeros:

2x^(5)+11x^(4)-2x^(3)-74x^(2)-60x+48 = 0

Using synthetic division, we can find that x = -4 is a zero of the function:

(-4) | 2  11  -2  -74  -60  48
    | 0  -8  20   88  104 -176
    ---------------------------
      2   3  18   14   44 -128

Now we can divide the original function by (x+4) to find the remaining zeros:

f(x) = (x+4)(2x^(4)-x^(3)+14x^(2)+10x-32)

Using the quadratic formula, we can find the remaining zeros of the function:

x = (-b ± √(b^(2)-4ac))/(2a)

x = (-(10) ± √((10)^(2)-4(2)(-32)))/(2(2))

x = (-10 ± √(164))/(4)

x = (-10 ± √(4*41))/(4)

x = (-10 ± 2√(41))/(4)

x = (-5 ± √(41))/(2)

So the zeros of the function are x = -4, x = (-5 + √(41))/(2), and x = (-5 - √(41))/(2).

The multiplicity of each zero is 1, since each zero appears only once as a solution.

Therefore, the zeros and their multiplicities of the function f(x)=2x^(5)+11x^(4)-2x^(3)-74x^(2)-60x+48 are:

x = -4, multiplicity 1

x = (-5 + √(41))/(2), multiplicity 1

x = (-5 - √(41))/(2), multiplicity 1

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The probability distribution for the possible outcomes can be created using the binomial distribution formula:
P(X=x) = (n choose x) * p^x * (1-p)^(n-x)
Where n is the number of trials (in this case, 8), x is the number of successes (returning for a second year), p is the probability of success (0.54), and 1-p is the probability of failure.

The probability distribution is as follows:
| X | P(X) |
|---|------|
| 0 | 0.010 |
| 1 | 0.059 |
| 2 | 0.167 |
| 3 | 0.282 |
| 4 | 0.313 |
| 5 | 0.223 |
| 6 | 0.106 |
| 7 | 0.033 |
| 8 | 0.005 |
To compute P(X≤3), we add the probabilities for X=0, X=1, X=2, and X=3:
P(X≤3) = 0.010 + 0.059 + 0.167 + 0.282 = 0.518
This means that there is a 51.8% chance that 3 or fewer of the randomly selected first-year students will return for a second year.
The expected number of students returning for a second year can be calculated using the formula:
E(X) = n * p = 8 * 0.54 = 4.32
This means that on average, 4.32 of the randomly selected first-year students will return for a second year.

The standard deviation can be calculated using the formula:
σ = √(n * p * (1-p)) = √(8 * 0.54 * 0.46) = 1.39

Finally, to determine if the advising program was a success, we can compare the observed number of students returning (7) to the expected number (4.32). Since 7 is greater than 4.32, it appears that the advising program may have had a positive effect on the students' decision to return for a second year. However, further analysis would be needed to determine if this difference is statistically significant.

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A baseball is hit so that its height, s, in feet after t seconds is S= - 16t² +60t+2. Based on the inequality, the ball is at least 46 feet above the ground in [0.20, 1.55] seconds.

To find the period of time the ball is at least 46 feet above the ground, we first need to solve the given inequality, which will be:

Simplifying, we find the following value:

Dividing both sides by -4, we find:

So, to solve this inequality, we can use the quadratic formula:

Plugging these values into the formula, we get:

Simplifying this expression, we get:

Therefore, it follows that the ball is at least 46 feet above the ground during the time period:

So, rounded to two decimal places, this is approximately:

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40 m³ is the volume of pyramid .

What is a pyramid defined as?

A three-dimensional shape is a pyramid. A pyramid's flat triangular faces and polygonal base all come together in a summit known as the apex. By fusing the bases together at the peak, a pyramid is created. The lateral face, a triangular feature formed by the connection of each base edge to the apex, is present.

  L= 5

h = 4

s = 6

V = 6 * 4 * 5/3

   = 120/3

   = 40 m³

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The solution to the equation -(2)/(7)u=-14 is u = 49.

Knowledge of inverse operations tells us that we need to multiply both sides of the equation by the reciprocal of -(2)/(7) to cancel out the fraction on the left side of the equation. The reciprocal of -(2)/(7) is -(7)/(2).

Multiply both sides of the equation by -(7)/(2):
u = -(7)/(2) * -(2)/(7)u = -(7)/(2) * -14

Simplify the left side of the equation:
u = 49

Solve for u:
u = 49

Therefore, the solution to the equation -(2)/(7)u=-14 is u = 49.

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The values of Ф solved using a unit circle, between 0 and 2π, for which tanФ =  - √3 is 2π/3 and 5π/3.

What is a circle?

A circle is a shape made up of all points in a plane that are at a specific distance from the centre point. In other words, it is the path a moving point in a plane takes to move around a curve while maintaining a constant distance from another point. The circle has an area and a perimeter and is a two-dimensional figure. The distance around a circle, or its circumference, is referred to as the perimeter of the circle. The region enclosed by a circle in a 2D plane is said to be its area.

The complete question is given below.

Given,

tan Ф = - √3

We have to find all values of Ф between 0 and 2π using a unit circle.

The unit circle for trigonometric calculations is given below.

from the unit circle,

tan 60 = √3

In quadrant 2,

tan ( 180 - 60) = - tan 60

tan 120 = - tan 160 = -√3

In quadrant 3,  tangent values are positive.

In quadrant 4

tan (360 - 60) = - tan 60

tan 300 = -tan 60 = -√3

Also,

tan 120 = tan 2π/3

tan 300 = tan 5π/3

Therefore the values of Ф solved using a unit circle, between 0 and 2π, for which tanФ =  - √3 is 2π/3 and  5π/3.

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The area of the parallelogram with base of 8cm and height of 9cm is 72cm²

It is important to note the following properties of a parallelogram;

From the information given, we have that;

Area = bh

Given that;

Now, substitute the values

Area = 8(9)

Area = 72cm²

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The equation we can use to find the number of people each car can hold "10n = 40".

This equation represents the fact that there are 10 cars on the Blazing Bullet, each holding n people, and that the total number of people on the ride is 40.

By solving this equation for n, we can find the number of people each car can hold.

Here is how to solve the equation:

10n = 40

Divide both sides of the equation by 10:

10n/10 = 40/10

n = 4

Therefore, in conclusion, it is stated that each car can hold 4 people.

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Answer:

10n = 40

Step-by-step explanation:

1. Subtract 10 from both sides of the equation:

2. 10n/10 = 40/10

3. n = 4

As a result, it can be concluded that each car can accommodate 4 passengers.

Answer:

enter the step by step answer u did and then add the number the match and enter them in the box and u shall be done

Step-by-step explanation:

If FG = 8, EH = x - 1: EG = x + 1​

In order to find EG, we need to use the fact that the sum of the lengths of the segments EF and FG is equal to the length of segment EG. That is,

EF + FG = EG

We are given that FG = 8, and we know that EH + HF = EF. Therefore,

EF = EH + HF

Putting this all together, we get:

EF + FG = EG

(EH + HF) + 8 = x + 1

EH + HF = x - 7

But we also know that EH = x - 1, so we can substitute that in:

x - 1 + HF = x - 7

Simplifying this equation, we get:

HF = -6

Now we can use the fact that the sum of the lengths of the segments EH and HF is equal to the length of segment EF. That is,

EH + HF = EF

(x - 1) + (-6) = EF

x - 7 = EF

Finally, we can substitute this value for EF into our original equation to find EG:

EF + FG = EG

(x - 7) + 8 = EG

x + 1 = EG

Therefore, EG = x + 1.

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The notation f(x) means:
- the value of the function at the number x
- f of x

These two options are the correct ones. The notation f(x) typically refers to a mathematical function named "f" that takes an input value "x" and returns an output value. The function f maps the input value x to a unique output value f(x). The symbol "f" is often used to represent a generic function, and the variable inside the parentheses, "x", represents the argument or input value of the function.

For example, if we have the function f(x) = x + 2, then f(3) = 3 + 2 = 5. This means that when we input the number 3 into the function, the output is 5.

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You are correct, every purchase of a product or service is an exchange of value. In a transaction, a product or service is traded for money that is equal to the value of the product or service.

This is known as the "exchange of value" and is a fundamental concept in economics. The buyer is willing to pay the seller for the product or service because they believe it is worth the amount of money being exchanged.

On the other hand, the seller is willing to provide the product or service in exchange for the money because they believe that the money is worth more than the product or service y are providing. In this way, both parties benefit from the exchange and the transaction is completed.

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The quadrilateral whose vertices are A(2, 3); B(4, -2);C(-1,-4); D(-3, 1) is a square.

Quadrilaterals are four sided polygons which also have four vertices and four angles.

Sum of all the interior angles of a quadrilateral is 360 degrees.

Given A(2, 3); B(4, -2);C(-1,-4); D(-3, 1)

Length of AB = [tex]\sqrt{(4-2)^2+(-2-3)^2}[/tex] = √29 units

Length of BC = [tex]\sqrt{(-1-4)^2+(-4--2)^2}[/tex] = √29 units

Adjacent sides are equal.

So the quadrilateral must be square or rhombus.

Now, find the length of diagonals.

If the diagonals are equal, then it is square. If they are not equal, then it is rhombus.

AC = [tex]\sqrt{(-1-2)^2+ (-4-3)^2}[/tex] = √58 units

BD = [tex]\sqrt{(-3-4)^2+(1--2)^2}[/tex] = √58 units

Diagonals are equal.

So it is a square.

Hence the quadrilateral is a square.

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Answer:

To find the angle for each category, we need to calculate the percentage of the total frequency for each category, and then multiply by 360 (the total number of degrees in a circle).

The total frequency is:

3 + 10 + 14 + 19 + 14 = 60

The percentage of the total frequency for each category is:

Pizza: 3/60 x 100% = 5%

Curry: 10/60 x 100% = 16.67%

Fish & chips: 14/60 x 100% = 23.33%

Kebab: 19/60 x 100% = 31.67%

Other: 14/60 x 100% = 23.33%

To find the angle for each category, we multiply the percentage by 360:

Pizza: 5% x 360 = 18 degrees

Curry: 16.67% x 360 = 60 degrees

Fish & chips: 23.33% x 360 = 84 degrees

Kebab: 31.67% x 360 = 114 degrees

Other: 23.33% x 360 = 84 degrees

So the table with the angles for each category is:

Take-away Frequency Angle

Pizza 3 18°

Curry 10 60°

Fish & chips 14 84°

Kebab 19 114°

Other 14 84°

Answer:

x = -4 + √15 and x = -4 - √15.

Step-by-step explanation:

To solve for x in the equation (x + 4)^2 = 15 using square roots, we can take the square root of both sides of the equation, remembering to include both the positive and negative square root:

(x + 4)^2 = 15

Taking the square root of both sides:

±(x + 4) = √15

Now we can isolate x by subtracting 4 from both sides of the equation:

x + 4 = ±√15

x = -4 ±√15

Therefore, the solutions to the equation (x + 4)^2 = 15 are x = -4 + √15 and x = -4 - √15.

a)age for the sample is 47

b)the variance is 49.67

c)standard deviation is 7.04

d)variation is 14.88%

(a) To determine the average age of the sample, use the formula:  Average age = (Sum of ages) / (Number of observations).

30-39: Add all ages between 30-39 and divide by 8.
40-49: Add all ages between 40-49 and divide by 7.
50-59: Add all ages between 50-59 and divide by 4.
60-69: Add all ages between 60-69 and divide by 5.
70-79: Add all ages between 70-79 and divide by 1.

Therefore, the average age for the sample is 47.

(b) To compute the variance, use the formula: Variance = (Sum of Squared Differences)/ (Number of Observations - 1).

30-39: Sum the squared differences between each age and the mean age, and divide by 7.
40-49: Sum the squared differences between each age and the mean age, and divide by 6.
50-59: Sum the squared differences between each age and the mean age, and divide by 3.
60-69: Sum the squared differences between each age and the mean age, and divide by 4.
70-79: Sum the squared differences between each age and the mean age, and divide by 0.

Therefore, the variance is 49.67.

(c) To compute the standard deviation, use the formula: Standard Deviation = √Variance.

Therefore, the standard deviation is 7.04.

(d) To compute the coefficient of variation, use the formula: Coefficient of Variation = (Standard Deviation/Mean)*100.

Therefore, the coefficient of variation is 14.88%.

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a) y = 1/2 √ −3(x + 1) + 4
- The function is multiplied by 1/2, indicating a vertical compression by a factor of 1/2.
- The function is multiplied by -3 inside the square root, indicating a horizontal compression by a factor of 1/3 and a reflection across the y-axis.
- The function is shifted 1 unit to the left, indicated by the (x + 1) inside the square root.
- The function is shifted 4 units up, indicated by the + 4 outside the square root.

a) y = (2(x − 1))^3 − 5
- The function is multiplied by 2 inside the cube, indicating a horizontal compression by a factor of 1/2.
- The function is shifted 1 unit to the right, indicated by the (x - 1) inside the cube.
- The function is shifted 5 units down, indicated by the - 5 outside the cube.

The transformations to f(x) = √x are as follows:
a) y = 1/2 √ −3(x + 1) + 4
- The function is multiplied by 1/2, indicating a vertical compression by a factor of 1/2.
- The function is multiplied by -3 inside the square root, indicating a horizontal compression by a factor of 1/3 and a reflection across the y-axis.
- The function is shifted 1 unit to the left, indicated by the (x + 1) inside the square root.
- The function is shifted 4 units up, indicated by the + 4 outside the square root.

The transformations to f(x) = x^3 are as follows:
a) y = (2(x − 1))^3 − 5
- The function is multiplied by 2 inside the cube, indicating a horizontal compression by a factor of 1/2.
- The function is shifted 1 unit to the right, indicated by the (x - 1) inside the cube.
- The function is shifted 5 units down, indicated by the - 5 outside the cube.

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The remainder when f(x)=3x^(5)+4 is divided by x-2 is 100.

The remainder when f(x)=3x^(5)+4 is divided by x-2 can be found using synthetic division.

Step 1: Set up the synthetic division by writing the coefficients of f(x) in a row and the value of x that makes the divisor equal to zero in a box to the left. In this case, the coefficients are 3, 0, 0, 0, 0, and 4 and the value of x is 2.

2|300004

Step 2: Bring down the first coefficient and multiply it by the value in the box. Write the result under the next coefficient and add them together. Repeat this process for all of the coefficients.

2|300004|612244896|36122448100

Step 3: The last number in the bottom row is the remainder. In this case, the remainder is 100.

Therefore, the remainder when f(x)=3x^(5)+4 is divided by x-2 is 100.

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A solution to the given compound inequality is m < 6 or m > 2.

In Mathematics, it is very important to note that you do not flip the inequality symbol (sign) when you are solving an inequality by multiplication or division with a positive numerical value (number).

In this context, we can logically deduce that you would flip the inequality symbol (sign) when you isolate the variable in an expression such as it's highlighted in the following steps:

4 - m < - 2

-m < -2 - 4

-m < -6

m < 6

For the second inequality, we have;

12 < -5m + 2

12 - 2 < -5m

10 < -5m

m > -10/5

m > 2.

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(a) To find the value of k, we need to use the fact that the sum of the probabilities of all possible outcomes is equal to 1. In this case, we have:
0.05 + 0.20 + 3k + 0.15 + k = 1
Solving for k, we get:
4k = 1 - 0.05 - 0.20 - 0.15
4k = 0.60
k = 0.15
Therefore, the value of k is 0.15.

(b) To find E(X), we need to multiply each value of x by its corresponding probability and sum the results. In this case, we have:
E(X) = (-1)(0.05) + (0)(0.20) + (1)(3k) + (2)(0.15) + (3)(k)
E(X) = -0.05 + 0 + 0.45 + 3k
E(X) = 0.40 + 3k
Substituting the value of k that we found in part (a), we get:
E(X) = 0.40 + 3(0.15)
E(X) = 0.85
Therefore, the expected value of X is 0.85.

(c) To find Var(X), we need to use the formula Var(X) = E(X^2) - (E(X))^2. First, we need to find E(X^2):
E(X^2) = (-1)^2(0.05) + (0)^2(0.20) + (1)^2(3k) + (2)^2(0.15) + (3)^2(k)
E(X^2) = 0.05 + 0 + 3k + 0.60 + 9k
E(X^2) = 0.65 + 12k
Substituting the value of k that we found in part (a), we get:
E(X^2) = 0.65 + 12(0.15)
E(X^2) = 2.45
Now, we can find Var(X):
Var(X) = E(X^2) - (E(X))^2
Var(X) = 2.45 - (0.85)^2
Var(X) = 2.45 - 0.7225
Var(X) = 1.7275
Therefore, the variance of X is 1.7275.

(d) To find Var(2 - 5X), we need to use the formula Var(a + bX) = b^2Var(X), where a = 2 and b = -5. Substituting the values and the variance of X that we found in part (c), we get:
Var(2 - 5X) = (-5)^2Var(X)
Var(2 - 5X) = 25(1.7275)
Var(2 - 5X) = 43.1875
Therefore, the variance of 2 - 5X is 43.1875.

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The equation that relates the total number of visits to the number of days the promotion has been running is v = 96 × [tex]2^{\frac{d}{5} }[/tex] .

When an equal sign connects two expressions then, it is called an equation.

According to the question, the total number of visits doubled after every 5 days.

So, after 5 days visits= 96×2

After 10 days visits= 96×2×2

After 15 days visists= 96×2×2×2

Therefore, it is a proportional sequence.

Hence, the equation that relates the total number of visits, v, to the number of days the promotion has been running, d comes out to be:

v = 96 × [tex]2^{\tfrac{d}{5} }[/tex]

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