Pre-Algebra Grade 8: Unit 1 Lesson 8 Plane Art (Alternative Portfolio) (Connections)Has anyone done this assignment? Just wanting to get some pointers to get started.

With 4.5% compounded semi-annually, Sarah will need to make installments for roughly 20.82 years in order to save $10,000. Hence, 20.82 is the answer.

Is every two years or semiannually?

Simply said, semiannual refers to events that occur twice a year. A couple might commemorate their nuptials each two years, a corporation might hold workplace celebrations each two years, as well as a family might go on vacation each two years. Every two years, something that happens twice a year does.

We can utilize the calculation again for future value of an annuity to resolve this issue:

FV = P × ((1 + r/n) - 1) / (r/n)

Where:

FV = Future value

P = Periodic payment

r = Annual interest rate

n = Compounding cycles per year, number

t = Number of years

In this instance, Sarah plans to deposit $300 each six months in order to save $10,000. She will so be required to make two payments each year. P thus equals $300, n equals 2, r equals 0.045 (decimalized 4.5%), and FV equals $10,000. The number of years, t, needs to be solved for.

$10,000 = $300 × ((1 + 0.045/2) - 1) / (0.045/2)

When we simplify this equation, we obtain:

20 = (1 + 0.0225) - 1

21 = (1 + 0.0225)

Using both sides' natural logarithms:

ln(21) = ln((1 + 0.0225))

ln(21) = 2×t × ln(1 + 0.0225)

t = ln(21) / (2 × ln(1 + 0.0225))

t = 20.82

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Answer:

[tex]\mathrm{D.\:\:\:f(x) = 2x^2 - 2x - 4; A = 8}[/tex]

Step-by-step explanation:

The formula for the area of a rectangle is A = l x w, where A is the area, l is the length, and w is the width. In this case, the length is 2x - 4 units and the width is x + 1 units. Therefore, the function that models the area of the rectangle is:

f(x) = (2x - 4)(x + 1)

f(x) = 2x^2 - 2x - 4

Therefore, option D is the correct answer.

To find the area when x = 3, we substitute x = 3 into the function:

f(3) = 2(3)^2 - 2(3) - 4

f(3) = 18 - 6 - 4

f(3) = 8

Therefore, when x = 3, the area of the rectangle is 8 square units.

Answer:

5. To find the time in which the loan of N3,900 at the rate of 5% yields N585, we can use the formula for simple interest:

I = P * r * t

where I is the interest, P is the principal (the amount borrowed), r is the interest rate (as a decimal), and t is the time (in years).

Plugging in the given values, we get:

585 = 3900 * 0.05 * t

Solving for t, we get:

t = 3 years

Therefore, it will take 3 years for the loan to yield N585 in interest.

6. To find the time it will take for the simple interest on N70,000 at the rate of 7% to be N20,000, we can again use the formula for simple interest:

I = P * r * t

Plugging in the given values, we get:

20000 = 70000 * 0.07 * t

Solving for t, we get:

t = 4 years

Therefore, it will take 4 years for the simple interest on N70,000 at the rate of 7% to be N20,000.

7. To find the time it takes for N300 to amount to N390 at the rate of 3%, we can use the formula for compound interest:

A = P * (1 + r/n)^(n*t)

where A is the final amount, P is the principal (the initial amount), r is the interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time (in years).

Plugging in the given values, we get:

390 = 300 * (1 + 0.03/1)^(1*t)

Simplifying, we get:

1.3^t = 1.3

Taking the logarithm of both sides, we get:

t = log(1.3) / log(1.3)

t ≈ 1.82

Therefore, it takes approximately 1.82 years for N300 to amount to N390 at the rate of 3%.

8. To find the rate percent per annum for a simple interest of N80 on N5600 in 1 year, we can use the formula for simple interest:

I = P * r * t

Plugging in the given values, we get:

80 = 5600 * r * 1

Solving for r, we get:

r = 0.0143 or 1.43%

Therefore, the rate percent per annum is 1.43%.

9. To find the year in which N5100 will yield an interest of N170 at the rate of 2 1/2 per annum, we can use the formula for simple interest:

I = P * r * t

Plugging in the given values, we get:

170 = 5100 * 0.025 * t

Solving for t, we get:

t = 2/3 years

Since the question asks for the year, we need to add 2/3 years to the current year. Assuming the current year is 2021, we get:

2021 + 2/3 ≈ 2022.67

Therefore, N5100 will yield an interest of N170 at the rate of 2 1/2 per annum in the year 2022.

10. To find the rate per annum at which N4,600 will yield an interest of N230 for 2 years, we can again use the formula for simple interest:

I = P * r * t

Plugging in the given values, we get:

230 = 4600 * r * 2

Solving for r, we get:

r = 0.025 or 2.5%

Therefore, the rate per annum is 2.5%.

11. To find the time it takes for N8100 to yield an interest of N729 at the rate of 9% per annum, we can use the formula for simple interest:

I = P * r * t

Plugging in the given values, we get:

729 = 8100 * 0.09 * t

Solving for t, we get:

t = 1 year

Therefore, it takes 1 year for N8100 to yield an interest of N729 at the rate of 9% per annum.

12. To find the rate of interest per annum for an interest of N23 on N2300 for 2 years, we can use the formula for simple interest:

I = P * r * t

Plugging in the given values, we get:

23 = 2300 * r * 2

Solving for r, we get:

r = 0.005 or 0.5%

Therefore, the rate of interest per annum is 0.5%.

Answer:

Let's use the formula:

distance = rate × time

Let's assume Latasha's rate of travel as "r". Then, Nathan's rate of travel would be "r + 8".

For Latasha:

distance = r × 9

For Nathan:

distance = (r + 8) × 9

According to the problem, the combined distance they traveled is 288 miles. So:

r × 9 + (r + 8) × 9 = 288

Simplifying the equation:

18r + 72 = 288

18r = 216

r = 12

So, Latasha's rate of travel is 12 mph, and Nathan's rate of travel is 20 mph (12 + 8).

Therefore, Latasha is traveling at 12 mph and Nathan is traveling at 20 mph.

Answer:

B (A rectangular prism in which BA = 30 and h = 5 has a volume of 150 units3; therefore, Shannon is correct)

Step-by-step explanation:

I took the test!

The solution is,: A rectangular prism in which BA = 30 and h = 5 has a volume of 150 units^3; therefore, Shannon is correct

In mathematics, volume is the space taken by an object. Volume is a measure of three-dimensional space. It is often quantified numerically using SI derived units or by various imperial or US customary units. The definition of length is interrelated with volume.

here, we have,

step 1

Find the area of the base of the rectangular pyramid

we know that:

The volume of the rectangular pyramid is equal to:

V = 1/3 * bh

where

B is the area of the base

H is the height of the pyramid

we have

V= 50

h = 5

substitute and solve for B

we get,

b= 30

step 2

Find the volume of the rectangular prism with the same base area and height

we know that

The volume of the rectangular prism is equal to

V = bh

we have

b = 30

h = 5

substitute

V = 30 * 5 = 150 unit^3

therefore

The rectangular prism has a volume that is three times the size of the given rectangular pyramid. Shannon is correct

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Therefore , the solution of the given problem of unitary method comes out to be $240 which will cover the cost of the new motorbike.

The task can be completed by multiplying the data gathered using this type of nanosection along with two individuals variable  who utilized the unilateral strategy. In essence, this signifies that perhaps the designated entity is defined or the colour of both mass production is skipped whenever a wanted item occurs. For forty pens, a variable charge of Inr ($1.01) could have been required.

Here,

The quantity supplied and the quantity requested are equal when there is equilibrium. As a result, we can equalise the two formulae and find the equilibrium quantity:

=> -57P + 364 = 25P + 26

=> 82P = 338

=> P = 4.12

We can use either equation to determine the equilibrium amount now that we know the equilibrium price. Use the supply calculation as an example:

=> Q = 25P + 26

=> Q = 25(4.12) + 26

=> Q = 129.5

So, rounded to two decimal places, the equilibrium amount of beer is roughly 129.5.

=> $60 + $180 = $240

which will cover the cost of the new motorbike.

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Therefore, approximately 110,294 ball bearings of radius 4mm can be made from the steel rod.

Volume is the amount of space that a three-dimensional object occupies. It is typically measured in cubic units, such as cubic meters, cubic centimeters, or cubic feet. The volume of an object can be calculated by multiplying its length, width, and height, or by using a specific formula for the shape of the object, such as the formula for the volume of a sphere, cylinder, or cone. Volume is an important concept in mathematics, physics, engineering, and many other fields, and it is used to describe the size and shape of objects, as well as their capacity, density, and other properties.

Here,

To solve this problem, we need to determine the total volume of the steel rod and then calculate how many ball bearings of radius 4mm can be made from this volume. First, we need to calculate the volume of the steel rod. We are given that one kilogram of the steel has a volume of 150 cm³. Therefore, the total volume of the steel rod is:

25 kg * 150 cm³/kg = 3750 cm³

Next, we need to calculate the volume of one ball bearing of radius 4mm. The formula for the volume of a sphere is:

V = (4/3)πr³

Substituting r = 4mm = 0.4cm, we get:

V = (4/3)π(0.4)³ = 0.034 cm³ (rounded to three decimal places)

To find the number of ball bearings that can be made from the steel rod, we divide the total volume of the steel rod by the volume of one ball bearing:

3750 cm³ / 0.034 cm³ ≈ 110,294

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Answer:

1,610,000

Step-by-step explanation:

To round to the nearest hundred, we need to go to the hundreds place. In 683, we can round this to the nearest hundred by using this rule:

5 or more, let it sore

4 or less, let it rest

683 rounded to the nearest hundred is 700

2329 rounded to the nearest hundred is 2300

Now we just need to multiply both of them to get the product

[tex]2300\times 700=1,610,000[/tex]

The three coordinates that satisfy the system of inequalities include the following:

(0, -2)

(-7, - 5)

(-2, -8).

In Mathematics, an ordered pair is sometimes referred to as a coordinate and it can be defined as a pair of two (2) elements or data points that are commonly written in a fixed order within parentheses as (x, y), which represents the x-coordinate (abscissa) and the y-coordinate (ordinate) on the coordinate plane of any graph.

By critically observing the graph of the given system of inequalities y < -2x + 4 and 2x - 4y ≥ 3, the three required solutions that satisfies it include following;

Ordered pair = (0, -2).

Ordered pair = (-7, - 5).

Ordered pair = (-2, -8).

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Answer: 345.6149

Step-by-step explanation:

Answer:

345.6149

Step-by-step explanation:

The two cars will draw level at a distance of 12 kilometers from the starting point, taking their speed and into consideration, as further explained below.

Since both cars are traveling towards the same destination, the distance between them will decrease over time until they draw level. Let's call the distance between the starting point and the meeting point "d" kilometers. We can use the formula:

distance = rate × time

Since car A has a three-minute head start, car B will have to travel for three minutes less than car A. Let's call the time it takes for car B to catch up to car A "t" hours. Then:

time for car A = t + 3/60 hours

time for car B = t hours

We want to find the distance between the starting point and the meeting point, which is the same for both cars. So we can set up an equation based on their distances:

distance traveled by car A = distance traveled by car B

(rate for car A) × (time for car A) = (rate for car B) × (time for car B)

60 × (t + 3/60) = 70 × t

3 = 10t

t = 3/10 hours

Now that we know the time it takes for car B to catch up to car A, we can use either car's rate and time to find the distance between the starting point and the meeting point. Let's use car A's rate and time:

distance = rate × time

distance = 60 × (t + 3/60)

distance = 60 × (3/10 + 1/20)

distance = 60 × (4/20)

distance = 12 kilometers

Therefore, the two cars will meet 12 kilometers from the starting point.

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Therefore, the total cost price of the goods is #5400.

Cost price refers to the amount of money that is spent by a business or individual to acquire or produce a product or service. It includes all the costs involved in the production or acquisition of the product or service, such as raw materials, labor costs, transportation costs, and other expenses. The cost price is usually used as a starting point for calculating the selling price of a product or service and is an important factor in determining the profitability of a business.

by the question.

Let's assume the cost price of the goods to be "x".

According to the problem, the trader sold the goods at a price of #5184 and incurred a loss of 4%.

we know that,

selling price = cost price - loss

Substituting the given values in the above formula, we get:

#5184 = x - 0.04x

Simplifying the equation, we get:

#5184 = 0.96x

Dividing both sides by 0.96, we get:

x = #5400

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Answer:

35.09 units

Step-by-step explanation:

This is a right-angled triangle where:

Hypotenuse = x units

With respect to angle 20°:

Opposite = 12 units

Use trigonometric function sinФ to solve for x:

sinФ = [tex]\frac{Opposite}{Hypotenuse}[/tex]

∴sin20° = [tex]\frac{12}{x}[/tex]

Cross-multiplication is applied:

[tex](x)(sin20) = 12[/tex]

x has to be isolated and made the subject of the equation:

∴[tex]x = \frac{12}{sin20}[/tex]

x = 35.09 units (Rounded to 2 decimal places)

Answer:

The length of x is 35.09 units to two decimal places.

Step-by-step explanation:

In the given right triangle, we have been given the length of the side opposite the angle and need to find the length of the hypotenuse.

To find the value of x, use the sine trigonometric ratio.

[tex]\boxed{\begin{minipage}{9 cm}\underline{Sine trigonometric ratio} \\\\$\sf \sin(\theta)=\dfrac{O}{H}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}[/tex]

Substitute θ = 20°, O = 12 and H = x into the ratio and solve for x:

[tex]\implies \sin 20^{\circ}=\dfrac{12}{x}[/tex]

[tex]\implies x=\dfrac{12}{\sin 20^{\circ}}[/tex]

[tex]\implies x=35.085652...[/tex]

[tex]\implies x=35.09\; \sf (2\;d.p.)[/tex]

Therefore, the length of x is 35.09 units to two decimal places.

Since line GF = line JK and < G = <J ; the minor arcs, the major arcs of the both circle are also similar and equal.

The minor arc of the given circle above can be defined as the part of the circle that is less than 180°. This is represented by <HGF in the first circle( with a red ink).

The major arc of the given circle above can be defined as the part of a circle that is greater than 180°. Thus is represented by <IJK in the second circle (with black ink).

Therefore in conclusion, there is a established similarity between the two circles since line GF = line JK and < G = <J, thus making their minor and major acre to be equal too.

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The expression for the area of the shaded region in its simplest form is x² + 23x + 49.

What is the area of the rectangle?

To find the area of a rectangle, we multiply the length of the rectangle by the width of the rectangle.

The shaded area is the difference of the areas of the rectangle and the square.

Rectangle:

A₁ = (x + 10)(2x + 5) = 2x² + 5x + 20x + 50 = 2x² + 25x + 50

Square:

A₂ = (x + 1)² = x² + 2x + 1

Shaded region:

A₁ - A₂ = 2x² + 25x + 50 - (x² + 2x + 1)

= 2x² + 25x + 50 - x² - 2x - 1

= x² + 23x + 49

Hence, the expression for the area of the shaded region in its simplest form is x² + 23x + 49.

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Complete question:

Enter your answer and show all the steps that you use to solve this problem in the space provided.

A rectangle is shown with length x plus 10 and width 2 x plus 5. The inside of the rectangle is shaded other than an unshaded square with length x plus 1 and width x plus 1.

Write an expression for the area of the shaded region in its simplest form. Show all of your steps.

Answer: She

bought a total of 5 notebooks and 13 pencils.

Step-by-step explanation:

Answer:

The answer is Ahope this helps

Answer:

honestly don't know but you got this bro {=•=}

Answer:

[tex]\huge\boxed{\sf x = -24}[/tex]

Step-by-step explanation:

[tex]\displaystyle -\frac{1}{2} x - 4 =8\\\\Add \ 4 \ to \ both \ sides\\\\-\frac{1}{2} x = 8+4\\\\-\frac{1}{2} x = 12\\\\Multiply \ both \ sides \ by \ -2\\\\-\frac{1}{2} x \times \ -2 = 12 \times -2\\\\\boxed{x = -24}\\\\\rule[225]{225}{2}[/tex]

Answer:

[tex]\boxed{\mathtt{1) \ x=4}}[/tex]

[tex]\boxed{\mathtt{2) \ x=2}}[/tex]

[tex]\boxed{\mathtt{3) \ x=-24}}[/tex]

Step-by-step explanation:

[tex]\textsf{For these problems, we are asked to solve for x in each equation.}[/tex]

[tex]\textsf{We should use similar steps for each of them.}[/tex]

[tex]\large\underline{\textsf{For Number 1:}}[/tex]

[tex]\textsf{We should first begin by adding 2 to both sides of the equation.}[/tex]

[tex]\mathtt{8x=32}[/tex]

[tex]\textsf{Now, divide by 8.}[/tex]

[tex]\boxed{\mathtt{1) \ x=4}}[/tex]

[tex]\large\underline{\textsf{For Number 2:}}[/tex]

[tex]\textsf{Let's begin by subtracting 5 from both sides of the equation.}[/tex]

[tex]\mathtt{-x=-2}[/tex]

[tex]\textsf{Now, divide by -1.}[/tex]

[tex]\boxed{\mathtt{2) \ x=2}}[/tex]

[tex]\large\underline{\textsf{For Number 3:}}[/tex]

[tex]\textsf{Beginning Number 3, let's multiply by -2 to both sides of the equation.}[/tex]

[tex]\mathtt{x+8=-16}[/tex]

[tex]\textsf{Now, subtract 8 from both sides of the equation.}[/tex]

[tex]\boxed{\mathtt{3) \ x=-24}}[/tex]

Answer:

To find the percentage decrease, we need to find the difference between the original price and the new price, divide that difference by the original price, and then multiply by 100 to get the percentage.

The difference between the original price and the new price is:

80 - 44 = 36

To find the percentage decrease, we divide this difference by the original price:

36 / 80 = 0.45

Finally, we multiply by 100 to get the percentage:

0.45 * 100 = 45%

Therefore, the percentage decrease from $80 to $44 is 45%.

Answer:  The percentage decrease is 45%.

Step-by-step explanation:

Percentage decrease = [tex](\frac{Oringinal Value-NewValue}{Oringinal Value} )*100[/tex]

                                    = [tex](\frac{80-44}{80} )*100[/tex]

                                    = 45%

The true statement are:

A. 15m3-6m=3m(5m2-6m),

B. 40m6-4=4(10m6-1),

C. 6m2+18m=6m2(1+3m),

Therefore the correct option is A, B, C.

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The correct question is:

Which statements are true

15m3-6m=3m(5m2-6m),

40m6-4=4(10m6-1),

6m2+18m=6m2(1+3m),

32m4+12m3(8m+3)

The value of x is 8 when GH is 3x-2, HI is 7x-4 and GI is 8x+10. It forms a straight line.

Segment addition is a concept in geometry that states that given three collinear points A, B, and C, with B between A and C, the length of AB added to the length of BC will give the length of AC. This can be expressed algebraically as:

AB + BC = AC

We can use the fact that GH + HI = GI (by the segment addition postulate) and substitute the given expressions to get:

(3x - 2) + (7x - 4) = 8x + 10

Simplifying the left side by combining like terms, we get:

10x - 6 = 8x + 10

Subtracting 8x from both sides, we get:

2x - 6 = 10

Adding 6 to both sides, we get:

2x = 16

Dividing by 2, we get:

x = 8

Therefore, x = 8.

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Answer:

  C

Step-by-step explanation:

You want to identify the set of ordered pairs corresponding to the given pattern definitions.

The x-pattern starts at 2.

The y-pattern starts at 0.

The first (x, y) ordered pair is (2, 0). This matches answer choice C.

__

Additional comment

The x-rule tells you the sequence is ...

  2, 2+6 = 8, 8+6 = 14, 14+6 = 20 . . . . . {2, 8, 14, 20}

The y-rule tells you the sequence is ...

  0, 0+5 = 5, 5+5 = 10, 10+5 = 15 . . . . . {0, 5, 10, 15}

Then the (x, y) pairs are ...

  (2, 0), (8, 5), (14, 10), (20, 15)

Note that when X and Y are the variables involved, we conventionally use the ordered pair (x, y). When other variables are involved, the ordered pair may be different, not always alphabetic order.

Generally, the independent variable is listed first. Here, neither depends on the other, so that criterion does not apply. Listing X first here is also suggested by the fact that the X pattern rule is defined first.

let's keep in mind that in the IV Quadrant, the cosine is positive whilst the sine is negative, so hmm

[tex]\cos(\theta )=\cfrac{\stackrel{adjacent}{8}}{\underset{hypotenuse}{9}} \qquad \textit{let's find the \underline{opposite side}} \\\\\\ \begin{array}{llll} \textit{using the pythagorean theorem} \\\\ a^2+o^2=c^2\implies o=\sqrt{c^2 - a^2} \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{9}\\ a=\stackrel{adjacent}{8}\\ o=opposite \end{cases}[/tex]

[tex]o=\pm\sqrt{ 9^2 - 8^2}\implies o=\pm\sqrt{ 81 - 64 } \implies o=\pm\sqrt{ 17 }\implies \stackrel{ IV~Quadrant }{o=-\sqrt{17}} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \tan(\theta )=\cfrac{\stackrel{opposite}{-\sqrt{17}}}{\underset{adjacent}{8}}~\hfill[/tex]

To find the value of tan(θ) when cos(θ) = 8/9 and θ is in the fourth quadrant, use the relation sin^2(θ) = 1 - cos^2(θ) to determine sin(θ). Then compute tan(θ) = sin(θ) / cos(θ). The result will be negative

Given the equation cos(θ) = 8/9 in the fourth quadrant of the Cartesian plane, you are required to determine the value of tan(θ). In the fourth quadrant, cosine values are positive, and sine values are negative. You can use the relation sin^2(θ) = 1 - cos^2(θ). So, sin(θ) would be -sqrt(1 - (8/9)^2). The tangent of an angle in any quadrant can be found by taking the ratio of the sine to the cosine, i.e., tan(θ) = sin(θ) / cos(θ). Substitute the values of sine and cosine to find the value of tan(θ). It turns out that tan(θ) will be negative in the fourth quadrant.

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The prοbability οf  that the number οf lοst-time accidents οccurring οver a periοd οf 9 days will be at least 4  is 0.3975.

Prοbability is a way οf calculating hοw likely sοmething is tο happen. It is difficult tο prοvide a cοmplete predictiοn fοr many events. Using it, we can οnly fοrecast the prοbability, οr likelihοοd, οf an event οccurring. The prοbability might be between 0 and 1, where 0 denοtes an impοssibility and 1 denοtes a certainty.

We can use here Pοisson distribution with λ=0.4 * 8=3.2

[tex]P[X\geq 4]=1-P[X < 4][/tex]

[tex]=1-{P[X=0] + P[X=1] + P[X=2] +P[X=3] =1-{0.0408+ 0.1304+ 0.2087 + 0.2226} =0.3975[/tex]

Where,

[tex]P[X=0]=\frac{e^{-3.2}(3.2)^0}{0!} = 0.0408[/tex]

[tex]P[X=1]=\frac{e^{-3.2}(3.2)^1}{1!} = 0.1304[/tex]

[tex]P[X=2]=\frac{e^{-3.2}(3.2)^2}{2!} = 0.2087[/tex]

[tex]P[X=3]=\frac{e^{-3.2}(3.2)^3}{3!} = 0.2226[/tex]

Hence the prοbability of  that the number of lost-time accidents οccurring over a period of 9 days will be at least 4  is 0.3975.

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We can say that after answering the offered question Therefore, the proportionality constant of proportionality in this scenario is 11.25.

Proportionate relationships are those that have the same ratio every time. For example, the average number of apples per tree defines how many trees are in an orchard and how many apples are in an apple harvest. Proportional refers to a linear relationship between two numbers or variables in mathematics. When the first quantity doubles, the second quantity doubles as well. When one of the variables decreases to 1/100th of its previous value, the other falls as well. When two quantities are proportional, it means that when one rises, the other rises as well, and the ratio between the two remains constant at all values. The diameter and circumference of a circle serve as an example.

In this case, we may use the following formula to calculate the proportionality constant:

proportionality constant = output/input

where the output is John's pay and the input is the number of hours he worked.

As a result, the proportionality constant is:

Paycheck/hours worked = proportionality constant

proportionality constant = 450/40

proportionality constant = 11.25

As a result, the proportionality constant in this scenario is 11.25.

In this case, we may use the following formula to calculate the proportionality constant:

proportionality constant = output/input

where the output is John's pay and the input is the number of hours he worked.

As a result, the proportionality constant is:

Paycheck/hours worked = proportionality constant

proportionality constant = 450/40

proportionality constant = 11.25

Therefore, the constant of proportionality in this scenario is 11.25.

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The equations that can be used to solve for y, the length of the room are: y² - 5y - 750 = 0 , (y - 30)(y + 25) = 0 , y(y - 5) + 750 = 0

An equation is a mathematical statement that asserts the equality of two expressions, often with variables and constants.

To solve for y, we can use the same equation we used for x, since y represents the length of the room. Therefore, the equation we can use to solve for y is:

y² - 5y - 750 = 0

This equation can be factored as (y - 30)(y + 25) = 0, which gives us two solutions: y = 30 or y = -25. However, since the length of a room cannot be negative, the only valid solution is y = 30.

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