Can someone help me please, I would appreciate it i need to answer the definitions of Theorems Formulas step by step and i need 4 examples in order please.
Definitions
Theorems
Formulas
Step-by-Steps
Etc.

And i need 4 examples please.

Example 1

Example 2

Example 3

Example 4

In the exponential decay, A) r = -0.0839 , B) r = -8.39% , C) Value=$11,800.

What is exponential decay?

The term "exponential decay" in mathematics refers to the process of a constant percentage rate reduction in an amount over time. It can be written as y=a(1-b)x, where x is the amount of time that has passed, an is the initial amount, b is the decay factor, and y is the final amount.

To find the annual rate of change between 1992 and 2006, we can use the formula:

r = [tex](V_2/V_1)^{1/n}-1[/tex]

where V1 is the initial value, V2 is the final value, and n is the number of years between the two values.

=>r = -0.0839

Therefore, the rate of change between 1992 and 2006 is -0.0839.

To express the rate of change in percentage form, we can multiply the result from part A by 100:

=>r = -0.0839 x 100

=> r = -8.39%

Therefore, the rate of change between 1992 and 2006 is a decrease of 8.39%.

To find the value of the car in the year 2009, we can assume that the value continues to drop at the same percentage rate as calculated in part A.

From 2006 to 2009, there are 3 years. So, using the formula for exponential decay, we have:

where V0 is the value in 2006, r is the rate of decrease, and n is the number of years between 2006 and 2009.

=>V = 11792.51

Therefore, the value of the car in the year 2009 would be approximately $11,800 (rounded to the nearest $50).

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Step-by-step explanation:

oil change. palm trees oil

The total cost of the metal which will be used to construct the metal tank would be = $82.5

To calculate the area of the metal tank, the formula for the area of the cylinder is used which would be;

= 2πr (h+r)

Where

R = 12/2 = 6 ft

h = 4ft

π = 3.14

area = 2×3.14×6(4+6)

= 37.68×10

= 3.75 ft²

But 1ft² = $22

3.75ft² = X

make X the subject of formula;

X = 22× 3.75 = $82.5

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Joseph must invest approximately $110840.16 today to reach his retirement goal of $2 million, assuming a consistent rate of inflation of 7.5%.

To calculate the amount Joseph needs to invest today to reach his retirement goal of $2 million, we can use the future value formula for a lump sum investment:

[tex]$FV = PV x (1 + r)^n[/tex]

where FV is the future value, PV is the present value, r is the rate of return (in this case, the inflation rate of 7.5%), and n is the number of years.

In this case, we want to solve for PV, which represents the amount Joseph needs to invest today. We know that:

Joseph has 40 years until he retires (65 - 25 = 40)

His retirement goal is $2 million

The inflation rate is 7.5%

Plugging these values into the formula, we get:

[tex]$2,000,000 = PV \times (1 + 0.075)^{40}[/tex]

Simplifying, we have:

[tex]$PV = \frac{2,000,000}{(1 + 0.075)^{40}}[/tex]

PV = $2,000,000 / 18.044

Therefore, Joseph must invest approximately $110840.16 today to reach his retirement goal of $2 million, assuming a consistent rate of inflation of 7.5%.

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Answer:

The answer is: m = B(3x + 1) degrees.

Step-by-step explanation:

In the given equation m/B= (3x + 1)°, we need to find the measure of m.

To find m, we need to isolate it on one side of the equation.

We can do this by multiplying both sides of the equation by B, which gives us m = B(3x + 1)°.

This means that m is equal to the product of B and (3x + 1)°.

We can simplify further by multiplying 3x + 1 by the degree symbol, which gives us m = B(3x + 1) degrees.

The formula used in this problem is m/B = angle measure, where m is the unknown side, B is the length of the known side, and the angle measure is given in degrees.

When solving problems like this, it is important to watch for units and make sure they are consistent throughout the equation.

For example, if B is measured in meters, then the units of m should also be in meters.

A real-world example of using this formula could be calculating the height of a building based on the length of its shadow and the angle of the sun's rays.

Answer: m = B(3x + 1) degrees.

Math: m = B(3x + 1)°

Formula: m/B = angle measure

Name of formula: Trigonometric ratio

Real-world example: Finding the height of a building based on the length of its shadow and the angle of the sun's rays.

chatgpt

Answer:

As the height of the rider increases by 1 inch, the bike frame increases by 0.29 inches

Step-by-step explanation:

Lets label what we know here:

b = size of the bike frame in inches

h = height of the rider in inches

We are also given the equation [tex]b=0.29h+1.35[/tex], which is in slope intercept form. This means that the y intercept is 1.35 and the slope is 0.29.

This means that, as the height of the rider increases by 1 inch, the bike frame will also increase. Lets put this to the test:

[tex]b=0.29(1) + 1.35[/tex]

[tex]b=1.64[/tex]

We will get a different size of the bike frame depending on what we set h equal too.

Step-by-step explanation:

on Monday:340ML

on Tuesday:1.5L

total millilitres?so we have to get the to get the total of water in Millilitres (ML)

step 2:we have to convert the 1.5L to ML

we know 1L=1000ML

and 1000ML=1ML

so 1.5L =1500ml

Step 3

340 +1500=1840ML

Side A and Side B are both equal to 32/3, which is 10 2/3 centimeters and the total number of hours the girls spent cleaning is 15 hours.

The perimeter of a shape can be found by adding up the lengths of all its sides.

The perimeter of a triangle is the sum of the lengths of its three sides, so in this case the equation to solve for the length of each of the two sides is:

Perimeter = 2 x Side A + Side B

32 = 2x + x

3x = 32

x = 32/3

Therefore, Side A and Side B are both equal to 32/3, which is 10 2/3 centimeters.

Kelsey and her 4 sisters spent an equal amount of time cleaning their home. To find the total number of hours the girls spent cleaning, we can write and solve a division equation. Let c be the total number of hours the girls spent cleaning.

c / 5 = 3

c = 15

Therefore, the total number of hours the girls spent cleaning is 15 hours.

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In the percentage , the statement is the best prediction about the scoops of ice cream the college will need is D)The college will have about 1,440 students who prefer ice cream.

What is percentage?

percentage. Divide the A value or ratio that may be stated as a fraction of 100 is referred to in mathematics as a number by the total and multiply by 100 to find the percent of a given number. Therefore, the percentage refers to a portion per hundred. Per 100 is what the word percentage signifies. The letter "%" stands for it.

Here then given, Total number of students = 4000

Sample number of students = 225.

In 225 students 81 students prefer ice cream.

Now to find percentage then,

=> [tex]\frac{81}{225}\times100[/tex]

=> 0.36*100

=> 36%.

Now Number of students who prefer ice cream = 36% of 4000

=> 36/100 * 4000

=> 1440.

Hence the correct option is D)The college will have about 1,440 students who prefer ice cream.

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Miss x baked 286 loaves of bread on Saturday and 208 loaves of bread on Sunday. To calculate the total loaves of bread baked on both days, we can subtract 78 from 286 to get the number of loaves of bread baked on Sunday.

Miss x baked 286 loaves of bread on Saturday. We know that she baked 78 fewer loaves of bread on Sunday than Saturday. To calculate the total loaves of bread baked on both days, we can subtract 78 from 286 to get the number of loaves of bread baked on Sunday. This gives us 208 loaves of bread baked on Sunday. Adding this to the 286 loaves of bread baked on Saturday, we get a total of 494 loaves of bread baked on both days. Therefore, Miss x baked 286 loaves of bread on Saturday and 208 loaves of bread on Sunday.

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The equation with SLOPE: -3/4 and Y-intercept (0,5) is Y = (-3/4)x + 5 and graphed below.

The pοint οn a line's graph where it crοsses the x-axis is knοwn as the x-intercept.

The pοint where a line's graph crοsses the y-axis is knοwn as the y-intercept.

On any graph, the x and y -intercepts are crucial lοcatiοns. The graphs οf linear equatiοns will be the main tοpic οf this chapter. Hοwever, at this pοint, we may utilize these cοncepts tο identify nοnlinear graph intercepts. Always keep in mind that intercepts are οrdered pairs that shοw where the graph and axes cοnnect.

The fοrmula οf slοpe-intercept is y=mx+b.

By using this fοrmula

Y=(-3/4)x+b

Using the given pοint we get,

5=(-3/4)0+b

Or, b=5

The answer is Y=(-3/4) x+5

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Answer:

a dilatation centred at the origin with scale factor 3

Step-by-step explanation:

under a dilation centred at the origin with scale factor k

a point (x, y ) → (kx, ky )

here

A (2, 4 ) → A' (3(2), 3(4) ) → A' (6, 12 )

B (1, 6 ) → B' (3(1), 3(6) ) → B' (3, 18 )

C (5, 3 ) → C' (3(5), 3(3) ) →C' (15, 9 )

thus the transformation for Δ ABC → Δ A'B'C'

is a dilatation centred at the origin with scale factor 3

Answer:o calculate the compound interest, we can use the formula:

A = P(1 + r/n)^(nt)

where:

A = the final amount

P = the principal amount (initial investment)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the time period (in years)

In this case, P = £8500, r = 5.7% = 0.057, n = 1 (since the interest is compounded annually), and t = 6.

So, plugging in the values:

A = £8500(1 + 0.057/1)^(1*6)

A = £8500(1.057)^6

A = £11260.23

Therefore, Brian will have £11260.23 in his bank account after 6 years with compound interest. Rounded to the nearest penny, the answer is £11,260.23.

Step-by-step explanation:

The relationship shown on the table is such that C. The relationship is not linear.

To determine if the relationship is linear, we can check if the differences in the y-values are constant for equal differences in the x-values.

Calculate the differences between the x-values and the y-values:

Δx1 = 7 - 1 = 6

Δy1 = -7 - (-4) = -3

Δx2 = 13 - 7 = 6

Δy2 = 10 - (-7) = 17

Δx3 = 19 - 13 = 6

Δy3 = -13 - 10 = -23

The differences in x-values are constant (Δx1 = Δx2 = Δx3 = 6), but the differences in y-values are not (Δy1 ≠ Δy2 ≠ Δy3). Therefore, the relationship is not linear.

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Step-by-step explanation:

Area of ENTIRE circle = pi r^2 = pi (18)^2 = 1017.876 cm^2

the shaded area is   221 out of 360    ( 360 is the ENTIRE circle area)

221 / 360   * area =   221/ 360 * 1017.876  = 624.8 = ~ 625 cm^2

The mean of the sampling distribution of all possible samples of size 16 will be 42.5, the correct option is D.

Normal distribution with mean μ = 42.5 inches and standard deviation σ = 2.5 inches. Let x₁, x₂, ..., xₙ be a random sample of size n = 16 from X, and let x be the sample mean

The sampling distribution of the mean x is also Normal, with mean μ and standard deviation σ/√n.

f(x) = (1/√(2π) × (σ/√n)) × exp[-(x - μ)² ÷ (2 × (σ/√n)²)]

we integrate

Mean of x = ∫(-∞ to ∞) x × f(x) dx

Mean of x = 2 × ∫(μ to ∞) x × f(x) dx

Next, we substitute

u = (x - μ) / (σ/√n):

Mean of x = 2 × ∫(0 to ∞) (u × (σ/√n) + μ) × (1/√(2π) × (σ/√n)) × exp(-u² / 2) du

Simplifying this expression gives:

Mean of x = μ

Therefore, the mean of the sampling distribution of all possible samples of size 16 is equal to the population mean μ, which is 42.5 inches.

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The complete questions is :

The heights of five-year-olds are Normally distributed with a mean of 42.5 inches and a standard deviation of 2.5 inches. A random sample of 16 five-year-olds is taken and the mean height is recorded. What would be the mean of the sampling distribution of all possible samples of size 16?

A 2.66

B 10.63

C 17

D 42.5

The area of the composite figure using area formula is 45unit².

Option b is the correct option.

The space that any composite shape occupies is referred to as the area of composite shapes. In order to create the desired shape, a few polygons are connected to create a composite shape. These figures or forms can be constructed using a variety of geometrical elements, including triangles, squares, quadrilaterals, and others.

To determine the area of a composite object, divide it into simple shapes like a square, triangle, rectangle, or hexagon. In essence, a composite shape is a combination of fundamental shapes. It goes by the name's "composite" or "complex" shapes.

In the question, we can see that the composite figure comprises of a triangle and a rectangle.

Now, length of the rectangle as per the vertices is, l = 6 units.

Breadth of the rectangle as per the vertices is, b = 6 units.

As the length and breadth of the rectangle as per the vertices are equal it's a square with side a = 6 units

Area of square = a²

= 6²

=36unit².

Now in the triangle,

Base, b = 6 units

Height, h = 3 units.

Area = 1/2 × b × h

= 1/2 × 6 × 3

= 9unit².

Therefore, the total area of the figure is 36 + 9 = 45unit².

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a. Using a standard normal distribution table,  p(z > 2.10) = 0.0188.

b.  p(z > -1.50) = 0.9332.

c. p(z < -0.55) = 0.2912.

The study of random occurrences or phenomena falls under the umbrella of the mathematic discipline known as probability. It is the measure of the likelihood or chance that an event will occur, expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

To find the probabilities for a normal distribution, we need to use a standard normal distribution table or a calculator with a normal distribution function.

a. p(z > 2.10)

Using a standard normal distribution table, we can find that the area to the right of z = 2.10 is 0.0188. Therefore, p(z > 2.10) = 0.0188.

b. p(z > -1.50)

The area to the right of z = -1.50 is the same as the area to the left of z = 1.50. Using a standard normal distribution table, we can find that the area to the left of z = 1.50 is 0.9332. Therefore, p(z > -1.50) = 0.9332.

c. p(z < -0.55)

Using a standard normal distribution table, we can find that the area to the left of z = -0.55 is 0.2912. Therefore, p(z < -0.55) = 0.2912.

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Answer: 41 degrees

Step-by-step explanation: To determine the angle at which the pizza was cut, we need to use the formula for the area of a sector of a circle:

A = (θ/360)πr²

where A is the area of the sector, θ is the central angle of the sector (in degrees), π is a constant (approximately equal to 3.14), and r is the radius of the circle.

In this case, we know that the area of the sector that corresponds to the slice of pizza that was eaten is 17 square inches. We also know that the pizza has a diameter of 14 inches, which means that the radius is 7 inches.

Substituting these values into the formula, we get:

17 = (θ/360)π(7²)

17 = (θ/360)49π

θ/360 = 17/(49π)

θ = (17/49π) * 360

θ ≈ 41 degrees (rounded to the nearest whole number)

Therefore, the pizza was cut at an angle of approximately 41 degrees.

One possible set of four lengths that meets the given requirements is:

3 inches, 4 inches, 5 inches, 7 inches

To show that this set of lengths meets all the requirements, we need to demonstrate that:

Each length is different: We can see that all four lengths are different.

Each length is a whole number: We can see that all four lengths are whole numbers.

No matter which three you choose, you will always be able to make a triangle: To show this, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

For example, if we choose the lengths 3, 4, and 5, we can see that 3 + 4 = 7, which is greater than 5. Therefore, we can make a triangle with sides of length 3, 4, and 5. Similarly, if we choose the lengths 3, 4, and 7, we can see that 3 + 4 = 7, which is still greater than 7. Therefore, we can make a triangle with sides of length 3, 4, and 7.

We can repeat this process with any combination of three lengths from the set, and we will always find that we can make a triangle. Therefore, we have shown that the set of lengths {3 inches, 4 inches, 5 inches, 7 inches} meets all the requirements.

The inequality or system of inequalities matched with the respective graphs are:


26) = Graph A
27) = Graph B
28) = Graph D

For y ≤ 3x² -
For any given value of x, the corresponding value of y can be found by squaring x and multiplying it by 3. Since the coefficient of x² is positive, the parabola represented by this inequality opens upward.

The graph of this inequality is a shaded region below or on the curve of the parabola. This means that any point (x, y) that lies on or below the curve satisfies the inequality, while any point above the curve does not.

For y ≥ - x²; y < x² + 3 -

The system of inequalities includes two quadratic functions in two variables, x and y. The first inequality y ≥ -x² represents a parabola that opens downward, while the second inequality y < x² + 3 represents a parabola that opens upward. The solution set for the system includes all points that satisfy both inequalities simultaneously. This solution set lies in the region above the first parabola and below the second parabola, and is bounded by the x-axis.

For y ≥ x²-5; y ≤ -2 x² + 3x + 3 -

The system of inequalities includes two quadratic functions in two variables, x and y. The first inequality y ≥ x² - 5 represents a parabola that opens upward, while the second inequality y ≤ -2x² + 3x + 3 represents a parabola that opens downward. The solution set for the system includes all points that satisfy both inequalities simultaneously. This solution set lies in the region between the two parabolas, and is bounded by the x-axis.


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To calculate the compound interest on Rs 16000 at 20% per annum for 9 months, compounded quarterly, we need first to calculate the quarterly interest rate and the number of compounding periods.

Quarterly interest rate = Annual interest rate / 4

= 20% / 4

= 5%

Number of compounding periods = (Time in months) / (Number of months per compounding period)

= 9 / 3

= 3

Using the formula for compound interest:

A = P(1 + r/n)^(nt)

where,

A = final amount

P = principal amount

r = annual interest rate

n = number of compounding periods per year

t = time in years

Plugging in the values, we get:

A = 16000(1 + 0.05/4)^(4*3/12)

A = 16000(1 + 0.0125)^1

A = 16000(1.0125)

A = 16180

Therefore, the compound interest on Rs 16000 at 20% per annum for 9 months, compounded quarterly is Rs 1800 (i.e., A - P = 16180 - 16000).

No, we cannot prove that war is near based on these propositions only.

In logic and mathematics, a proposition is a statement or assertion that is either true or false. It is also referred to as a declarative sentence. Propositions are often used as the building blocks for logical reasoning and mathematical proofs.

Here,

The given propositions are a chain of conditional statements, also known as if-then statements. To prove that war is near, we would need a statement or evidence that directly supports this claim. None of the given propositions directly state or imply that war is near. Even if we assume that all the given propositions are true, the only thing we can conclude is that Notre Dame de Paris cathedral started burning. We cannot make any conclusions about whether war is near or not.

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Answer:

The given statement is False. When the interest is compounded half yearly the number of conversion periods will be two because a year comprises 12 months and has two periods of six months each.

Step-by-step explanation:

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Thus, the amount of money Richard owe (in dollars) after 1 year on his credit card is $2,016.19.

Compounding interest on both the principal and the accrued interest is expressed by the term "monthly compound interest," which refers to interest that is compounded over month. The principal amount times one plus the interest rate divided by a number of periods, raised to the power of such number of periods, is how monthly compounding is computed.

Formula for amount after compounding:

A = P[tex](1 + r/n)^{nt}[/tex]

A = amount after compounding

P is principal (=$1,597.57)

r is rate of interest (23.5%)

n is the number of times compounded (= 12)

t is time in years (1 year)

A = 1597.57* [tex](1 + 0.235/12)^{12*1}[/tex]

A = 1597.57* 1.26

A = 2,016.19

Thus, the amount of money Richard owe (in dollars) after 1 year on his credit card is $2,016.19.

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Answer:

2, 3, 3, 7, 10

Which values are the same?

Mean- 5

Mode- 3

Median- 3

Thus the mode and the median are the same

Step-by-step explanation:

You're welcome.

Answer: Mode and Median...

Step-by-step explanation: Mode and Median: Mode(Number that you see most) 3, Median(Middle number in least to greatest order) 3

Mode and Mean: Mode( Number you see most) 3, Mean(   Add up all the values in the set, then divide the sum by how many values there are. ) 2+3+3+7+10=25 25 divided by 5=5  (3 and 5)

Range and Mean: Range( Largest value minus the Smallest value) 10-2=8 Mean(Add up all the values in the set, then divide the sum by how many values there are.) 2+3+3+7+10=25 25 divided by 5=5 (8 and 5)

So the probability of it not snowing today is 0.7 or 70%.

what is probability?

Probability is a measure of the likelihood or chance of an event occurring. It is a numerical value between 0 and 1, where 0 indicates an impossible event and 1 indicates a certain event. Probability theory is a branch of mathematics that deals with the study of random events and their properties.

In the given question,

The complement of an event is the probability that the event does not occur. In this case, the complement of "it snows today" would be "it does not snow today".

To calculate the complement of an event, you can subtract the probability of the event from 1. So if the probability of it snowing today is 0.3 (or 30%), then the probability of it not snowing today would be:

1 - 0.3 = 0.7

So the probability of it not snowing today is 0.7 or 70%.

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Answer: Since the parabola opens down and has a focus at (0, -12), its directrix is a horizontal line located 12 units above the vertex. Let the vertex of the parabola be (h, k).

Since the focal diameter is 20, the distance between the focus and the directrix is also 20. Therefore, the directrix is the horizontal line y = k + 12, and the distance from the vertex to the focus is equal to the distance from the vertex to the directrix, which is 10.

Using the definition of a parabola, we can write:

sqrt((x - h)^2 + (y - k)^2) = 10 + (y - k - 12)

Squaring both sides, we get:

(x - h)^2 + (y - k)^2 = (10 + y - k - 12)^2

Expanding the right-hand side and simplifying, we get:

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

Simplifying further, we get:

x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = y^2 - 4y + 4

Rearranging the terms, we get:

x^2 - 2hx + h^2 + 2ky - 4y + k^2 - 4 = 0

Since the parabola opens down, the coefficient of x^2 must be negative. Therefore, we can write:

-(x^2 - 2hx + h^2 + 2ky - 4y + k^2 - 4) = 0

Multiplying out the negative sign, we get:

-h^2 + 2hx - 2ky + 4y - k^2 + 4 = 0

Therefore, the equation of the parabola that opens down, has focus (0, -12), and a focal diameter of 20 is:

-h^2 + 2hx - 2ky + 4y - k^2 + 4 = 0

Step-by-step explanation:

The value of the two numbers are 43 and 218

From the information given, we have;

Let the numbers be x and y

Then,

The difference between the numbers = x - y

The sum of the numbers = x + y

Substitute the values

x + y = 261

x - y = 175

Let's solve the simultaneous equations

Make 'x' the subject from 1

x = 261 - y

Substitute the value in 2

261 - y - y= 175

collect the like terms

-2y = - 86

y = 43

Substitute to determine the value of x

x = 261 - y = 261 - 43 = 218

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