Part a: Assume that the height of your cylinder is 8
inches. Consider A
as a function of r
, so we can write that as A(r)=2πr2+16πr
. What is the domain of A(r)
? In other words, for which values of r
is A(r)
defined?



Part b: Continue to assume that the height of your cylinder is 8
inches. Write the radius r
as a function of A
. This is the inverse function to A(r)
, i.e., to turn A
as a function of r
into r
as a function of A
.



r(A)=






Change entry mode
Hints:

To calculate an inverse function, you need to solve for r
. Here, you would start with A=2πr2+16πr
. This equation is the same as 2πr2+16πr−A=0
which is a quadratic equation in the variable r
, and you can solve that using the quadratic formula. You will want to keep A
as a variable when you plug the values into the quadratic formula.
If you want to type in 3π+1x+1
in Mobius, in text mode you can type in (3*pi+1)/(x+1). There is more information in the Introduction to Mobius unit.


Part c: If the surface area is 300
square inches, then what is the radius r
? In other words, evaluate r(300)
. Round your answer to 2 decimal places.

Hint: To compute a numeric square root such as 17.3−−−−√
, you could

Use a spreadsheet such as Microsoft Excel or OpenOffice Calc and type in =sqrt(17.3)
Use a browser to connect to the Internet and type in sqrt(17.3) into a search field
Use a calculator


The radius is
Number
inches if the surface area is 300
square inches.

Answer:

s = 8 mi

Step-by-step explanation:

We know that the ratio of side lengths of a 30-60-90 triangle is:

short leg : long leg : hypotenuse

     x       :    x√3     :        2x

We can apply this information to the given triangle. We know that the length of the short leg is 4 mi. This can be substituted for x:

x = 4 mi

We are trying to solve for s, which is the triangle's hypotenuse. So, we can apply the ratio to x.

s = 2x = 2(4 mi) = 8 mi

The value of r'(1) = 60 for the given function, using the chain rule.

By adding the derivatives of f(x) with regard to g(x) and g(x) with respect to the variable x, one may get the derivative of the composite function h(x) = f(g(x)). The chain rule of differentiation can be used to derive derivatives of composite functions. Now, let's go over what composite functions are. When a function is expressed in terms of another function, it is said to have a composite function. This suggests that a function can be changed into another function in a composite function.

Using the chain rule of derivatives we have:

r'(x) = f'(g(h(x))) * g'(h(x)) * h'(x)

Substituting the values we have:

r'(1) = f'(g(h(1))) * g'(h(1)) * h'(1)

r'(1) = f'(g(3)) * g'(3) * 3

r'(1) = f'(4) * 4 * 3

r'(1) = 5 * 4 * 3

r'(1) = 60

Hence, the value of r'(1) = 60 for the given function, using the chain rule.

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Based on the information provided, it can be concluded that the length of the two shorter sides combined is half the length of the longest side.

The longest side, which is the green bar is 10 centimeters long. On the other hand, by adding the two shortest sides (orange bar 3 centimeters + blue bar 2 centimeters) the total length would be 5 centimeters.

Considering the longest side is 10 centimeters and the shortest side is 5 centimeters then the length of the two shorter sides combined is half the length of the longest side.

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Monique needs to invest $50,354.37 today to have $500,000 in 30 years at an annual interest rate of 7.8% compounded half-yearly. Option D

We can use the formula for future value of an annuity to solve this problem:

FV = P * [(1 + r/n)^(n*t) - 1]/(r/n)

Where:

FV = future value (desired amount) = $500,000

P = the amount invested today

r = annual interest rate = 7.8%

n = number of compounding periods per year = 2 (half-yearly)

t = number of years = 30

Substituting the values:

$500,000 = P * [(1 + 0.078/2)^(2*30) - 1]/(0.078/2)

Solving for P using a financial calculator or an online TVM calculator, we get:

P = $50,354.37

Therefore, Monique needs to invest $50,354.37 today to have $500,000 in 30 years at an annual interest rate of 7.8% compounded half-yearly.

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The relationship "d varies directly with the square of w" can be expressed mathematically as: d = k w² where k is the constant of proportionality.

An equation is a mathematical statement that shows the equality of two expressions, usually with an equal sign "=" in between them. An equation can contain variables, constants, and operators. The variables are represented by letters and can take on different values, while constants are fixed values that do not change. Operators include mathematical symbols like plus, minus, multiplication, and division, as well as exponents and logarithms.

Here,

This equation states that d is directly proportional to the square of w, meaning that if we double w, d will be four times larger (2² = 4). The constant of proportionality, k, depends on the specific situation and values of d and w. It can be determined by measuring d and w for a given situation, and then solving for k using the equation above.

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Based on the information in the graph, we can infer that the tax that Melvin and Sylvia must pay if they pay federal income taxes when filling it out together would be 11,631.

To identify the value of the federal income taxes we must take into account how much money the income of the two of them is. In this case, on the left side, they indicate that they are $77,000.

So, once we know the income we must look at the right side of the graph. In this case we must look for the value that matches the row that says "Married filling jointly" and the value of income (77,000). According to the above, these values coincide in the box that says 11,631.

From the above, we can infer that the tax they must pay would be 11,631

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The prοbability that the mean gain fοr the sample was between 400 and 800 is apprοximately 0.603.

The distributiοn οf sample means apprοaches a nοrmal distributiοn with mean equal tο the pοpulatiοn mean and standard deviatiοn equal tο the pοpulatiοn standard deviatiοn divided by the square rοοt οf the sample size.

standard errοr οf the mean = standard deviatiοn / square rοοt οf sample size

[tex]s = 1539 / \sqrt{(34)}\\\\s = 263.8[/tex]

we can standardize the sample mean using the fοrmula:

z = (sample mean - pοpulatiοn mean) / standard errοr οf the mean

z1 = (400 - 651) / 263.8 ≈ -0.953

z2 = (800 - 651) / 263.8 ≈ 0.564

P(-0.953 < z < 0.564) ≈ 0.603

Therefοre, the prοbability that the mean gain fοr the sample was between 400 and 800 is apprοximately 0.603.

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Based on the information provided, the value of x in this triangle will be 0.9646 if solved by this letter.

In this case, we are required to calculate the base of the triangle by using the angle provided 16° and the length of the hypotenuse which is 3.5. Based on this, a possible equation considering the law of the sinuses would be:

Sin 16° = x/3.5

Now using this equation, let's solve it by x to find the x value:

x = Sin 16° × 3.5

x = 0.2756 × 3.5

x = 0.9646

This means that the value of x is 0.9646

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The answers are:

3 a). 02:25 is time when aircraft arrives

b). The lower bound of the distance is 3150 km

4). The bottle weight is 200 g

3 a). Aircraft leaves at 22:35

As per the given,

From 22:35 and takes 3h 50min is 02:25 will time when aircraft arrive

b). Because the distance is corrected to the nearest 100 km

The lower bound of the distance is = 3200 - 100/2

                                                           = 3150 km

4). Assume that the container weighs "x" and the honey weighs "y."

We are aware that a 1000 g container of honey weighs in total.

So Equation 1 is x + y = 1000 g

We can calculate the overall weight to be 600g by using half of the honey.

Equation 2: x + (y/2) = 600 g

y/2=600 - x

y= 2 (600 - x)

y= 1200 - 2x

We can replace “y” with 1200 - 2x in Equation 1:

x + 1200 - 2x = 1000

x = 200 g

The container weighs 200 grammes.

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Hence, in response to the provided question, we can say that We discard the t = 0.003 solution since time cannot be negative. As a result, the discus takes around 2.375 seconds to strike the ground.

Mathematicians research numbers and their variants, equation and related structures, objects and their locations, and prospective locations for these things. The term "module" is used to describe the connection that exists in between set of inputs, each of which has a corresponding output. A function is an input-output connection in which each inputs results in something like a single, distinct return. A domain, codomain, or scope is assigned to each function. Functions are usually denoted by the letter f. (x). An x is used for entry. On capabilities, one-to-one capabilities, multiple prowess, in capabilities, and on capabilities are the four major types of accessible functions.

The specified function appears to have a typo. I believe it should be:

[tex]h = -16t^2 + 38t + 5[/tex]

a) The discus's initial height can be calculated by entering t = 0 into the function:

[tex]h = -16(0)^2 + 38(0) + 5 = 5[/tex]

As a result, the discus's starting height is 5 feet.

[tex]-16t^2 + 38t + 5 = 0[/tex]

[tex]t = (-b \sqrt(b2 - 4ac)) /[/tex]

t = 0.003 or 2.375

We discard the t = 0.003 solution since time cannot be negative. As a result, the discus takes around 2.375 seconds to strike the ground.

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Answer:

Without knowing the number of pieces that the pizzas were cut into, it is not possible to determine the fraction of a whole pizza that is left.

Step-by-step explanation:

In linear equation, $38.25 are each coin are there .

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.

                                 Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.

 A jar of dimes and quarters  = $2.55

 There are 15 coins in all.

  total coins = $2.55 * 15

                    = $38.25

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The final amount after 5 years is $ 49521.98.  

Compound interest refers to the interest that is earned on the initial amount of money deposited or borrowed and also on the accumulated interest from previous periods.  

The formula for calculating compound interest is:

Where

Where:

A = Final amount

P = Principal amount

r = Annual interest rate

n = Number of times the interest is compounded per year

t = Time period

Here we have

The deposited amount, P= $200.00  

Rate of interest, r = 9% = 9/100 = 0.09%

Number of compounds per year, n = 1

Time period, t = 5 years

Using the formula,

A = 2000(1 + 0.09/1)⁽¹⁽⁵⁾⁾

A = 2000(1.09)⁵

A = 49521.98  

Therefore,

The final amount after 5 years is $ 49521.98.  

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Answer:

Step-by-step explanation:

We can use SOH, CAH, TOA to solve this.

sin(55) = 23/x

x = 23/sin(55)

x = 28.08 cm

1. The rational expression representing the charge each attendee will have to pay is 300 + 55x.

2. Each attendee will have to pay $65 in order to cover the total cost.

The solution has been obtained by using arithmetic operations.

What are arithmetic operations?

All real numbers are thought to be adequately explained by the four basic operations, sometimes known as "arithmetic operations." The mathematical operations quotient, product, sum, and difference come after division, multiplication, addition, and subtraction.

1. We are given that it costs $300 plus $55 per person attending to host an event.

Let the number of attendees be x.

We get the expression as 300 + 55x.

2. We are given that 30 people are attending the event.

So, substituting x = 30 in the expression, we get

⇒ Total cost = 300 + (55 * 30)

⇒ Total cost = 300 + 1650

⇒ Total cost = $1950

Now, using the division operation, we get

⇒ Per person charge = 1950 ÷ 30

⇒ Per person charge = $65

Hence, the required solutions have been obtained.

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Question: To host an event at a popular convention center, it costs $300 plus $55 per person attending.

1. Write a rational expression that represents how much you would need to charge each attendee in order to cover the cost of hosting the event. Let x be the number of attendees.

2. In the previous step you determined that for x number of people, the total cost for hosting the event is 300+55x. Now, adjust the expression to represent what each attendee would need to be charged to cover the total cost if 30 people were attending.

P(inside larger square and outside smaller) = [tex]\dfrac{16}{100}[/tex]

Probability is the result of the division of the number of possible outcome by the number of an event.

In the question, for a point chosen, the point can be in the small square only or in the area or region between the small square and the big square as such,

Area of larger square = area of region between both squares + area of smaller square

Where the area of a square is S × S where S is the side of a square

Area of larger square = 5 × 5

                                   = 25 cm square

Area of smaller square = 3 × 3

                                     = 9 cm square

Area of the region between  both squares

                                     = 25 - 9

                                     = 16 cm square

The probability that a dot selected is inside the larger square and outside the smaller is

P(inside larger square and outside smaller) = Area of region between both square/ Area of larger square

P(inside larger square and outside smaller) = [tex]\dfrac{16}{100}[/tex]

The cuboidal prism has a volume of 40.625 cubic centimeters

Multiplying the cuboidal prism's length, width, and height yields its volume. Assuming that the width b and height h are both 2.5 cm and the length l is 6.5 cm. The volume of the cuboidal prism is given by multiplying these dimensions:

V = l b h = 6.5cm x 2.5cm x 2.5cm

 = [tex]40.625cm^3.[/tex]

As a result, the cuboidal prism has a volume of 40.625 cubic centimeters. As a result, the prism can be utilized for a variety of purposes, including packaging, storage, and transportation, and it can accommodate 40.625 cubic centimeters of material inside its boundaries.

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Therefore, the range of the function is all real numbers except for 0. In interval notation, this can be written as (-∞, 0) U (0, ∞).

In mathematics, a function is a relation between two sets, where each element in the first set (called the domain) is associated with exactly one element in the second set (called the range). Formally, a function f from a set A to a set B is a rule that assigns to each element a in A a unique element b in B, denoted by f(a). The set A is called the domain of the function, and the set B is called the range.

Here,

The given function is y = x/(8-x).

To find the range of this function, we need to determine the possible values of y for all possible values of x.

First, we note that the function is undefined when the denominator (8-x) equals zero. Therefore, x cannot be equal to 8.

Next, we consider the behavior of the function as x approaches positive infinity and negative infinity.

As x approaches positive infinity, the value of y approaches zero. This is because the x term in the numerator grows much more slowly than the 8-x term in the denominator, causing the overall value of the function to approach zero.

As x approaches negative infinity, the value of y approaches negative infinity. This is because the x term in the numerator becomes very negative while the 8-x term in the denominator approaches 8, causing the overall value of the function to become very negative.

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Answer:

y = x + 6

Step-by-step explanation:

We can put the equation of the line in slope intercept form [tex]y=mx+b[/tex] where m is the slope and b is the y intercept.

The line passes through (0,6) which means that is our y intercept. Now we have to count the slope. To do this, we can either evaluate the slope given 2 points or count it if given a graph. We have a graph, so the convenient thing to do is to count it. If we count, we see that our slope is [tex]\frac{1}{1}[/tex] or simply 1. All we have to do now is write it in slope intercept form.

[tex]y=x+6[/tex]

After answering the presented question, we can conclude that Yes, equation formal rules is a characteristic that fits with the mechanistic factor in organizational design.

In mathematics, an equation is a proposition that states the equivalence of two expressions. An equation consists of two sides separated by a system of equations (=). For instance, the statement "2x + 3 = 9" states that the word "2x Plus 3" equals the integer "9". The goal of solving equations is to find the value or amounts of the variable in the model) that will permit the calculation to be accurate. Mathematics can be simple or complex, linear or nonlinear, and contain one or more parts. For example, in the equation "x2 + 2x - 3 = 0," the variable x is raised to the power of 2. Lines are used in many areas of mathematics, including algebra, arithmetic, and geometry.

Yes, formal rules is a characteristic that fits with the mechanistic factor in organizational design.

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Answer:

y - 1 = 2(x + 2)

Step-by-step explanation:

Point-slope form is y - y₁ = m(x - x₁)

y - 1 = 2(x - -2)

=> y - 1 = 2(x + 2)

Using the equation, A = 6,000(1+0.029t) The number of years that is the investment period is: D 8 years.

Investment period refers to the length of time during which an individual or organization invests money in a particular investment vehicle or asset. It can also be referred to as the holding period, and it is the duration for which an investment is held or maintained before it is sold or liquidated.

We can start by setting the equation for the final account balance, A, to $7,392:

7,392 = 6,000(1 + 0.029t)

Divide both sides by 6,000:

1.232 = 1 + 0.029t

Subtract 1 from both sides:

0.232 = 0.029t

Divide both sides by 0.029:

t ≈ 8

Therefore, the investment period is approximately 8 years.

So the answer is option D) 8 years.

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Expenses wοuld be same and in bοth cases there expenses wοuld exceed, but in career 1 the οver expense wοuld be $500 while in career 2 the οver expense wοuld be $1000.

Tο determine whether a cοntract is lucrative οr nοt, the phrases prοfit and lοss are utilised. These wοrds are frequently used in οrdinary cοnversatiοn. If the selling price exceeds the cοst price, the difference between the twο amοunts is knοwn as the prοfit. The difference between the cοst price and the selling price, if the selling price is less than the cοst price, is referred tο as a lοss. A prοduct's cοst price is the cοst at which it is purchased. The selling price οf a thing is the cοst at which it is sοld. In this essay, let's study mοre abοut prοfit and lοss.

Expenses                                        Career Choice 1         Career Choice 2

Monthly income                              $10,000                       $20,000              

Rent: 30%*                                       $3,000                        $6,000

Utilities: 10%*                                    $1,000                        $2,000

Car Insurance: 5%*                          $500                           $1,000

Cell Phone: 10%*                              $1,000                        $2,000

Occasional Spending: 10%*            $1,000                        $2,000

Savings: 10%*                                   $1,000                        $2,000

Food: 15%*                                       $1,500                        $3,000

Car Loan: 10%                                  $1,000                        $2,000

Entertainment: 5%                           $500                          $1,000

Total:                                               $10,500                      $21,000              

Thus, after comparing both the careers, we have come to find that expenses would be same and in both cases there expenses would exceed, but in career 1 the over expense would be $500 while in career 2 the over expense would be $1000.

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Answer:

4 Outcomes.

Step-by-step explanation:

Both land tails.

Both land heads.

First coin lands Tails while the Second lands Heads.

First lands Heads while the Second lands Tails.

The area of the carpet for a room with length 7.5 m and breadth 3 cm less than the length is 562.275m².

The area refers to the two-dimensional space of occupied by an object.

The area is computed as the number of unit squares that cover the surface of a closed figure or object.

We measure area in square units like cm² and m².

The area is the squared product of the length and breadth or width.

1m = 100cm

Length = 7.5m

Breadth = 7.5m - 3 cm

= 7.5m - 0.03m

=74.97m

Area = Length x Breadth

= 7.5 x 74.97

= 562.275m²

Thus, the carpet should be 562.275m² in area.

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Answer:

Step-by-step explanation:

Descriptive as the information provided is factual and there is no inferences made using the data provided.

Answer: 275,400

Step-by-step explanation:

- First we do 18 multiplied by 20 which would get us the answer of 360

- Next we do 45 multiplied by 17 which would get us the answer of 765

- Now we multiply both 360 and 765 and get the answer of 275,400

The statement "Given any two numbers x, y, there is a third number w between them" can be expressed using symbolic logic and quantifiers as follows:

∀x∀y∃w (w is between x and y)

Here, the universal quantifier ∀x∀y indicates that the statement is true for any two numbers x and y. The existential quantifier ∃w indicates that there exists a third number w that is between x and y.

the examination of the links between statements that are used to express precise mathematical concepts. Formal logic is another name for symbolic logic.

Given any two digits x, y, there exists a third integer w between them. This proposition can be expressed in symbolic logic as follows:

∀x∀y(x<y → ∃w(x<w<y))

Where ∀x means "for all x", ∀y means "for all y", → means "implies", ∃w means "there exists a w", and (x<y → ∃w(x<w<y)) means "if x is less than y, then there exists a w such that w is between x and y".

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The statement can be expressed using symbοlic lοgic and quantifiers as in ∀x∀y∃w (w is between x and y)

The examinatiοn οf the links between statements that are used tο express precise mathematical cοncepts. Fοrmal lοgic is anοther name fοr symbοlic lοgic.

Given any twο digits x, y, there exists a third integer w between them. This prοpοsitiοn can be expressed in symbοlic lοgic as fοllοws:

∀x∀y(x<y → ∃w(x<w<y))

Where ∀x means "fοr all x", ∀y means "fοr all y", → means "implies", ∃w means "there exists a w", and (x<y → ∃w(x<w<y)) means "if x is less than y, then there exists a w such that w is between x and y".

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Answer:

The equation x = 5 represents a vertical line passing through the point (5, 0) on the x-axis. A vertical line has an undefined slope because the rise (change in y) is undefined when the run (change in x) is zero. Therefore, the slope of the line x = 5 is undefined.