(a) find the critical numbers of (if any), (b)
find the open interval(s) on which the function is increasing or
decreasing, (c) apply the First Derivative Test to identify all
relative extrema, and (d) use a graphing utility to confirm your
results

After answering the presented question, we can conclude that As a equation result, for any x and y values, these expressions will be identical to 24x + 36y.

Expressions can be used to represent numerical or algebraic quantities, and can be used to perform calculations. For example, the expression "2 + 3" represents the sum of the numbers 2 and 3, which is 5. Similarly, the expression "x + 5" represents the sum of the variable x and the number 5, and can be evaluated to a specific value depending on the value of x.

[tex]12(2x-y)[/tex]

[tex]-6(4x-3y)[/tex]

[tex]3(x+12y)[/tex][tex]12(2x-y)=24x-12y[/tex]

So,[tex]24x-12y+24=24x+36y-6(4x-3y)=-24x+18y[/tex]

So,[tex]-24x+18y+42x+18y=24x+36y[/tex]

[tex]3(x+12y)=3x+36y[/tex]

So,[tex]3x+36y+21x-24y=24x+36y[/tex]

As a result, for any x and y values, these expressions will be identical to 24x + 36y.

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After answering the presented question, we can conclude that variable

a) equation to help Sofia figure out how much money was charged in fees is 365 = 59 × 5 + f

b) the amount of money charged for fees is $70.

A variable is something that can be altered in the context of a mathematical notion or experiment. Variables are frequently denoted by a single symbol. The characters x, y, and z are frequently used as generic variables symbols. Variables are qualities with a wide range of values that can be investigated. Size, age, money, where you were born, academic position, and type of housing are just a few examples. Variables can be classified into two major groups using both numerical and categorical methods.

a) Total cost = Cost per night × Number of nights + Fees

365 = 59 × 5 + f

b) Solving for f, we get:

f = 365 - 295

f = 70

Therefore, the amount of money charged for fees is $70.

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Answer:

To find the slope of a line parallel to 3x - y = -1, we need to first rearrange this equation into slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Starting with 3x - y = -1, we can add y to both sides to get: 3x = y - 1 Then, we can add 1 to both sides to get: 3x + 1 = y This is now in slope-intercept form, with y = 3x + 1, so we can see that the slope of the line is 3. Since we want to find the slope of a line parallel to this one, we know that it must have the same slope of 3. Therefore, the answer is: Slope = 3.

The domain of F(d) is: {d ∈ R | d ≥ 0}

The domain is a mathematical concept that refers to the set of all possible input values (independent variables) for which a function is defined. In other words, the domain of a function is the range of values that can be used as input to the function.

Here we have

A large passenger airplane is making the 9,537-mile trip from New York to Singapore. During this flight, the plane will use 47,685 gallons of fuel or 5 gallons per mile of flight.

The function F(d) represents the amount of fuel, in gallons, consumed by the airplane on this trip after d miles of flight.

Therefore, we can express F(d) as:

F(d) = 5d

The domain of F(d) represents the set of all valid values of d for which the function is defined.

In this case, d represents the distance flown by airplane, and it must be a non-negative number since the plane cannot fly a negative distance.

Therefore,

The domain of F(d) is: {d ∈ R | d ≥ 0}

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Answer: We can simplify the expression as follows:

sin theta / (1 + sin theta) - sin theta / (1 - sin theta)

= [(sin theta)(1 - sin theta) - (sin theta)(1 + sin theta)] / [(1 + sin theta)(1 - sin theta)]

= [(sin theta - sin^2 theta) - (sin theta + sin^2 theta)] / [1 - sin^2 theta]

= -2 sin theta / cos^2 theta

= -2 tan theta

So the simplified expression is -2 tan theta, with no quotients.

Step-by-step explanation:

As a result, the range of the population's mean height for male Swedes is between 70.331 and 73.669 inches.

The range of a function or relation in mathematics is the collection of all feasible output values (also known as the dependent variable). It is the collection of all possible values that such a function can have. For instance, since its function can output anything non-negative real integer, f(x) = x2 would have a range of all non-negative real numbers. The range is frequently stated as a collection of numbers or as a circle, and it may be either finite or infinite.

given

For the population mean height of male Swedes, the following formula provides a 95% confidence interval:

where n is the sample size, x is the sample mean, y is the standard deviation of height, n* is the z-score corresponding to a 95% confidence level, which is 1.96, and z* is the z-score.

By entering the specified values, we obtain:

CI = 72 ± 1.96 * (3/√48)

CI = (70.331, 73.669) (70.331, 73.669)

As a result, the range of the population's mean height for male Swedes is between 70.331 and 73.669 inches.

By multiplying the critical value (1.96), by the standard error, one may determine the error bound:

EBM = 1.96 * (3/√48) ≈ 1.382

The error bound is therefore roughly 1.382 inches.

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The graph that shows the solution of the inequality 7 ≥ n + 5 is the graph of option D.

What is the solution to the inequality 7 ≥ n + 5?

The solution to the inequality 7 ≥ n + 5 is as follows:

7 ≥ n + 5

7 - 5 ≥ n

2 ≥ n

n ≤ 2

2, Let p be the number of pages Eduardo wrote before today. Then, the inequality that represents the situation is:

p + 16 > 50

To solve for p, first subtract 16 from both sides of the inequality:

p + 16 - 16 > 50 - 16

p > 34

Therefore, Eduardo wrote at least 34 pages before today.

Let r be the number of rows of seeds the farmer needs to plant. Then, the equation that represents the situation is:

6.5r ≥ 52

divide both sides of the equation by 6.5:

r ≥ 8

Therefore, the farmer needs to plant at least 8 rows of seeds to harvest at least 52 bushels of tomatoes.

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Answer:

Leonhard Euler was a Swiss mathematician who lived from 1707-1783. He made significant contributions to many areas of mathematics, including number theory, calculus, and geometry.

One of Euler's significant contributions to the field of sequences and series is his work on the summation of infinite series. In particular, he developed a method for finding the sum of an infinite geometric series, which has the form:

a + ar + ar^2 + ar^3 + ...

where a is the first term, r is the common ratio, and the series continues infinitely.

Euler's method for finding the sum of this series is as follows:

If |r| < 1, then the sum of the series is:

a / (1 - r)

For example, let's consider the infinite geometric series:

2 + 4 + 8 + 16 + ...

In this series, a = 2 and r = 2. Since |r| < 1, we can use Euler's formula to find the sum:

sum = a / (1 - r)

= 2 / (1 - 2)

= -2

Therefore, the sum of the infinite geometric series 2 + 4 + 8 + 16 + ... is -2.

Euler's formula is just one example of his many contributions to mathematics. His work on sequences and series laid the foundation for many important concepts in calculus, and his ideas continue to influence the field of mathematics today.

P(stop at P|stop at Q) = P(stop at P and stop at Q) / P(stop at Q) is 28/47

How to solve questions?

(a) Tree diagram:

                 | P stop 7/20

                 |

          -------|-------

         |              |

   Q stop|  P stop 2/5  | P not stop 3/5

         |              |

    ------|-------      |    

         |              |

  Q not stop| P stop 1/10 | P not stop 9/10

         |              |

          -------|-------

                 |

                 | P not stop 13/20

(b) The probability that he has to stop at both P and Q is:

P(stop at P) * P(stop at Q|stop at P) = (7/20) * (2/5) = 7/50

(c) The probability that he has to stop at least once is:

P(stop at P and/or stop at Q) = 1 - P(not stop at P) * P(not stop at Q|not stop at P)

                             = 1 - (13/20) * (9/10) = 11/20

(d) If he has to stop at Q, the probability that he would have stopped at P is:

P(stop at P|stop at Q) = P(stop at P and stop at Q) / P(stop at Q)

                       = (7/50) / (13/20 * 2/5 + 7/50)

                       =28/47

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Answer:

she does not have enough!

Step-by-step explanation:

thirty percent of $28 is $8.4, so you would subtract the two leaving you with $19.6

2.5 sales tax of $19.6 is $0.49

Adding those two the total comes to $20.09 which is just over the budget!

Answer:  She does not have enough money.

She needs 9 cents (aka $0.09)

========================================================

Explanation:

A discount of 30% means Mary would pay the remaining 70% (because 100-30 = 70)

The decimal form of 70% is 0.70

Let A = 0.70 since we'll use it later.

An increase of 2.5% means we will also have the multiplier 1.025; which you can think of it like saying 100% + 2.5% = 1 + 0.025 = 1.025

Let B = 1.025 since we'll use it later

Multiply the values of A and B to get: 0.70*1.025 = 0.7175

This is the net multiplier when combining the 30% discount and 2.5% tax.

The original price $28 then becomes 0.7175*28 = 20.09 which is 9 cents (aka $0.09) over the goal of $20

Therefore, Mary does not have enough money to buy the book. She'll need that remaining 9 cents.

Answer:

a = b = 3√2

Step-by-step explanation:

Use trigonometry:

[tex] \sin(45°) = \frac{a}{6} [/tex]

Cross-multiply to find a:

[tex]a = 6 \times \sin(45°) = 6 \times \frac{ \sqrt{2} }{2} = \frac{6 \sqrt{2} }{2} = 3 \sqrt{2} [/tex]

Use the Pythagorean theorem to find b:

[tex] {b}^{2} = {6}^{2} - {a}^{2} [/tex]

[tex] {b}^{2} = {6}^{2} - ( {3 \sqrt{2}) }^{2} = 36 - 9 \times 2 = 36 - 18 = 18[/tex]

[tex]b > 0[/tex]

[tex]b = \sqrt{18} = 3 \sqrt{2} [/tex]

Answer:3x4x5 = 60
but how r angles Cm did u mean sides or lxhxw

Step-by-step explanation:

Answer:

It's not possible to determine the area of a triangle based on the given information about the angles.

To find the area of a triangle, we need to know at least one side length in addition to the angles. The formula for the area of a triangle is:

Area = (1/2) * base * height

where the base is any side of the triangle, and the height is the perpendicular distance from that side to the opposite vertex.

Without knowing any side lengths, we cannot calculate the height and therefore cannot find the area of the triangle.

To find the greatest possible age of the skeleton, we need to add the error to the estimate:

60 centuries + 0.8 centuries = 60.8 centuries

Therefore, the greatest possible age of the skeleton is 60.8 centuries.

Answer: To find the margin of error and construct a 90% confidence interval for the population mean, we can use the following formula:

Margin of error = t-value x (standard deviation / square root of sample size)

where t-value is the value obtained from the t-distribution table with n-1 degrees of freedom and a confidence level of 90%.

For a sample size of 8 and a confidence level of 90%, the degrees of freedom is 7 and the t-value is 1.895 (obtained from a t-distribution table).

Using the given values, we can calculate the margin of error as:

Margin of error = 1.895 x (7.9 / sqrt(8)) ≈ 5.69 (rounded to one decimal place)

So the margin of error is approximately 5.69 miles.

To construct a 90% confidence interval for the population mean, we can use the formula:

Confidence interval = sample mean ± margin of error

Substituting the values, we get:

Confidence interval = 18.5 ± 5.69

So the 90% confidence interval for the population mean is (12.81, 24.19).

Interpretation:

We are 90% confident that the true mean driving distance to work for the population lies between 12.81 and 24.19 miles. This means that if we were to take many samples of 8 people and calculate the confidence interval for each sample, 90% of those intervals would contain the true population mean. The margin of error is the amount by which the sample mean may differ from the true population mean.

Step-by-step explanation:

The measure of angle A is 59°

A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It is also sometimes called inscribed quadrilateral. The circle which consists of all the vertices of any polygon on its circumference is known as the circumcircle or circumscribed circle.

The sum of opposite angle in a cyclic quadrilateral is 180. This means, 121+<A = 180°

therefore,

= angle A = 180-121

= 59°

therefore the measure of angle A is 59°

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The rate of interest is 7.7 %  if £9000 has a final value of £13102 in 5 years

When interest is compounding, it indicates that the balance as a whole, rather than just the principal, is taken into account when the following interest period begins.

For instance, after a year, a $100 loan with 5% annual compound interest will have a debt of $105 due. The interest will be applied to the entire $105, creating a new amount of $110.25 the next year, rather than being taken as 5% of $100.

The interest will be added to that $110.25 the following year, and so on throughout the loan.

This is distinct from simple interest, which is a periodic payment to the loan holder of a fixed sum of money derived from a percentage of the principal

We have given that the final amount is 13102 and the principal amount is 9000 and time n is 5 years we have to find the rate

We have the formula

[tex]A=p(1-\frac{r}{100})^n[/tex]

A= 13102

P=9000

n=5

r = ?

[tex]13102=9000(1-\frac{r}{100})^5\\\\1.455=(1-\frac{r}{100})^5\\\\1.077=1-\frac{r}{100}\\\\r=7.7[/tex]

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Answer:  (6, -2)

Step-by-step explanation:

         First, we will graph these equations. See attached. One has a y-intercept of -5 and then moves 2 units right for every unit up (we get this from the slope of 1/2). The other has a y-intercept of 4, and moves right one unit for every unit down (we get this from the slope of -1).

         The point of intersection is the solution, this is the point at which both graphed lines cross each other. Our solution is:

                         (6, -2)     x = 6, y = -2

Rounding this to the nearest whole number, we can estimate that about 407 customers would pay with a credit card next week.

How to estimate number of customers?

To estimate the number of customers who would pay with a credit card next week, we can use the proportion of customers who paid with a credit card in the last week as a guide.

First, we need to find the total number of customers who visited Omar's business in the last week:

Total customers = Cash customers + Debit card customers + Credit card customers

Total customers = 91 + 8 + 27

Total customers = 126

Next, we can find the proportion of customers who paid with a credit card:

Proportion of credit card customers = Credit card customers / Total customers

Proportion of credit card customers = 27 / 126

Proportion of credit card customers = 0.2143

Now, we can estimate the number of customers who would pay with a credit card next week by multiplying the total number of expected customers by the proportion of credit card customers from the last week:

Expected credit card customers = Total expected customers x Proportion of credit card customers

Expected credit card customers = 1900 x 0.2143

Expected credit card customers = 407.17

Rounding this to the nearest whole number, we can estimate that about 407 customers would pay with a credit card next week.

However, it is important to note that this estimate is based on the assumption that the proportion of customers paying with a credit card remains the same next week. Factors such as changes in consumer behavior or market conditions may affect this proportion, and hence, the accuracy of the estimate.

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Answer:

Step-by-step explanation:

The solution to the set of equations can be found by solving the system:

3x + 3y = 12

x + y = 4

We can simplify the second equation by solving for y:

y = 4 - x

Substituting this expression for y into the first equation, we get:

The solution to the set of equations can be found by solving the system:

3x + 3y = 12

x + y = 4

We can simplify the second equation by solving for y:

y = 4 - x

Substituting this expression for y into the first equation, we get:

3x + 3(4 - x) = 12

3x + 12 - 3x = 12

12 = 12

This is a true statement, which means that the system is consistent and has infinitely many solutions. Therefore, the correct answer is:

There are infinitely many solutions.

Answer:

Step-by-step explanation:

To find:-

Answer:-

We are here given that there are two linear equations, namely,

[tex]\begin{cases} 3x+3y = 12 \\ x+y = 4 \end{cases}[/tex]

These can be rewritten as ,

[tex]\begin{cases} 3x+3y - 12 =0\\ x+y -4=0 \end{cases}[/tex]

Before we precede we must know that,

Conditions for solvability :-

If there are two linear equations namely,

[tex]\begin{cases} a_x + b_1y+c_1 = 0 \\ a_2x+b_2y+c_2=0\end{cases}[/tex]

Then ,

Case 1 :-

If we have,

[tex]\longrightarrow \boxed{ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2} } \\[/tex]

Then , the lines are coincident and there are infinitely many solutions .

Case 2 :-

If we have,

[tex]\longrightarrow \boxed{ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2} } \\[/tex]

Then, the linear equations are inconsistent and have no solutions , thus the lines are parallel .

[tex]\rule{200}2[/tex]

So here with respect to angle standard form of pair linear equations, we have;

Hence here we have,

[tex]\longrightarrow \dfrac{a_1}{a_2} = \dfrac{3}{1} \\[/tex]

[tex]\longrightarrow \dfrac{b_1}{b_2}=\dfrac{3}{1} \\[/tex]

[tex]\longrightarrow \dfrac{c_1}{c_2}=\dfrac{-12}{-4}=\dfrac{3}{1} \\[/tex]

Therefore we can clearly see that,

[tex]\longrightarrow \boxed{ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2} =\boxed{\dfrac{3}{1}}} \\[/tex]

Hence there are infinitely many solutions and the lines are coincident .

Answer:

$60

Step-by-step explanation:

2*25.50=51

25% of 16=4

16-4=12

12*2=24

51+24=75

20% of 75=15

75-15 =

60

Answer:

Step-by-step explanation:

I am assuming the triangles are similar.

You can make a proportion between the two triangles and solve for x but you must match the sides properly.

RS ≈ MN

ST ≈ NO

Making a proportion between the sides:

[tex]\frac{x}{3}[/tex] = [tex]\frac{6}{4}[/tex]  

cross multiplyand then divide

6 · 3 = 18

18 ÷ 4 = 4.5

x = 4.5

If you check the ratios - they are all the same.

For example  6 ÷ 4 = 1.5

7.5 ÷ 5 = 1.5

4.5 ÷ 3 = 1.5

this checks out

So, the corresponding point for g(x) on the graph of f(x) is (-3/2, F (-3/2)).

In mathematics, a function is a relation between a set of inputs (the function's domain) and a set of possible outputs (the function's range) with the property that each input is associated with exactly one output.

Functions are often described using a formula or an equation, such as [tex]f(x) = x^2,[/tex] where "f" is the name of the function, "x" is the input variable, and "[tex]x^2[/tex]" is the output. The value of the function for a specific input value of "x" can be found by substituting that value into the equation.

Functions are used in many areas of mathematics, science, engineering, and economics to model real-world phenomena, analyze data, and solve problems. They are also important in computer science, where they are used to write algorithms and design software.

if we know the point (x, y) on the graph of a function f(x), we can find the corresponding point for the function g(x) = F(-1/2x) by substituting -1/2x for x in the expression for f(x) and simplifying. That is, we can find the point (u, v) on the graph of g(x) such that u = -1/2x and v = F(-1/2x), where (x, y) is a point on the graph of f(x).

For example, if f(x) = x^2 and the point (3, 9) is on the graph of f(x), then we can find the corresponding point for g(x) = F(-1/2x) by substituting -1/2x for x in the expression for f(x) and simplifying:

[tex]u = -1/2x = -1/2(3) = -3/2\\v = F(-1/2x) = F(-1/2(3)) = F(-3/2)[/tex]

So, the corresponding point for g(x) on the graph of f(x) is (-3/2, F(-3/2)). However, without knowing the specific definition of fix,

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Answer:

[tex](x+1)^{2} +(y+2)^{2} =20[/tex]

Step-by-step explanation:

Using the mid point formula to find the center of the circle:

midpoint = [tex](\frac{x_{1} +x_{2} }{2} ,\frac{y_{1} +y_{2} }{2} )[/tex]

Midpoint = [tex](\frac{3+-5}{2} ,\frac{0+-4}{2} )[/tex]

Midpoint = [tex](-1,-2)[/tex]

The midpoint is the same as the centre of the circle

Find the distance(the diameter of the circle) between those two points to find the radius:

Distance formula = [tex]\sqrt{(y_{2}-y_{1} )^{2} +(x_{2} -x_{1} ) ^{2} }[/tex]

Distance formula = [tex]\sqrt{(-4-0)^{2} +(-5-3)^{2} }[/tex]

Distance formula = [tex]\sqrt{16+64}[/tex]

Distance formula = [tex]\sqrt{80}[/tex]

So,the diameter is [tex]\sqrt{80}[/tex] and to find the radius we need to divide the diameter by two

Radius = [tex]\frac{\sqrt{80} }{2}[/tex]

Radius = [tex]2\sqrt{5}[/tex]

the equation of circle:

Radius = [tex]2\sqrt{5}[/tex]

Center = (-1,-2)

[tex](x-h)^{2} +(y-k)^{2} =r^{2}[/tex]

[tex](x- -1)^{2} +(y- - 2)^{2} =(2\sqrt{5} )^{2}[/tex]

[tex](x+1)^{2} +(y+2)^{2} =20[/tex]

By answering the presented question, we may conclude that Therefore, equation the two numbers are either 3 and 4, or 4 and 3.

An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the phrase "2x Plus 3" equals the number "9." The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. The variable x is raised to the second power in the equation "x2 + 2x - 3 = 0." Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.

the system of equations:

[tex]x + y = 7\\x * y = 12 \\y = 7 - x\\x * (7 - x) = 12\\7x - x^2 = 12\\x^2 - 7x + 12 = 0\\(x - 3)(x - 4) = 0\\If x = 3, then y = 4\\If x = 4, then y = 3\\[/tex]

Therefore, the two numbers are either 3 and 4, or 4 and 3.

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0.2525 is the probability that a simple random sample.

What is the simple definition of probability?

A probability is a number that expresses the possibility or likelihood that a specific event will take place. Probabilities can be stated as proportions with a range of 0 to 1, or as percentages with a range of 0% to 100%. Probability is a measure of how likely or likely-possible something is to happen.

Since the distribution of the life expectancies of a certain protozoan is normal, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = life expectancies of the certain protozoan.

µ = mean

σ = standard deviation

n = number of samples

From the information given,

µ = 46 days

σ = 10.5 days

n = 49

The probability that a simple random sample of 49 protozoa will have a mean life expectancy of 47 or more days is expressed as

P(x ≥ 47) = 1 - P(x < 47)

For x = 47

z = (47 - 46)/(10.5/√49) = 0.67

Looking at the normal distribution table, the probability corresponding to the z score is 0.0.75

P(x ≥ 47) = 1 - 0.75 = 0.25

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The height of the poster storage tube is approximately 40 inches.

Vοlume is the amοunt οf space that a three-dimensiοnal οbject οccupies. It is measured in cubic units such as cubic meters, cubic feet, οr cubic inches. The fοrmula fοr calculating the vοlume οf a sοlid οbject depends οn its shape. Fοr example, the vοlume οf a cube can be calculated using the fοrmula V = s³, where s is the length οf a side, while the vοlume οf a cylinder can be calculated using the fοrmula V = πr²h, where r is the radius οf the base and h is the height.

The formula for the volume of a cylinder is V = πr²h, where r is the radius of the base and h is the height of the cylinder.

Since the diameter of the cylinder is given as 3.5 inches, the radius is half of that or 1.75 inches. Also, the volume is given as 122.5π cubic inches.

Therefore, we can use the formula to solve for the height:

V = π[tex]r^{2}[/tex]h

122.5π = π(1.75)(1.75)h

122.5 = 3.0625h

h = 122.5/3.0625

h ≈ 40

Therefore, the height of the poster storage tube is approximately 40 inches.

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the actual area of the park is 19,200 square feet.

Marcus drew a scale drawing of the rectangular park in his neighborhood. He drew the length of the park as 8 inches and the width of the park as 6 inches. The key on his drawing shows 1 inch=20 feet. To find the actual area of the park, we need to convert the measurements in Marcus's drawing to the actual measurements in feet.

Since 1 inch on Marcus's drawing represents 20 feet in actuality, we can use a scale factor of 1 inch : 20 feet to convert the measurements. We can then multiply the actual length and width of the park in feet to find the actual area of the park in square feet.

To convert the length of 8 inches to feet, we multiply it by the scale factor:

8 inches × (20 feet/1 inch) = 160 feet

Similarly, we can convert the width of 6 inches to feet:

6 inches × (20 feet/1 inch) = 120 feet

So the actual length of the park is 160 feet and the actual width of the park is 120 feet.

To find the actual area of the park, we multiply the actual length by the actual width:

160 feet × 120 feet = 19,200 square feet

Therefore, the actual area of the park is 19,200 square feet.

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Answer: 165 degrees

Step-by-step explanation:

Let's start off and name the angle between the hour and minute hand x, representing the number of degrees between the hands. Next, let's look at the minute hand first. After exactly an hour, the minute hand does not change its position(since after one hour, it comes back to where it was previously). As for the hour hand however, its position differs by exactly "1 hour tick", which is equivalent to 30 degrees (360 / 12 hours). What that implies is that x would have to equal x + 30, which is clearly impossible. However, we can think about the problem in a different way: by looking at the acute angle only between the hands, that would indeed be possible. Assuming that x + 30 is greater than 180(since if it wasn't, that wouldn't be possible), we can write the following equation:

x = 360 - (x+30)

, with the right hand side coming from the fact that the obtuse angle of x + 30 must have an acute counterpart, of which both add up to 360.

Then, we can simplify to:

x = 330 - x

or:

2x = 330

x = 165.

This gives us our only answer of 165 degrees