2. Pequot Lakes, Minnesota has a spherical water tower that is known as “the world’s largest fishing bobber.” The water tower holds 6,684 cubic feet of water. The water tower needs to be repainted, and it will have to be painted twice to get two coats of coverage. One gallon of paint can cover 350 square feet. How many gallons of paint will be needed to paint the water tower?

Answer:

1 - [tex]e^{inx}[/tex]

2 - [tex]e^{5in}[/tex]

3 - [tex]e^{in^{3} x}[/tex]

Step-by-step explanation:

The new arc length function for the curve after reparametrization is as follows

[tex]\\\\r(s) = ( 4 - \frac{s}{\sqrt{26}})i + (\frac{4s}{\sqrt{26} } + 4)j+(\frac{3s}{\sqrt{26} } + 3)k[/tex]

What is meant by the arc length of a curve?

The distance between two places along a segment of a curve is known as the arc length.

The formula for arc length is used to determine how far something is along the curved line that makes up an arc. The arc length is, to put it simply, the distance that passes through the curved line of the circle that forms the arc.

Given,

Vector equation of curve  r(t) = (5 - t) i + (4t - 3) j + 3t k

    Given P = (4, 1, 3)

5 - t = 4

So, t = 1

Now differentiate r(t)

 r'(t) = -i + 4j + 3k

Magnitude of r'(t) = |r'(t)| = √[(-1)² + 4² + 3²] = √26

The arc length function for the curve

 [tex]s = \int\limits^t_1 { |r'(t)| } \, dt = \int\limits^t_1 { \sqrt{26} } \, dt = \sqrt{26} (t-1)[/tex]

  [tex]t = \frac{s}{\sqrt{26} } + 1[/tex]

Now to reparametrize the curve with respect to arc length, substitute t with (s/√26) + 1 in r(t).

[tex]r(s) = (5 - ( \frac{s}{\sqrt{26} } + 1)) i + (4( \frac{s}{\sqrt{26} } + 1) - 3) j + 3( \frac{s}{\sqrt{26} } + 1) k\\\\r(s) = ( 4 - \frac{s}{\sqrt{26}})i + (\frac{4s}{\sqrt{26} } + 4)j+(\frac{3s}{\sqrt{26} } + 3)k[/tex]

Therefore the new arc length equation of the curve when  [tex]t = \frac{s}{\sqrt{26} } + 1[/tex] is as follows

[tex]\\\\r(s) = ( 4 - \frac{s}{\sqrt{26}})i + (\frac{4s}{\sqrt{26} } + 4)j+(\frac{3s}{\sqrt{26} } + 3)k[/tex].

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a) The domain of the function is 0 to 15 miles

b) The range of the function is 0 to 270 miles

The domain of a function is the set of values that we are allowed to plug into our function. This set is the x values in a function such as f(x). The range of a function is the set of values that the function assumes. This set is the values that the function shoots out after we plug an x value in.

The range is the set of outputs of a relation or function. In other words, it's the set of possible y values. Recall that ordered pairs are of the form (x,y) so the y coordinate is listed after the x. The output is listed after the input.

Given data ,

Let the mileage of the car = 18 miles per gallon

The amount of gasoline the car can hold = 15 gallons

Let the distance to empty be a function of mileage of the car × amount of gas in gallons

Now , the domain of a function is the set of values that we are allowed to plug into our function which ranges from 0 to 15

So , distance to empty = 18 x 15 = 270

So , the range of the function is the set of output values which ranges from 0 to 270

Hence , the domain of the function is 0 to 15 miles and the range of the function is 0 to 270 miles

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The original cost of the sweater that is on sale for a 20% discount will be $60.

The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100. The phrase % translates to one hundred percent. It is symbolized by the character '%'.

The percentage is given as,

Percentage (P) = [Final value - Initial value] / Initial value x 100

A sweater is on sale for a 20% discount. The amount of the discount is $12.

Let 'C' be the original cost. Then the equation is given as,

20% = ($12 / C) x 100

Simplify the equation, then we have

20% = ($12 / C) x 100

0.20 = ($12 / C)

C = $12 / 0.20

C = $60

The original cost of the sweater that is on sale for a 20% discount will be $60.

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I think that answer is yes

The number of ways in which a committee of four Democrats and three Republicans be formed from a group of ten Democrats and eleven Republicans is 34650 ways.

By choosing some items from a set and creating subsets, permutation and combination are two approaches to represent a group of objects. It outlines the numerous configurations for a particular set of data. Permutations are the selection of data or objects from a set, whereas combinations are the order in which they are represented. Both ideas are critical to mathematics.

Here, the differences between permutation and combination are described in detail.

The number of ways in which we can select a committee of 4 democrats from ten democrats is:

P = 10C 4 = 10! / (4)! (10 - 4)!

P = 210

The number of ways in which we can select a committee of 3 Republicans from 11 Republicans is:

P = 11C 3 = 11! / 3! (11 - 3)!

P = 165

The number of ways in which a committee of four Democrats and three Republicans be formed from a group of ten Democrats and eleven Republicans is:

P = (210) (165) = 34650

The number of ways in which a committee of four Democrats and three Republicans be formed from a group of ten Democrats and eleven Republicans is 34650 ways.

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The standard form of the polynomial function y = x(x + 5)(x - 3) is

y = x³ + 2x² - 15x

Function is a relation between a set of inputs and a set of outputs which are permissible. In a function, for particular values of x we will get only a single image in y. It is denoted by f(x).

Vertical line test:-

Whenever we want to check whether a given expression is a function or not we can apply a vertical line test, in this test we check for a single image of x , we are getting a single image or more.

If we get more images then it will not be a function.

For example, let us take, y² = 4ax

y = ±√4ax

For single value of x we get two values of y

Hence, it will not be a function.

Given that,

The polynomial function,

y = x(x+5)(x−3)

To obtain the standard form, we need to multiply out the expression and then arrange the terms in decreasing order of the degree of the monomials. The degree of a polynomial is the highest degree of its monomials.

x(x + 5)(x - 3) = x³ + (5x²) - (3x²) + (5x²) + (-15x) = x³ + 2x² - 15x

In the standard form, the coefficient of the monomial with the highest degree is 1, and the rest of the terms are arranged in decreasing order of the degree.

Thus, the standard form is x³ + 2x² - 15x

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About 0.04% percent of quokkas weigh over 15.5 pounds.

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Given that, the weights of quokkas can be described by the Normal model N(10, 2.7).

Let Y the weights of quokkas

Given that, Y~N(μ= 10, σ²=2.7)

So, σ=1.643

a) The probability, P(Y＞15.5) denotes chances that quokkas weight over 9.3 pounds, and it is,

P(Y>9.3) = P(Z> (15.5-μ)/σ)

= P(Z> (15.5-10)/1.643)

= P(Z>3.347)

= 0.0004

= 0.0004×100 =0.04%

b) The probability, P(Y＜11.9) denotes chances that quokkas weight under 11.9 pounds, and it is,

P(Y>9.3) = P(Z> (11.9-μ)/σ)

= P(Z> (11.9-10)/1.643)

= P(Z>1.156)

= 0.1250

= 0.1250×100 =12.5%

c) The probability, P(11.9≤Y≤15.5) denotes chances that quokkas weight will be between 11.9 and 15.5 pounds, and it is,

P(11.9≤Y≤15.5) = P((15.5-μ)/σ ≤ Z ≤ (11.9-μ)/σ)

= P((15.5-10)/1.643) ≤ Z ≤ (11.9-10)/1.643)

= P(3.347≤ Z ≤1.156)

= P(Z≤3.347)-PZ ≤1.156)

= 0.0004-0.1250

= 0.1246

= 0.1246×100

= 12.46%

Therefore, about 0.04% percent of quokkas weigh over 15.5 pounds.

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The correct option regarding the solution of the system of equations is given as follows:

One solution at (-1, 0).

We want to obtain the graphic solution for the system of equations, hence we obtain the point of intersection of the two equations.

For this problem, we have a single equation, hence the solution is obtained when the graph crosses the x-axis.

The graph crosses the x-axis when x = -1 and y = 0, hence the solution is given as follows:

(-1,0).

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Answer:

17

Step-by-step explanation:

Find the absolute value of -8, which is 8, and add 9 to that, which gives you 17.

Considering the definition of an equation, a mathematical statement for calculating the total labor cost is Total labor cost= 60×S + 30×PD + 19×T and the total labor cost for the recent project is $3,160.

Definition of equation

An equation is the equality between two algebraic expressions connected by the equals sign in which some known data are included together with one or more unknown values, also known as unknowns.

The terms of an equation are the addends that make up the members of an equation, whereas the members of an equation are each of the expressions that appear on both sides of the equal sign.

Finding the value that satisfies an equation is what is meant by finding its solution. By converting the unknown to the answer in this manner, the equality must be true.

This case

In this case, you consider:

S= hours of work from a senior artist.

PD= Production designer labor hours.

T= hours of work from a trainee.

On the other hand, you know that Girilla Graphics charges:

For the work of its most senior graphic artists, they charge $60 per hour.

For production designer work, the hourly rate is $30.

$19 per hour for a trainee's labor.

Therefore, the following equation can be used to determine the project's overall labor costs:

Total labor cost= 60×S + 30×PD + 19×T

A recent project required:

A senior artist's 34 hours of labor.

13 hours from a production designer.

35 hours from a trainee.

So, the following formula can be used to determine the project's overall labor costs:

Total labor cost= 60×34 + 35×13 + 19×35

Solving:

Total labor cost= $3,160

In summary, a mathematical statement for calculating the total labor cost is Total labor cost= 60×S + 30×PD + 19×T and the total labor cost for the recent project is $3,160.

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Based on the given system of equations, the value of x and y are undefined.

In order to solve the given system of equations, we would apply the substitution method. From the information provided in the image attached above, we have the following system of equations:

x = 5 - 2y      .......equation 1.

6y = 5 - 3x            .......equation 2.

By using the substitution method to substitute equation 1 into equation 2, we have the following:

6y = 5 - 3(5 - 2y)

6y = 5 - 15 + 6y

6y = -10 + 6y

6y - 6y = -10 (no solution).

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Therefore, since the total of all the angles is 180°, the value of the other angle is 100°, 40°, and 40°.

In Trigonometry, an angle is a curve made up of two rods, referred to as the angle's sides, which thus converge at a center point regarded as the ball's vertex. Two rays can create an angle in the region where they are arranged. Additionally, when airplane collide, an angle is created. Dihedral angles are what they are called. An offset is a possible shape spanning two radiation or lines with an identical termination in planar geometry. The English term "angle" comes from the Latin word "angulus," which meaning "horn." The vertex is the common endpoint of the two rays, also know as the faces of the angle.

given

one angle = 40°

other angle = (11x - 18)°

in triangle sum of all angle = 180°

40° + 11x - 18 + 40  = 180

80 + 11x - 18 = 180

62 + 11x = 180

11x = 180 - 62

11x = 118

x = 10 . 2

Therefore, since the total of all the angles is 180°, the value of the other angle is 100°, 40°, and 40°.

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According to the model, the current population of Algeria is 38 million.

The model for the population in t years from now is given by the model P = 38 ( 1 + r ) ^ t.

This means that when given the model, the current population of Algeria is 38 million because the 38 in the model represents the value of population when no years have elapsed.

We can solve for this to get :

= 38 ( 1 + r ) ^ t

= 38 ( 1 + r ) ⁰

= 38 x 1

= 38

= 38 million

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Answer:

Can you please post the picture of the question, then I can help.

Therefore, the project progresses and the unitary method wood becomes smoother, finer steel wool (i.e. grades 3, 2, 1, and 0000) can be used to achieve a smoother finish..

To finish a task using unitary procedure, multiply this same sum by two after finding the measurements of one tiny slice. To differentiate a coded unit from a specific group or collection of groups, use the unit variable method, which only needs an expression. For instance, 40 pencils could cost. 4,000, which really is equivalent to $1.01 as well as 4000 pounds.

Here,

The information in the graphic that supports the advice to use coarse steel wool at the start of a major wood refinishing project is:

"Grade 4 steel wool is used to remove paint and other materials, while grade 0000 is used for the final finish on fine woodwork."

This suggests that coarse steel wool (i.e. grade 4) is more effective at removing paint and other materials, which is often necessary at the beginning of a wood refinishing project.

As the project progresses and the wood becomes smoother, finer steel wool (i.e. grades 3, 2, 1, and 0000) can be used to achieve a smoother finish..

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The total mass of the necklace is 20 grams.

Multiplication of two numbers is defined as the addition of one of the number repeatedly until the times of the other number.

a × b means that a is added to itself b times or b is added to itself a times.

Given that,

Ariana loves making things she can wear. She makes a necklace by joining together 10 large pink paper clips. Each paper clip has a mass of 2 grams.

Total number of paper clips used to make necklace = 10

Mass of each clip = 2 grams

To find the total mass of the necklace, multiply the number of clips used to make necklace with the mass of each clip.

Mass of necklace = 10 × 2 grams = 20 grams

Hence 20 grams is the mass of the necklace.

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The subsets are {}, {0}.

What is the subset in mathematics?

In mathematics, a subset is a set consisting of elements that are also elements of another set. A set A is said to be a subset of set B if and only if every element of A is also an element of B. In other words, if you have a set A and another set B, and all the elements of set A are present in set B, then set A is considered a subset of set B.

In mathematics, a subset is a collection of elements that belong to a larger set. The elements in a subset can be any type of mathematical objects, such as numbers, vectors, or functions. A set is a subset of itself, and the empty set is a subset of every set.

The subsets of the given set {0} are the empty set {} and the set itself, {0}. So, the subsets of the given set 0 are {}, {0}.

Hence, the subsets are {}, {0}.

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Answer:

The equivalent circuit can be constructed using two NAND gates. The NAND gate is universal and can be used to implement all other logic gates including AND, OR, NOT, etc.

Here's the construction of the equivalent circuit:

First NAND gate: The first NAND gate implements the expression (~P ∧ ~Q). The inputs to the gate are P and Q, and the output is connected to the first input of the second NAND gate.

Second NAND gate: The second NAND gate implements the expression (~P ∧ (output of first NAND gate)) ∨ (P ∧ ~Q). The second input of the second NAND gate is connected to the output of the first NAND gate. The output of the second NAND gate is the final output of the circuit.

This circuit implements the given Boolean expression and has at most two logic gates.

The triangular table that Mrs. Gomez is buying for her classroom is a right triangle table because 3² +4² = 5². That is option C.

A right angle triangle is the type of triangle that has one side of its angle perpendicular to another and the value of the three sides can be calculated through the use of the Pythagorean formula.

The Pythagorean formula is expressed as, c2 = a2 + b2, where 'c' = hypotenuse of the right triangle and 'a' and 'b' are the other two legs.

Since, 5= c

b= 3

a = 4

Therefore, 3² +4² = 5² shows that the table is a right angle triangle.

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The function at the given value of the independent variable, f(x) = 1/√(x-4)f(x) - f(5) / x - 5 is (x-5).

Each element of X is given precisely one element of Y by the function from a set X to a set Y. The sets X and Y are collectively referred to as the function's domain and codomain, respectively. Initially, functions represented the idealised relationship between two changing quantities.

We have given function as ,

f(x) = 1/√(x-4)

We have to find f(x) - f(5)/x-5

f(x) = f(5)/x-5 = 1/√(5-4)/(x-5)

= 1/√(5-4)/(x-5)

= (x-5)/√(5-4)

= (x-5)

Therefore, The function at the given value of the independent variable, f(x) = 1/√(x-4)f(x) - f(5) / x - 5 is (x-5).

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Answer: The equation for the area of the garden is x(x - 2) = 48. To find the length, we need to solve for x:

x^2 - 2x - 48 = 0

Using the quadratic formula:

x = (-(-2) ± √((-2)^2 - 4(1)(-48)))/2(1)

x = (2 ± √(4 + 192))/2

x = (2 ± √196)/2

x = (2 ± 14)/2

x = 8 or x = -6

Since the length cannot be negative, the length of the garden is 8 feet.

Step-by-step explanation:

It is true that if points F and G are contained in a plane, then the line segment FG that connects them is entirely contained in that plane.

A two-dimensional plane, also known as a plane in two dimensions, is a flat surface that has length and width, but no thickness. It is an abstract mathematical concept that can be represented graphically.

Here,
In geometry, if two points are contained in a plane, then any line segment connecting those two points is also contained in that plane. So, if points F and G are contained in a plane, then the line segment FG that connects them is entirely contained in that plane.

Thus, it is true that if points F and G are contained in a plane, then the line segment FG that connects them is entirely contained in that plane.

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Using Compound Interest, The initially invested amount was $5400.

The interest you earn on interest is known as compound interest. Simple math may be used to demonstrate this: if you have $100 and it generates 5% interest annually, you will have $105 at the end of the first year. You'll have $110.25 after the second year is over.

The formula for compound interest is

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

Where C is the compound interest value after n years.

P is the principal amount.

I is interest.

n is the number of years.

It is given that percentage of interest is 5% so I=0.05. C=$5676.27.

n=7.

According to the formula of compound interest,

5676.27 = P(1+0.05)7 P = 5676.27 1.051084

P = 5400

Therefore the amount that was invested initially was $5400.

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We are given the following differential equation, [tex]\frac{dx}{dt}-4x=cos(5t)[/tex], with the initial condition [tex]x(0)=1[/tex]. This is a linear DE in the form, [tex]\frac{dx}{dt}+P(t)x=Q(t)[/tex], where [tex]\mu(t)=e^{\int\ {P(t)} \, dt }[/tex].

First, finding [tex]\mu(t)[/tex].

[tex]\Longrightarrow\mu(t)=e^{\int\ {P(t)} \, dt }\Longrightarrow\mu(t)=e^{\int\ {-4} \, dt }[/tex]

[tex]\mu(t)=e^{-4t }[/tex]

Multiply the DE by [tex]\mu(t)[/tex].

[tex]\Longrightarrow\frac{dx}{dt}-4x=cos(5t)\Longrightarrow e^{-4t} [\frac{dx}{dt}-4x=cos(5t)][/tex]

[tex]\Longrightarrow (e^{-4t}) \frac{dx}{dt}-(e^{-4t})4x=(e^{-4t})cos(5t)[/tex]

[tex]\Longrightarrow e^{-4t} \frac{dx}{dt}-4e^{-4t}x=e^{-4t}cos(5t)[/tex]

Verify that the L.H.S is a product rule of [tex]e^{-4t}x[/tex].

[tex]\Longrightarrow \frac{d}{dt}[e^{-4t}x ] =(e^{-4t}*-4)x+(e^{-4t})(\frac{dx}{dt} )[/tex]

[tex]\Longrightarrow \frac{d}{dt}[e^{-4t}x ] =-4e^{-4t}x+e^{-4t}\frac{dx}{dt}[/tex]

This matches, so we can move forward by integrating both sides of the DE.

[tex]\int\ {[e^{-4t}x ]'} \, =\int\ {e^{-4t}cos(5t)} \, dt[/tex]

The integral on the R.H.S is pretty nasty to do using integration by parts and u-sub. I will use a table of integrals.

[tex]\int\ {e^{ax}cos(bx) } \, dx=\frac{e^{ax}(acos(bx)+bsin(bx))}{a^2+b^2}[/tex]

Let [tex]a=-4[/tex] and [tex]b=5[/tex]

[tex]\int\ {[e^{-4t}x ]'} \, =\int\ {e^{-4t}cos(5t)} \, dt[/tex]

[tex]\Longrightarrow e^{-4t}x = \frac{e^{-4t}(-4cos(5t)+5sin(5t))}{(-4)^2+(5)^2}+C[/tex]

[tex]\Longrightarrow e^{-4t}x = \frac{e^{-4t}(-4cos(5t)+5sin(5t))}{41} +C[/tex]

Now for the initial condition, [tex]x(0)=1[/tex].

[tex]\Longrightarrow e^{-4(0)}(1) = \frac{e^{-4(0)}(-4cos(5(0))+5sin(5(0)))}{41} +C[/tex]

[tex]\Longrightarrow (1)(1) = \frac{(1)(-4(1)+(0))}{41} +C[/tex]

[tex]\Longrightarrow 1 = -\frac{4}{41} +C[/tex]

[tex]\Longrightarrow 1+\frac{4}{41} = C[/tex]

[tex]\Longrightarrow C=\frac{45}{41}[/tex]

Now we have, [tex]e^{-4t}x = \frac{e^{-4t}(-4cos(5t)+5sin(5t))}{41} +\frac{45}{41}[/tex], solve for x.

[tex]\Longrightarrow x = \frac{\frac{e^{-4t}(-4cos(5t)+5sin(5t))}{41}}{ e^{-4t}} +\frac{\frac{45}{41}}{e^{-4t}}[/tex]

[tex]\Longrightarrow x = \frac{(-4cos(5t)+5sin(5t))}{41}} +\frac{45e^{4t}}{41}[/tex]

Thus, the answer to the given differential equation with an initial condition is, [tex]x = \frac{(-4cos(5t)+5sin(5t))}{41}} +\frac{45e^{4t}}{41}[/tex].

Answer:

745.57

Step-by-step explanation:

[tex]691.95 \times \frac{7.75}{100} = 53.62[/tex]

[tex]53.62 + 691.95 = 745.57[/tex]

The area of the tabletop in square inches is 1071 square inches.

An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure. The square unit, which is frequently expressed as square inches, square feet, etc., is the accepted unit of area.

Given that, there are 3 inches between each side of the aquarium and each side of the tabletop.

The length of the tabletop will be:

l = 48 + 3 = 51 inches.

The width of the tabletop will be:

w = 18 + 3 = 21 inches.

The area of the table top will be:

A = (l)(b)

A = (51)(21)

A = 1071 square inches.

The area of the tabletop in square inches is 1071 square inches.

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The correct answer is to Reject the null hypothesis. This is strong evidence that the mean amount of sleep is less than 8 hours.

The test statistic of t = -2.03 and P-value of approximately 0.025 suggest that the mean amount of sleep for students is significantly less than 8 hours.

The P-value of 0.025 is less than the significance level of 0.05, which means that there is enough evidence to reject the null hypothesis that the mean amount of sleep is 8 hours.

Therefore, the appropriate conclusion at the a = 0.05 significance level is that the mean amount of sleep for students is less than 8 hours.

Hence, the correct answer is C.

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