R = 2 + 4sin(theta)

Domain is [0,2pi)


a. Write the integral to determine the area enclosed by the inner loop. Show your work that leads to the integral.

b. Write the integral to find the area between the loops

The law of exponential and logarithmic function is that the logarithmic functions are the inverses of the exponential functions.

According to the first law, we can multiply two exponential functions with the same base by adding their respective exponents. According to the second law, the exponents must be subtracted in order to divide two exponential functions with the same base.

According to the third law, we must multiply the exponents in order to raise a power to a new power. According to the fourth and fifth laws, we must raise each factor to the same power in order to raise a product or quotient to that power.

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The amount of money that Alex will use to purchase the 5 x 7 print would be = $11.89.

An online printing service is the service that is rendered online that helps interested individuals to prepare their printed paper works.

The number of 4 x 6 prints(X) purchased by Alex = 29

The number of 5 x 7 prints (y) purchased by Alex = 16

The total amount he paid for the whole service = $22.89

The cost of 5 x 7 prints can be calculated using this expression;

22.89 - 0.84 = 22.05/2 = 11.025

Therefore the cost of 5×7 prints = 11.05+ 0.84 = $11.89.

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1.59C + 2.25A = 786.75

C + A = 33    (even if all were the higher paying  adults, that would only be 33x2.25=74.25  far less than <<786.75)

1.59C + 1.59A = 33(1.59) = 52.47   subtract to eliminate C

     (2.25-1.59)A = 786.75 -52.47 = 734.28

       .66A = 734.28

            A = 73428/66 = 1112.55 = about 1113  adults

C = 33-1113 = -1080 children

which makes no sense.  over a thousand  "negative" children?  the public pool gave refunds to over a thousand prepaid children who didn't show up?  but a public swimming pool, no matter how large, probably wouldn't accomodate 1113 adults.

320 peoples are fishing for their peir.

A statement that demonstrates the link between various variables using mathematical processes is called an equation. It is applied to problem-solving and forecasting. The same relationship can be expressed in an equation in a number of ways, and it can be used to obtain the value of an unknown variable by setting it to a known value.

length of fishing pier 3200 feet

Requirement of one person = 10 feet

so the let x person can accomodate in the given length

so 10 * x =3200

so , x = 3200/10

=> x =320

so 320 are fishing for their pier.

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When you solve an equation for one variable and then substitute the resulting expression in for that variable in the other equation its is known as Substitution method.

I will demonstrate how the substitution method works. Suppose we have the following systems of linear equations with two variables:

Equation 1:  3x + 2y = 11

Equation 2:  -x + y = 3

By solving these two equations we get x=1 and y=4.

Equation 2:  -x + y = 3

-x + x + y = x + 3

y = x + 3

Next, substitute the value of y in Equation 2 (from the previous step), and substitute its value into Equation 1 to solve for x:

Equation 1:  3x + 2y = 11

Substitute y = x + 3 into Equation 1:

3x + 2(x + 3) = 11

3x + 2x + 6 = 11

Combine like terms:

5x + 6 = 11

5x + 6 - 6 = 11 - 6

5x = 5

Divide both sides by 5 to solve for x:

[tex]\frac{5x}{5} = \frac{5}{5}[/tex]

x = 1

Substitute the value of x into Equation 2 to solve for y:

Equation 2:  -x + y = 3

-1 + y = 3

y = 3 + 1

y = 4

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[tex]4a^4-8a^2+4=0\implies 4(a^4-2a^2+1)=0\implies a^4-2a^2+1=0 \\\\\\ (a^2)^2-2a^2+1=0\implies (a^2-1)(a^2-1)=0\implies a^2=1 \\\\\\ a=\pm\sqrt{1}\implies a=\pm 1[/tex]

The area of the carpeted part of the patio is 1 [tex]\frac{11}{16}[/tex] square yards.

Mixed fractions are type of fractions which consist of a whole number part and a fractional part.

Mixed fractions are of the form a b/c where a is the whole number part and b/c is the fractional part.

Given that,

Area of the patio = 2 1/4 square yards

                             = (2 × 4 + 1) / 4 square yards

                             = 9/4 square yards

Also given that, 3/4 of the patio is carpeted.

Area of the carpeted part of the patio = 3/4 (9/4)

                                                               = 27/16 square yards

                                                               = 1 11/16 square yards

Hence the area of the carpeted part is 1 [tex]\frac{11}{16}[/tex] square yards.

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The complete question is as follows :

A patio has an area of 2 1/4 square yards. 3/4 of the patio is carpeted. What is the area of the carpeted part of the patio?

The given expression is expressed in an algebraic expression as: y^5 + 11

An equation is the equality between two algebraic expressions, which have at least one unknown or variable.

In mathematics, when we express “sum” we mean an addition. So the expression "the sum of y... and eleven"  is represented by:

y + 11

When we refer "fifth power" we mean an exponent equal to five. So the expression “y to the fifth power” is represented by:

y^5

The complete expression "the sum of y to the fifth power and eleven" in an algebraic expression represented by: y^5 + 11

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Correctly written question:

Write an algebraic expression that represent the sum of y to the fifth power and eleven

The angle between the ladder and the ground is decreasing at the rate of 1 ft/sec.

According to the question,

A sliding ladder of length let L=13ft.

Let the ladder touches the house at height h and touches the ground

x ft from the house. This is shown in the figure.

In ΔABC,

 θ = [tex]cos^{-1} \frac{AB}{BC}[/tex]

⇒ θ  = [tex]cos^{-1} \frac{x}{BC}[/tex]

⇒ dθ/dt= [tex]\frac{d}{dt} cos^{-1} \frac{x}{BC}[/tex]

⇒ dθ/dt=  [tex]\frac{-1}{\sqrt{1 - (\frac{x}{13} )^{2} } } \frac{1}{13} \frac{dx}{dt}[/tex]

substituting the value of x=12ft and [tex]\frac{dh}{dt} = 5ft/sec[/tex] , we have

⇒ dθ/dt= [tex]\frac{-1}{\sqrt{1 - (\frac{12}{13} )^{2} } } \frac{1}{13} (5)[/tex]

⇒ dθ/dt=  [tex]\frac{-1}{\frac{5}{13} } \frac{1}{13} (5)[/tex]

dθ/dt= -1

so the angle between the ladder and the ground is decreasing at the rate of 1 ft/sec.

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The complete question is:

A 13 ft. ladder is leaning against a wall when its base starts to slide away At the instant when the base is 12 ft. away from the wall, the base is moving away from the wall at the rate of 5 ft/sec. The rate at which the angle θ between the ladder and the ground is changing is -

While the moving platform rotates, the shape of the quadrilateral ABCD does not change, only its shape. Therefore, ABCD is a parallelogram by the Parallelogram Opposite Sides Converse because both pairs of opposite sides are congruent. A parallelogram has the following dimensions: — AB — DC.

The Complete Question is Parallelogram.

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Samuel needs a total of 6 containers for his plan

The complete question is added as an attachment

From the question, we have the following parameters that can be used in our computation:

1 cup of rubbing alcohol = 1/(1/2) = 2 shares

Also, we have

1 cup of rubbing alcohol contains = 1 + 1/2 + 16 = 17.5

For the 2 shares calculated above, we have

2 shares = 17.5 * 2

Evaluate

2 shares = 35

Given that each can hold a maximum of 6 cups, the number of containers needed is

Container = 35/6

This gives

Container = 5.83

Round up

Container = 6

Hence, he needs 6 containers

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The difference between the largest 5-digit number and the smallest 6-digit number is 1.

The aim is to find the largest 5-digit and the smallest 6-digit numbers, and then we need to identify the difference between these two values.

Let's start by finding the smallest 6-digit number.

We know that 0 is the smallest digit that can not be used in the highest place value, so we will consider the next smallest digit; that is, 1 and 1 can be used in the highest place value.

Thus, we can create the required 6-digit smallest number with 0 and 1.

Hence, the smallest 6-digit number is 100000.

Now, we will discover the largest 5-digit number.

As we know, the largest digit is 9, which can be used anywhere. So, we will form the largest 5-digit number with the digit 9.

Thus, the largest 5-digit number is 99999.

Next, we need to find the difference between these two values.

So, we can determine the difference between the smallest 6-digit number and the largest 5-digit number by subtracting the smallest from the largest.

Thus, we get,

⇒100000 - 999999

⇒1

Hence, 1 is the difference between the smallest 6-digit and the largest 5-digit numbers.

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∫(f(x) + g(x)) dx = ∫f(x) dx + ∫g(x) dx is the sum rule in integration.

The sum rule in integration states that if f(x) and g(x) are two functions that are both integrable over a given interval, then the integral of the sum of these functions over that interval is equal to the sum of the integrals of the individual functions over that interval. Mathematically, this can be expressed as:

∫(f(x) + g(x)) dx = ∫f(x) dx + ∫g(x) dx

The sum rule of integration is: Integral of the sum of two functions is equal to the sum of integration of individual functions.

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Answer:

The given expression is -7xy^3z^2

The degree of x is 1

The degree of -7 is 0

The degree of y^3 is 3

The degree of z^2 is 2

The degree of a polynomial is the highest exponent of its variables. In the case of the polynomial -7xy^3z^2, the degree is the highest exponent among x, y, and z.

The highest exponent is 3, which is the exponent of y. So the degree of the polynomial -7xy^3z^2 is 3.

It's worth noting that the coefficient -7 has no effect on the degree of the polynomial. The degree of a polynomial is determined by the exponents of its variables, not its coefficients.

Answer:

3(5a + 4)

Step-by-step explanation:

Given expression: 15a + 12

The common factor for both is 3.

Factorize:

=> 3(5a) + 3(4)

Take out the common factor: (3)

=> 3(5a + 4)

Answer:

3(5a+4)

Step-by-step explanation:

Firstly, find the highest common factor between 15a and 12, which is 3 and write it in brackets, which is:

3(5a+4)

The brackets are 5a+4 because 3×5a=15a and 3×4=12, which gives you the algebraic expression 15a+12

The solution is,  A’B’ =  12 units, when AB = 12.

A transformation is a dramatic change in form or appearance. An important event like getting your driver's license, going to college, or getting married can cause a transformation in your life. A transformation is an extreme, radical change.

here, we have,

Given

Line segment AB = 12 units

Translation of AB → A'B' by the rule T(x +0, y+5)

This translation won't change the length of A'B' as the difference in (x2-x1) and (y2-y1) stays same

because x-coordinate stays as is and y-coordinate's change cancelled when subtracting.

So,  A'B' = 12 units.

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Answer:

Step-by-step explanation:From the issue, we may deduce that:

x + y = 50 She desires to produce 50 ounces of the combination (because to this).

(0.35x) + (0.60y) = 0.55(x+y) (because we're calculating the amount of salt in each solution and the combination contains 55% salt)

We may take the first equation and replace it in the second equation:

(0.35x) + (0.60y) = 0.55(50) (50)

0.35x + 0.60y = 27.5 when x is substituted.

A system of two equations with two variables is now available:

x + y = 50\s0.35x + 0.60y = 27.5

To determine one of the variables in terms of the other, we may utilize the first equation. Let's figure out y:

y = 50 - x

We now change y in the second equation to the following expression:

0.35x + 0.60(50-x) = 27.5

Finding the value of x:

0.35x + 30 - 0.6x = 27.5

0.25x= -2.5

x= -10

Since the number of ounces cannot be negative, the answer is illogical. As a result, you should double-check your calculations or determine if the issue statement contains a mistake.

To get the answer, you can also try an alternative approach like substitution or elimination. Then, double-check your calculations.

Step 1: Think about this as four separate division questions:

     [tex]\dfrac{10c^3de^2}{2cde} =\dfrac{10}{2} \cdot \dfrac{c^3}{c} \cdot\dfrac{d}{d}\cdot \dfrac{e^2}{e}[/tex]

Step 2: Simplify each of the four, one at a time:

     [tex]\begin{aligned}\dfrac{10c^3de^2}{2cde} &=\dfrac{10}{2} \cdot \dfrac{c^3}{c} \cdot\dfrac{d}{d}\cdot \dfrac{e^2}{e}\\[0.5em]&=5 \cdot c^{3-1} \cdot1\cdot e^{2-1}\\[0.5em]&=5 c^{2} e\end{aligned}[/tex]

To find the sine of an angle in a right triangle, we can use the following relationship:

[tex]sin(angle) = opposite side / hypotenuse\\[/tex]

What is right triangle?

A right triangle is a type of triangle that has one angle that measures exactly 90 degrees (also known as a right angle). This angle is formed by the intersection of two legs of the triangle, which are not the hypotenuse of the triangle. The side opposite the right angle is called the hypotenuse and it is always the longest side of the triangle.

To find the sine of an angle in a right triangle, you can use the following relationship:

[tex]sin(angle) = opposite side / hypotenuse\\[/tex]

This relationship holds true for any angle in a right triangle, not just the acute angles.

For example, if you have a right triangle with an angle of 30 degrees and the opposite side is 5 units and the hypotenuse is 12 units, you can find the sine of the angle by dividing the opposite side by the hypotenuse:

[tex]sin(30) = 5/12 = 0.4166\\[/tex]

It's also important to note that the sine of an angle is always between -1 and 1, and the sine of an acute angle is always positive.

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To find the sine of an angle in a right triangle, we can use the following relationship:

sine = opposite side / hypotenuse side

What is right triangle?

A right triangle is a type of triangle that has one angle that measures exactly 90 degrees (also known as a right angle). This angle is formed by the intersection of two legs of the triangle, which are not the hypotenuse of the triangle. The side opposite the right angle is called the hypotenuse and it is always the longest side of the triangle.

To find the sine of an angle in a right triangle, you can use the following relationship:

sine = opposite side / hypotenuse side

This relationship holds true for any angle in a right triangle, not just the acute angles.

For example, if you have a right triangle with an angle of 30 degrees and the opposite side is 5 units and the hypotenuse is 12 units, you can find the sine of the angle by dividing the opposite side by the hypotenuse:

sin(30) = 5/12 = 0.4166

It's also important to note that the sine of an angle is always between -1 and 1, and the sine of an acute angle is always positive.

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By using the multiplication formula, the number of apples in all 4 boxes are 40.

Aside from addition, subtraction, and division, multiplication is also considered to be one of the four fundamental mathematical operations. Multiplying in mathematics refers to the process of repeatedly adding together groups of the same size.

The multiplication formula is expressed as:

Multiplicand × Multiplier = Product

Where:

Multiplicand: The first number (factor)

Multiplier: The second number (factor)

Product: The final result after multiplying the multiplicand and multiplier

In this case, we are given that:

Multiplicand = 4 boxes

Multiplier = 10 apples

Thus, the number of apples in 4 boxes are:

Product = 4 x 10

Product = 40

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To produce g, function was reflected over the x-axis and | shifted up 3 units . Function g can be defined as g(x)=f(x-1)+4.

What is axis ?

An axis is a straight line around which an object or a geometric shape can rotate or be symmetrical. In a coordinate system, an axis is a line that serves as a reference for the location of points in space.

When a function f(x) is reflected over the x-axis, it changes the sign of the y-coordinate values. By reflecting the graph of f(x) over the x-axis, the new function g(x) will have the same shape as f(x) but with all y-coordinate values inverted.

Additionally, by shifting the function up 3 units, it means that the y-coordinate values of the new function g(x) will be 3 units greater than the y-coordinate values of the original function f(x).

So the transformation can be represented as:

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

It is the horizontal shift of 1 unit to the right, and a vertical shift of 3 units up

To produce g, function was reflected over the x-axis and | shifted up 3 units . Function g can be defined as g(x)=f(x-1)+4.

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Answer:

Undefined

Step-by-step explanation:

The function is defined in the domain (1, +∞)

Any value in the range of (-∞, 1] is undefined

Quadratic equations' solutions are  x=0, x=32 , y=3, y=-2 , [tex]y=-\frac{5}{2}, y=4$$[/tex] , z=0, z=-10 , and x=0, x=3

[tex]$$x(x-32)=0 \quad: \quad x=0, x=32$$[/tex]

Steps

x(x-32)=0

If a b=0, then either a=0 or b=0, according to the Zero Factor Principle.x=0 or x-32=0

Solve x-32=0:  x=32

The following are the quadratic equation's solutions:

x=0, x=32

2 . [tex]$$(y-3)(y+2)=0 \quad: \quad y=3, y=-2$$[/tex]

Steps

(y-3)(y+2)=0

If a b=0, then either a=0 or b=0, according to the Zero Factor Principle. y-3=0 or y+2=0

Solve y-3=0 :  y=3

Solve y+2=0 :  y=-2

The following are the quadratic equation's solutions:

y=3, y=-2

3. [tex](2 y+5)(y-4)=0 \quad: \quad y=-\frac{5}{2}, y=4$$[/tex]

Steps

(2 y+5)(y-4)=0

If a b=0, then either a=0 or b=0, according to the Zero Factor Principle. 2 y+5=0 or        y-4=0

Solve 2 y+5=0:  y=-[tex]\frac{5}{2}[/tex]

Solve y-4=0 :  y=4

The following are the quadratic equation's solutions:

[tex]y=-\frac{5}{2}, y=4$$[/tex]

4.[tex]$$2 z^2+20 z=0 \quad: \quad z=0, z=-10$$[/tex]

steps

[tex]$$2 z^2+20 z=0$$[/tex]

The following are the quadratic equation's solutions:

[tex]$$z_{1,2}=\frac{-20 \pm \sqrt{20^2-4 \cdot 2 \cdot 0}}{2 \cdot 2}$$$$\sqrt{20^2-4 \cdot 2 \cdot 0}=20$$$$z_{1,2}=\frac{-20 \pm 20}{2 \cdot 2}$$[/tex]

Distinguish the answers

[tex]$$\begin{aligned}& z_1=\frac{-20+20}{2 \cdot 2}, z_2=\frac{-20-20}{2 \cdot 2} \\& z=\frac{-20+20}{2 \cdot 2}: 0 \\& z=\frac{-20-20}{2 \cdot 2}:-10\end{aligned}$$[/tex]

The following are the quadratic equation's solutions:

z=0, z=-10

5 .

[tex]$$\begin{aligned}& 9 x^2=27 x: x=0, x=3 \\& \text { Steps } \\& 9 x^2=27 x\end{aligned}$$[/tex]

Move 27 x to the left side

[tex]$$9 x^2-27 x=0$$[/tex]

[tex]$$\begin{aligned}& \text { Factor } 9 x^2-27 x: \quad 9 x(x-3) \\& 9 x(x-3)=0\end{aligned}$$[/tex]

If a b=0, then either a=0 or b=0, according to the Zero Factor Principle. x=0 or x-3=0

Solve x-3=0: x=3

The following are the quadratic equation's solutions

x=0, x=3

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It is option D (f-¹(x) = x-3/5)

All questions are solved using BODMAS rule.

BODMAS stands for Brackets, Orders (of operations), Division, Multiplication, Addition, and Subtraction. It is a commonly used acronym in mathematics to remember the order in which mathematical operations should be performed. When solving mathematical expressions, operations within brackets should be done first, followed by any exponents or roots.

Next, perform any divisions or multiplications from left to right, followed by any additions or subtractions from left to right. This order ensures that the expression is evaluated correctly and avoids confusion or errors.

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Correct option is A,  f(x) -2 has changed the parent function f(x) = (0.5)x to its new appearance.

The parent function f(x) = log5x has been modified by reflecting it over the x-axis, extending it vertically by a factor of three, and moving it down by three units.

In the picture below you can see the blue line is the graph of the function

f(x) = log(5x) and the green line is the graph of the function

g(x) = log[5(x + 4)] - 2

Since the function f passes at point (2, 1), we must reduce it by 2 units to ensure that it also passes at position (-2, -1). To do this, we add 2 to the function f.

We only need to add 4 to the variable x to have the function go left when we obtain log(5x) - 2.

The function shifts to the left when you add a number to x;

The function moves to the right when you remove a number from x;

The function increases when you add a number to it;

The function decreases when you take a number away from it.

Then we get a function g(x) = log[5 (x + 4)] - 2.

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Answer:

23.6 units.

Step-by-step explanation:

Since quadrilateral DEFG is similar to quadrilateral HIJK, the ratios of corresponding sides are equal. We can use this information to find the measure of IJ.

We know that:

DF/HJ = EF/IJ = 6/23.6 = 2/8 = 1/4

So we can set up the proportion:

IJ/EF = HJ/DF

IJ/6 = 23.6/6

IJ = 23.6/6 * 6 = 23.6

Answer: 30+1+.002

Step-by-step explanation:

Just break it up into number groups. Like tens, ones, or hundreds

Answer: Yes, it does.

Step-by-step explanation: Let x represent the price of gas today. 200% of x=200/100*x, which can be written as 2x because 200/100=2. 2x is x doubled, and therefore, it does mean the price has doubled.

Answer:

yes, the price has doubled