express x=e−3t, y=4e4t in the form y=f(x) by eliminating the parameter.

To solve a piecewise defined function, you should follow these steps:

1. Understand the definition of the function. A piecewise defined function is a function that is defined differently on different intervals of its domain.

2. Identify the different intervals on which the function is defined. These intervals are usually separated by vertical asymptotes, which are points where the function is undefined.

3. Determine the formula for the function on each interval. This is usually given in the problem statement, but if not, you will need to figure it out from the context.

4. Find the domain and range of each formula. The domain of a function is the set of all input values for which the function is defined, and the range is the set of all output values.

5. Determine any discontinuities in the function. These are points at which the function is defined differently on different intervals, and they will be vertical asymptotes.

6. Sketch the graph of the function by plotting points and connecting them with line segments.

7. If the function is a piecewise continuous function, Analyze it further by finding its limits, derivatives, integrals, or other properties of interest as you would with any other function.

8. Evaluate the function for a given input value, you need to determine which interval the input value belongs to and then use the appropriate formula for that interval to find the output value.

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The solutions of x + 2 = 2ˣ + 1 are x = 0 and 1.

The answers are given in the attached image.

Coordinates are a pair of integers (Cartesian coordinates), or occasionally a letter and a number, that identify a certain place on a grid, often referred to as a coordinate plane.

Given:

Two functions,

f(x) = x + 2 and g(x) = 2ˣ + 1.

The graph of the functions on the same coordinate plane is given in the attached image.

The blue line represents the graph f(x) and the green represents g(x).

Then f(x) = g(x)

x + 2 = 2ˣ + 1.

The solution is the x-value of the point of intersection.

That means,

x = 0 and 1

Therefore, the solutions of the f(x) = g(x) are x = 0 and 1.

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Answer:

Opens up

a =  1

b =  0

c = -6

Step-by-step explanation:

y=x^2-6

This is a parabola that opens upwards (see attached graph).  

I will assume the a, b, and c are the coefficients of a parabolic equation of the form:

y = ax^2 + bx + c

Since y = x^2 - 6, the values are:

a = 1

b = 0

c = -6

The geometric series which diverges is shown in the option number c as the absolute value of the common ratio of this series 4 is ∑∞n=1 (2/3)(-4)ⁿ⁻¹

The next term in a geometric sequence is produced by multiplying the previous term by the same number across the whole series.

It can be given as,

a+ar+ar²+ar³+...

Here,

is the sequence's (a) initial term, and (r) stands for common ratio.

It should therefore be a series as geometric series diverge.

|r|>1

the first choice is,

3/5+3/10+3/20+3/40+ ...

The common ratio in this case is,

r = (3/10)/(3/5)

r= 1/2

The average ratio is below one. Option an is therefore incorrect.

second option given as,

-10+4-8/4+18/25- ...

Here, the common ratio is,

r = (4/-10)/(-2/5)

r= -2/5

The usual ratio for the second choice is (-2/5), which is below one. This alternative is also incorrect.

The common ratio in option number c is equal to -4. The common ratio's exact value is,

|r|=|-4|

|r|= 4

This common ratio has a greater absolute value than one. This is the best choice as a result.

The absolute value of this series' common ratio, or option number c, represents the geometric series that diverges.

∑∞n=1 (2/3)(-4)ⁿ⁻¹

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The probability of 45 heads in 100 tosses of a coin given the probability of heads is 50% is 7.1%.

The binomial probability formula is used to calculate the probability of getting a certain number of heads (or success) in a certain number of coin tosses (or trials) when the probability of getting a head (or success) in each individual coin toss is known. The formula is:

P(x) = (n choose x) * (pˣ) * (1-p)^(n-x)

Where,

P(x) is the probability of getting x heads (or success) in n coin tosses (or trials)

n is the number of coin tosses

x is the number of heads (or success)

p is the probability of getting a head (or success) in a single coin toss

1-p is the probability of getting a tail (or failure) in a single coin toss

(n choose x) is the binomial coefficient, which is the number of ways to choose x heads (or success) from n coin tosses (or trials)

In this case, the probability of getting 45 heads in 100 tosses of a coin with a probability of heads of 50% can be calculated using the formula: (100 choose 45) * (0.5⁴⁵) * (0.5⁵⁵) which is approximately 0.071, or 7.1%.

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By solving the given expression when x=12 we get a = 11-12b

what is an algebraic expression ?

Algebraic expression can be defined as the combination of  constants , variables and operators.

Algebraic expressions are the concept of expressing numbers using letters or alphabets without specifying their real values. The basics of algebra taught us a way to specific an unknown value the usage of letters which includes x, y, z, and many others. those letters are called here as variables. An algebraic expression may be a combination of each variables and constants. Any value this is positioned before and increased through a variable is a coefficient.

Given,

expression ,

x = (1+a) / (1-b )

By putting x =12

we get,

12 = (1+a) / (1-b)

12 * (1-b) = 1+a

12- 12b =1+a

a = 12-12b-1

a = 11 - 12b

Hence , By solving the given expression we get a = 11-12b

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She spends a total of $ 180 in copayments, $20 for medical visits and $10 for generic prescriptions.

The system process is a strategy for addressing issues that entails first figuring out the price of a single item, than multiplication that value to figure out the needed value. Simply put, the unit technique is employed to extract a single unit quantity from a multiple that is supplied. For example, 40 pens would cost 400 rupees, which is equal to the cost of one pen. This procedure could follow a regular procedure. a single nation. anything has a distinct identity. Its adjoint and inverse are equivalent in terms of mathematics, algebra, linear algebra, mathematical, or math and science of matrices or operators.

given

$20 for copay for doctor’s visits

$10 copay for generic prescription

for 5 doctor visits = 20*5 = $100

for 8 prescriptions = 10 * 8 = $80

so total = 100+ 80 = $180

She spends a total of $ 180 in copayments, $20 for medical visits and $10 for generic prescriptions.

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today fever is 100.7 F, today is 2.5 higher than yesterday

yesterday is 2.5 lower than today.

x = temperature of yesterday

x + 2.5 = 100.7

x = 98.2 F

Answer:

B)  Let x = yesterday's body temperature

         x + 2.5° = 100.7 °F

         x = 98.2 °F

Step-by-step explanation:

We want to determine Dud's body temperature yesterday.  Therefore, the unknown variable is:

If Dud's body temperature today is 2.5° higher than yesterday's body temperature, then his body temperature today is the sum of yesterday's body temperature and the increase:

To solve the equation, subtract 2.5° from both sides of the equation:

Therefore, Dud's body temperature yesterday was 98.2 °F.

An inequality is a mathematical statement that compares two expressions and expresses the relationship between them. It is represented by a symbol such as ">", "<", "≥", or "≤" and can be thought of as a generalization of an equation.

Unlike an equation, which has one unique solution, an inequality can have multiple solutions or no solution at all. The solutions of an inequality are the values that make the inequality true.

Inequalities are used to model real-world situations where there are constraints or limitations, such as determining the range of possible values for a variable. They are also used to describe relationships between variables, such as when one variable is greater than or less than another variable.

There are different types of inequalities, such as linear inequalities, quadratic inequalities, and absolute value inequalities, each with their own set of rules and methods for solving them.

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Acres of land needed by the developer to build a community center that will use up 9/25 of the property is 15.66 acres, option C.

The size of land required to build a community center is calculated as follows:

First plot size = 24.6 acres

Field size = 18.9 acres

The local developer purchases both (24.6 + 18.9) acres to have a total of 43.5 acres.

The amount needed to build a community center is 9/25

Therefore 43.5 x 9/25 = 15.66 acres

9/25 of a 43.5 acres purchased by local developer will require only 15.66 acres.

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Answer: likely

Step-by-step explanation:

50/50

Answer: 50/50

Step-by-step explanation: It Is equally likely because it is 50% of the total.

Answer: 6

Step-by-step explanation:

Each year it will lose $160. So v=160

The transformations that result in similar triangles are dilations, rotations, reflections, and translations.

A transformation that conserves the shape, but not necessarily the size, of a figure is called a similarity transformation. Rotation, reflection, and translation are rigid motions, which means they preserve both size and shape, on the other hand, dilation only ensures that the shape is preserved. As the rigid motions are congruence transformations, all congruent figures are also similar.

A similarity transformation is a dilation or a combination of rigid motions and dilations. Two geometric figures are similar if and only if there is a similarity transformation that maps one of the figures onto the other. Similar figures have the same shape but necessarily not the same size.

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A lossless compression algorithm preserves all of the original data, while a lossy compression algorithm may discard some of the original data.

A lossless compression algorithm is a type of data compression algorithm that preserves all of the original data without any loss. This means that when the original data is uncompressed, it is exactly the same as when it was compressed. On the other hand, a lossy compression algorithm is a type of data compression algorithm that discards some of the original data in order to reduce the size of the file. This means that when the original data is uncompressed, it may not be exactly the same as when it was compressed.

The advantage of a lossless compression algorithm over a lossy compression algorithm is that it preserves all of the original data, while a lossy compression algorithm may discard some of the original data.

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Therefore , the solution of the given problem of unitary method comes out to be  A.$10.5 total cost and $ 0.12 .

By first multiply the value with one unit by the number of units needed, one can use the unit method to solve a problem. Simply put, the unit function is used to distinguish a single entity unit from a given many. 40 needles may be purchased for 400 rupees, or the price of one pen. There may be a standard method for figuring this out. only one nation. Everything has a unique quality. Algebra, its dialectic and inversion, and matrix logic are all analogous concepts.

Here,

Given:

A) Estimate :

70 * 1.25

80*1.25

=> 80 + 20 = 100

and

100/30 = 3.33

thus, approx 4 gallons of gas felicia needs

=> 4 * 3.5 = 14

thus, $14 total cost

Exact :

=> 70 * 1.25 = 87.5 miles

=> 87.5/30 = 2.92 gallons

thus, approx 3 gallons of gas felicia needs

=> 3*3.5 =10.5

thus, $10.5 total cost

B)

=> per cost / miles

=> 3.5 / 30 = 0.116

=>$ 0.12

Therefore , the solution of the given problem of unitary method comes out to be  A.$10.5 total cost and $ 0.12 .

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the explanation of how heart disease progresses over time in part of her presentation would be best suited to a video.

Several different heart disorders are referred to as "heart disease." Coronary artery disease (CAD), which affects the blood flow to the heart, is the most typical kind of heart disease in the United States. A heart attack may result from reduced blood flow.

here,

Chest pain or discomfort, pain in the upper back or neck, heartburn, nausea, vomiting, extreme exhaustion, pain in the upper body, dizziness, and shortness of breath are all symptoms of a heart attack.

Heart palpitations are caused by an arrhythmia (palpitations). Heart failure symptoms include shortness of breath, exhaustion, or swelling of the legs, abdomen, or neck veins.

The three main risk factors for heart disease are excessive blood pressure, high cholesterol, and smoking. Nearly half of Americans (47%) have at least one of these three risks.

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Answer:

d

Step-by-step explanation:

The required area of the shaded region is 141.3  sq. in.

A circle's area is the area that it takes up in a two-dimensional plane. It can be simply calculated using the formula A = r2, (Pi r-squared), where r is the circle's radius. The square unit, such as m^2, cm^2, etc., is the unit of area. Area of a circle is equal to r2 or d2/4, where = 22/7 or 3.14.

inner radius = 6 in, outer radius= 9 in

are of small circle = πr^2

Area = π(6)^2 = 36π sq. in

Similarly

Area of big circle = πr^2

Area = π(9)^2 = 81π sq. in

So,

Area of shaded region =  81π - 36π

Area of shaded region = 45π

Area of shaded region = 141.3  sq. in

Thus, required area is 141.3  sq. in.

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If we evaluate the ∫(2to1)(v^6+v^5)/4 dv  the answer would be 7.16

An integral gives numerical values to functions to represent concepts like volume, area, and displacement that result from combining infinite small amounts of data. Integration is the action of locating integrals.

The signed area of the region in the plane that is circumscribed by the graph of a specific function between two points on the real line can be used to represent definitive integrals.

Traditionally, portions above the plane's horizontal axis are positive, and those below it are negative.

First, calculate the corresponding indefinite integral.

∫(v^6+v^5)/4 dv= v^6(6v+7)/168

According to the Fundamental Theorem of Calculus, ∫(b to a) F(x)dx=f(b)−f(a), so just evaluate the integral at the endpoints

(v^6(6v+7)/168)at (v=2)=152/21.

(v^6(6v+7)/168)at (v=1)=13/168

∫(2to1)(v^6+v^5)/4 dv= (v^6(6v+7)/168)at (v=2) - (v^6(6v+7)/168)at (v=1) = 401/56 = 7.16

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Answer: The diagonal and the height of the right triangle form the legs of the triangle, and the unknown length is the hypotenuse. To find the length of the hypotenuse, we can use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs).

So, if the legs of the right triangle measure 12 inches and 15 inches, the hypotenuse c can be calculated as follows:

c² = a² + b²

c² = 12² + 15²

c² = 144 + 225

c² = 369

Therefore, the length of the hypotenuse c is the square root of 369.

c = √369

c = 19.1 inches

Therefore, the top of the spider web is 19.1 inches long.

Step-by-step explanation:

By solving the given expression we get this simplified expression 12x³ - 7x²y - 8xy² + 3y³.

What is the linear equation?

A linear equation is an algebraic equation of the form y=mx+b. where m is the slope and b is the y-intercept.

To simplify the expression (3x² - 4xy + y²)(4x + 3y), we can use the distributive property of multiplication to expand the expression as follows:

(3x² - 4xy + y²)(4x + 3y) = 3x²(4x) - 4xy(4x) + y²(4x) + 3x²(3y) - 4xy(3y) + y²(3y)

Simplifying each term, we get:

12x³ - 16x²y + 4xy² + 9x²y - 12xy² + 3y³

Combining like terms, we get:

12x³ - 7x²y - 8xy² + 3y³

Therefore, the simplified expression is 12x³ - 7x²y - 8xy² + 3y³.

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If Mike pitches a ridge tent. The tent is 16 yards wide and the slope of the tent measures 23 yards, the height of the tent is 28.07 cm

The solution for the height would be gotten with the Pythagorean theorem. The theorem states that in a right angled triangle the sum of the square on the hypotenuse is equal to the sum of the square of the two other sides of same triangle.

This is illustrated mathematically as

C² = A² + B²

where C is the height

A is the width in yards

B is the slope.

Then we have

C² = 16² + 23²

C = √785

C = 28.07 yards

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A "pair of 3" is not a commonly used term.

A set of 3 numbers can be called a triplet, or a set of 3 items can be called "a set of three".

Please let me know if you want me to clarify about something else.

sqrt( A / pi ) =r is the solution for r and r is called as radius in the formula  A=πr^2

A large part of the field of operations research is the use of statistical equations to solve problems in business or military operations.

Music, linguistics, and philosophy all use mathematical models (for example, intensively in analytic philosophy).

A model is, for example, the method for the area of a circle, πr2.

Radius of a circle is the distance from the center of the circle to any point on it's circumference. It is usually denoted by 'R' or 'r'. This quantity has importance in almost all circle-related formulas. The area and circumference of a circle are also measured in terms of radius.

Α= pi r^2

Divide each side by pi

A / pi = r^2

Take the square root of each side

sqrt( A / pi ) = sqrt(r^2)

sqrt( A / pi ) =r

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

Explanation:

The weight of A is 2/3 that of B. This flips to saying "the weight of B is 3/2 that of A". Because 3/2 = 1.5, we can then have this equation

B = 1.5A

The statement "C is 75% the weight of D" leads to the equation

C = 0.75D

----------------------------

The sum of A and B is triple that of the sum of C and D, giving us this equation:

A+B = 3(C+D)

Plug in the previous two equations we found in the previous section. Afterward, solve for the variable "A".

A+B = 3(C+D)

A+1.5A = 3(0.75D+D)

2.5A = 3(1.75D)

A = 3(1.75D)/2.5

A = 2.1D

----------------------------

Let's revisit the first equation we set up, and then apply substitution like so:

B = 1.5A

B = 1.5*2.1D

B = 3.15D

----------------------------

There's a lot of equations to keep track of. The key equations are these:

A = 2.1D

B = 3.15D

C = 0.75D

The one common factor of all these equations is the variable "D". This will be useful when setting up the ratio A:B:C:D

We'll go from the ratio

A:B:C:D

to

2.1D:3.15D:0.75D:1D

afterward we can divide all four parts by the GCF variable D, to arrive at

2.1:3.15:0.75:1

----------------------------

This last section will focus on converting each part of that ratio into integers. This is done by multiplying all four parts by 100 (as shown in the 2nd step).

Then we divide all parts by the GCF 5 to get the final reduced ratio.

2.1:3.15:0.75:1

100*2.1:100*3.15:100*0.75:100*1

210:315:75:100

5*42:5*63:5*15:5*20

42:63:15:20

The number of maths books sold is 70 and the number of history books sold is 210.

Let no of maths book sold be x.

According to the question, the number of history books sold is three times the number of maths books sold.

Therefore the number of history books sold is 3x.

Given: Total books sold = 280

So, x + 3x = 280

Solving the equation, we get:

4x = 280

x = [tex]\frac{280}{4}[/tex]

x = 70 (Number of maths books sold)

3x = 3 × 70 = 210 (Number of history books sold)

Thus the number of maths books sold is 70 and the number of history books sold is 210.

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The solution to the system of equations, x = 7y - 3 and x + 7y = 8, is: (5/2, 11/14).

One way to solve a system of equations is to use substitution.

We can start by substituting the first equation into the second equation:

x + 7y = 8

(7y - 3) + 7y = 8

7y + 7y - 3 = 8

14y - 3 = 8

14y = 11

y = 11/14

Now that we know the value of y, we can substitute it back into the first equation to find x:

x = 7y - 3

x = 7(11/14) - 3

x = 11/2 - 3/2

x = 8/2 - 3/2

x = 5/2

So the solution to the system of equations is (x, y) = (5/2, 11/14)

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The value of cos(Z) is 12/13.

Since both triangles are similar, △ABC ~ △XYZ, the corresponding angles are congruent. It means:

m∠C = m∠Z, and

cos C = cos Z

The ratio for cosine is adjacent/hypotenuse. In a right triangle, the hypotenuse is the longest side, while an opposite side is the one across from a given angle, and an adjacent side is the one next to a given angle.

Since the side marked 12 is adjacent to angle C and 13 is the hypotenuse of this triangle; so, we know that:

cos C = 12/13

Thus, because of the angles and cosines for C and Z will be the same, it makes:

cos Z = cos C = 12/13

Hence, the value of cos(Z) is 12/13.

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Answer:

2.44444

but rounded to 2.4 (tens place)

Step-by-step explanation:

hope this helps!:)

The price of one pound of apples is represented in the equation by the number 4.3 by the price in dollars of x pounds of apples, where x is pounds.

A mathematical equation is a formula that connects two statements using the equal sign (=) to denote equivalence. An algebraic equation is a mathematical statement that establishes the equality of two mathematical expressions. For example, the variables 3x + 5 and 14 are separated by an equal sign in the equation 3x + 5 = 14.

given

y = 4.3x

the price in dollars of x pounds of apples, where x is pounds.

The price of one pound of apples is represented in the equation by the number 4.3.

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