Simplify the rational expression
7x/14x^6

The domain and range of the graph above in interval notation include the following:

Domain = [-6, 3]

Range = [-3, 3]

In Mathematics and Geometry, a domain refers to the set of all real numbers (x-values) for which a particular function (equation) is defined.

In Mathematics and Geometry, the horizontal portion of any graph is used to represent all domain values and they are both read and written from smaller to larger numerical values, which simply means from the left of any graph to the right.

By critically observing the graph shown in the image attached above, we can reasonably and logically deduce the following domain and range:

Domain = [-6, 3] or -6 ≤ x < 3.

Range = [-3, 3] or -3 < y < 3

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Answer:

Mean: 27

Median: 22

Mode: 22

Standard deviation: 17

No outlier (61 might be one just check with people you know and see if they have an outlier or not)

Step-by-step explanation: Give a definition of mean, median, mode…and just add your answers.

The function representing the number of bacteria after t hours is N(t) = 5300 * (1 + 0.0592)^t, and the growth rate per hour is approximately 5.92%.

To represent the number of bacteria after t hours, we can use the exponential growth formula:

N(t) = N₀ * (1 + r)^t,

where N(t) is the number of bacteria after t hours, N₀ is the initial number of bacteria, r is the hourly growth rate, and t is the time in hours.

In this case, the initial number of bacteria is 5300, and the hourly growth rate can be determined from the doubling time of 12 hours. The growth rate can be calculated using the formula:

r = 2^(1/t_double) - 1,

where t_double is the doubling time in hours.

Substituting the given values, we have:

r = 2^(1/12) - 1 ≈ 0.0592.

Now we can write the function for the number of bacteria after t hours:

N(t) = 5300 * (1 + 0.0592)^t.

To determine the percentage of growth per hour, we can calculate the relative growth rate as a percentage:

percentage_growth = r * 100 ≈ 5.92%.

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y intercept will be 0,-6 .

Given,

Amplitude = 6

Time period = pi

Phase shift = pi/4

In trigonometry, the sine function can be defined as the ratio of the length of the opposite side(perpendicular) to that of the hypotenuse in a right-angled triangle.

The sine function is used to find the unknown angle or sides of a right angled triangle.

Mathematically,

sinФ = p/h

Now ,

Let us assume the phase shift to be on right side.

So the graph will be at negative y axis.

Thus the points will be 0 , -6 .

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Answer:

x+4y=7

Step-by-step explanation:

a) The  two equations in terms of x and y: one for the perimeter and one for the area of the rectangle are:

y = 0.5x + 3.5

6y + 3 = x² + 14x + 48

b) The area and perimeter of the rectangle are:

Perimeter = 30 m

Area = 15 m²

The formula to find the area of a rectangle is:

Area = Length * Width

The formula to find the perimeter of a rectangle is:

Perimeter = 2(Length + Width)

We are given that:

Perimeter = 8y meters

Area = (6y + 3) square meters

From the image, we see that:

Length = x + 6

Width = x + 8

Thus:

Perimeter equation is:

8y = 2(x + 6 + x + 8)

8y = 4x + 28

y = 0.5x + 3.5

Area equation is:

6y + 3 = (x + 6)(x + 8)

6y + 3 = x² + 14x + 48

Thus:

6(0.5x + 3.5) + 3 = x² + 14x + 48

3x + 24 = x² + 14x + 48

x² + 11x + 24 = 0

Using quadratic equation calculator gives:

x = -8 or -3

Thus, we will use x = -3 and we have:

Length = -3 + 6 = 3 m

Width = -3 + 8 = 5 m

Perimeter = 2(3 + 5) = 30 m

Area = 3 * 5 = 15 m²

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a. It will take 7 months to pay off the credit card.

b. it will take 4 months to pay off the credit card.

Since, APR stands for Annual Percentage Rate. It is the interest rate charged on a loan or credit card, expressed as a yearly percentage rate. The APR takes into account not only the interest rate, but also any fees or charges associated with the loan or credit card.

a. If you put half of the available money each month toward the credit card, then you are paying $150.00 per month towards the credit card balance.

We can use the formula for the present value of an annuity to find how many months it will take to pay off the credit card:

PV = PMT × ((1 - (1 + r)⁻ⁿ) / r)

where:

PV is the present value of the debt

PMT is the payment amount per period

r is the monthly interest rate

n is the number of periods

Substituting the values, we get:

754.43 = 150 × ((1 - (1 + 0.011333)⁻ⁿ) / 0.011333)

Simplifying and solving for n, we get:

n = log(1 + (PV ×r / PMT)) / log(1 + r)

n = log(1 + (754.43×0.011333 / 150)) / log(1 + 0.011333)

n = 6.18

Therefore, it will take 7 months to pay off the credit card if you put half of the available money each month toward the credit card.

b. If you put all of the available money each month toward the credit card, then you are paying $300.00 per month towards the credit card balance.

754.43 = 300 ×((1 - (1 + 0.011333)⁻ⁿ) / 0.011333)

Simplifying and solving for n, we get:

n = log(1 + (PV × r / PMT)) / log(1 + r)

n = log(1 + (754.43× 0.011333 / 300)) / log(1 + 0.011333)

n = 3.43

Therefore, it will take 4 months to pay off the credit card if you put all of the available money each month toward the credit card.

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The volume of the tarp shelter is 65.37 ft³ .

Given,

Conic tarp shelter with radius 5.1 ft and height 7.2 ft .

Now,

Volume of cone = 1/3 × π × r² × h

Substitute the values in the formula,

Volume of cone = 1/3 ×3.14 × (5.1)² × 7.2

Volume of cone = 65.37 ft³ .

Hence volume of tarp shelter will be 65.37 ft³ .

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Answer:  A    -6x² + 2x -4

Step-by-step explanation:

[tex]\frac{-12x^{3}+ 4x^{2} -8x}{2x}[/tex]                             >Divide each of the top terms by 2x[tex]=\frac{-12x^{3}}{2x} +\frac{4x^{2} }{2x} -\frac{8x}{2x}[/tex]

= -6x² + 2x -4

The function A(x) = 100 + 4x describes the scenario of an ant colony that starts with an initial population of 100 and experiences a constant rate of increase of 4 ants per year.

We have,

The function A(x) = 100 + 4x represents a linear relationship between the variable x (representing time in this case) and the variable A(x) (representing the population of the ant colony).

The term 100 in the function represents the initial population of the ant colony.

It indicates that at the starting point (x = 0), the population is 100.

The term 4x in the function represents the rate at which the population increases over time. Since the coefficient of x is positive (4), it indicates that the population is increasing.

For every unit increase in x (in this case, for every year that passes), the population increases by 4.

Therefore,

The function A(x) = 100 + 4x describes the scenario of an ant colony that starts with an initial population of 100 and experiences a constant rate of increase of 4 ants per year.

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ΔFSH ≅ ΔFSI by the rule of angle-angle-side theorem, or AAS.

The angle-angle-side theorem, or AAS, tells us that if two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent.

If we consider triangle FSH and triangle FSI, we will observe the following;

So based on the  angle-angle-side theorem, or AAS, we can see that triangle FSH is congruent to triangle FSI.

Thus, our answer will be; ΔFSH ≅ ΔFSI by the rule of angle-angle-side theorem, or AAS.

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Answer:

Step-by-step explanation:

3x3x3= 27

The instantaneous rate of change of y = √x at x = 2 is √2 / 2.

To find the instantaneous rate of change of y = √x at x = 2, we need to calculate the derivative of the function with respect to x and evaluate it at x = 2.

Taking the derivative of y = √x using the power rule for differentiation, we have:

dy/dx = (1/2) × x^(-1/2)

Now, substituting x = 2 into the derivative expression:

dy/dx = (1/2) × 2^(-1/2)

= (1/2) × √2

= √2 / 2

Therefore, the instantaneous rate of change of y = √x at x = 2 is √2 / 2.

None of the provided options (-1/2, -1/4, -1) are the correct instantaneous rate of change at x = 2.

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Answer: Unfactorable

Step-by-step explanation:

5v^2-30v+70

5(v^2-6v+14)

You can't factor v^2-6v+14 with rational numbers

Answer: each base angle in an isosceles right triangle measures 45 degrees.

Step-by-step explanation:

In an isosceles right triangle, two sides have equal lengths, and one angle is a right angle (90 degrees).

Since it is an isosceles triangle, the two base angles (the angles opposite the equal sides) must be congruent, meaning they have the same measure.

To find the measure of each base angle, we can use the fact that the sum of the angles in a triangle is 180 degrees.

Let's denote the measure of each base angle as "x". Then, the sum of the three angles in the triangle can be expressed as:

x + x + 90 = 180

Simplifying the equation:

2x + 90 = 180

Subtracting 90 from both sides:

2x = 90

Dividing both sides by 2:

x = 45

Answer: [tex]3x^4yz^3[/tex]

I'm hoping this is your equation: [tex](9x^{8}y^{2}z^{6})^{1/2}[/tex]

and not: [tex](9x^{8y^{2z^{z^6}}})[/tex]

Step-by-step explanation:

The square root of a number can be shown as a 1/2 power

We'll use the exponent rule:

= [tex]9^{1/2}(x^8 )^{1/2}(y^2)^{1/2}(z^6){1/2}[/tex]

Then we'll do each term individually

the square root of 9 is 3

for [tex](x^8)^{1/2}[/tex] the exponents multiply which give us [tex]x^4[/tex]

[tex](y^2)^{1/2}[/tex] gives us y

[tex](z^6)^{1/2}[/tex] gives us z^3

After doing all this we get [tex]3x^4yz^3[/tex]

Part A:

A(t) is a linear function, and 12∙J(t) is an exponential function. The straight line nature of A(t) indicates linear growth, while the curve of 12∙J(t) suggests exponential growth.

Part B:

The intersection of the 12∙J(t) curve and the A(t) curve represents the point where the accumulated profit from sales is sufficient to pay off the entire loan in one lump-sum payment.

Part C:

As P increases while keeping r and J(t) constant, the relationship between the two curves shifts, indicating a higher profit requirement to cover the increased loan amount. This is dangerous for business financial health and banks have safeguards to prevent excessive debt and defaults.

Part A:

From the given information, we have two functions: A(t) and 12∙J(t). Let's analyze their features to determine their types.

A(t) is a linear function:

A linear function is characterized by a constant rate of change and forms a straight line on a graph.

In the given graph, A(t) is represented by a straight line, indicating a linear relationship between the variables.

This suggests that A(t) is a linear function.

12∙J(t) is an exponential function:

An exponential function is characterized by a constant ratio or base and shows exponential growth or decay.

In the given graph, 12∙J(t) is represented by a curve that exhibits exponential growth.

The increasing rate of change as time progresses indicates an exponential relationship.

Therefore, 12∙J(t) is an exponential function.

Part B:

The intersection of the 12∙J(t) curve and the A(t) curve represents the point at which the profit generated from sales is sufficient to pay off the entire loan in one lump-sum payment. In other words, it represents the point in time when the accumulated profit matches the amount of the loan.

At this intersection point, the profit generated from sales has reached a level where it can fully cover the principal and interest owed on the loan. This indicates that the business has generated enough funds to repay the loan in its entirety.

Part C:

As the principal (P) is increased while keeping the interest rate (r) and 12∙J(t) constant, the relationship between the two curves changes. Specifically, the intersection point between the curves shifts to the right on the graph.

This change in the relationship between the curves signifies that the business needs a higher level of profit to cover the increased loan amount. It suggests that the business has taken on a larger loan, which requires a higher profit to repay.

This situation is considered dangerous for the financial health of the business because it increases the risk of not generating sufficient profit to repay the loan. If the business fails to generate enough profit to cover the loan payments, it may lead to financial instability and potential default on the loan.

To mitigate such risks, banks put safeguards in place to prevent businesses from taking on excessive debt. These safeguards include conducting thorough evaluations of the business's financial health, setting limits on loan amounts based on the business's income and creditworthiness, and assessing the repayment capacity of the borrower. These measures aim to minimize the risk of defaults and protect both the borrower and the lender.

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Step-by-step explanation:

you just have to put that value in place of x in the function.

don't be so lazy you a hole do it yourself

The solution of the given equation is x = -10.

We are given that;

The equation 1/27^{4-x}=9^{2x-1}

Now,

This is an exponential equation that can be solved by using the properties of exponents and logarithms. Here are the steps to solve it:

Rewrite both sides of the equation using the same base. Since 27 and 9 are both powers of 3, we can use 3 as the base. We have:

(3(-3))(4-x) = (32)(2x-1)

Apply the power rule of exponents to simplify the expressions. The power rule states that (ab)c = a^(bc). We have:

3^(-12+3x) = 3^(4x-2)

Since the bases are equal, we can set the exponents equal to each other and solve for x. We have:

-12 + 3x = 4x - 2 -10 = x

Therefore, by the given equation the answer will be x = -10.

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Answer:

Step-by-step explanation:

V = 3.9V

I = 0.12A

Ohm's Law, V = IR

Rearranging Ohm's Law, R = V/I

R = 3.9/0.12 = 32.5Ω

Answer:

Look Below

Step-by-step explanation:

The statement "definition of finiteness uses the notion of a natural number" means that when defining what it means for a set or collection to be finite, the concept of a natural number is employed.

In mathematics, a set is said to be finite if it can be put into a one-to-one correspondence with a specific subset of the natural numbers. This subset typically starts with the number 1 and includes a finite sequence of consecutive natural numbers, depending on the size of the set.

For example, consider a set A with three elements: A = {a, b, c}. To show that A is finite, we can establish a one-to-one correspondence between the elements of A and the set {1, 2, 3} as follows: a ↔ 1, b ↔ 2, c ↔ 3. This mapping shows that each element of A corresponds to a unique natural number, and vice versa. Since there is a bijection (a one-to-one correspondence) between the elements of A and a subset of the natural numbers, we conclude that A is finite.

Thus, in the definition of finiteness, the notion of a natural number is used to provide a precise and rigorous characterization of what it means for a set to be finite. It allows us to establish a clear distinction between finite sets and infinite sets, which do not have a one-to-one correspondence with any subset of the natural numbers.

Answer:

At the time the ball was thrown, it was 8 feet above the ground.

h'(x) = -32x + 64 = 0, so x = 2

The ball reaches its maximum height after 2 seconds.

The description of the image is one that has a four-horned star. The description is as follows:

It comprises of four equilateral triangles and One Square.

Next,

The image or shape has twelve (12) symmetrical lines.

Next,

It must be rotated 90° (clockwise or anticlockwise) in order for the image to match itself.

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May Kawasaki's required minimum distribution (RMD) is $3,808. This is calculated by dividing her IRA balance of $98,000 by the distribution period of 26.2, which is found in the Uniform Life Table on page 45 for a 72-year-old.

b) If May Kawasaki fails to take her RMD, she will incur a 50% penalty on the amount she should have withdrawn. In this case, the penalty would be $1,904.

c) If May Kawasaki had made an early withdrawal of $10,000 to take a vacation, she would have paid a 10% penalty on the amount withdrawn. In this case, the penalty would have been $1,000.

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1. May Kawasaki is 72 and has an IRA with a fair market value of $98,000. a) Use the Uniform Life Table on p. 45 to determine her required minimum distribution. b) What penalty would she incur if she failed to take the distribution? c) What penalty would she have paid if she had made an early withdrawal of $10,000 to take a vacation?

The solution to the equation 3x = 8 - 2x is x = 1.6.

To solve the equation 3x = 8 - 2x by graphing, we can plot the two sides of the equation as functions of x and find the point(s) where they intersect. Here's a step-by-step explanation:

Express the equation in the form of y = f(x). Rearrange the equation:

3x + 2x = 8

5x = 8

x = 8/5 or 1.6

Graph the functions y = 3x and y = 8 - 2x on the same coordinate plane. The line represented by y = 3x is upward sloping, and the line represented by y = 8 - 2x is downward sloping.

Plot the points (1.6, 3(1.6)) and (1.6, 8 - 2(1.6)) on the graph.

The point of intersection represents the solution to the equation. In this case, the lines intersect at (1.6, 4.8).

Therefore, the solution to the equation 3x = 8 - 2x is x = 1.6.

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As you can see below, both expressions result in the same value of 105. This demonstrates the application of the associative law in regrouping terms and maintaining the equivalence of the expression.

The associative law in mathematics states that the grouping of numbers in an addition or multiplication operation does not affect the result. In other words, you can regroup terms within parentheses without changing the value of the expression.

Using the associative law, we can regroup the terms in the expression s + (r + 75) by removing the parentheses and rearranging the terms:

s + (r + 75) = (s + r) + 75

The expression (s + r) + 75 is equivalent to s + (r + 75) because the addition operation is associative.

Let's take an example to illustrate this:

Suppose s = 10 and r = 20.

Using the original expression:

s + (r + 75) = 10 + (20 + 75) = 10 + 95 = 105

Using the expression with regrouped terms:

(s + r) + 75 = (10 + 20) + 75 = 30 + 75 = 105

As you can see, both expressions result in the same value of 105. This demonstrates the application of the associative law in regrouping terms and maintaining the equivalence of the expression.

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One main disadvantage of piece rate or commission pay is low quality.

Piece rate or commision pay involves paying workers based on coverage or performance. This way the company is able to cut short excess salary especially for low performing workers.

However, a notable disadvantage of piece-rate pay is that it can lead to employees sacrificing quality for quantity. When employees are paid based on the number of units they produce, they may be tempted to produce low-quality products in order to meet their quotas. This can damage the company's reputation and lead to customer dissatisfaction.

Therefore, piece rate pay will often lead to quality reduction.

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The length of PQ is 3√5 and its slope is -2

The length of SR is 3√5 and its slope is -2

The length of SP is 5√2 and its slope is -7

The length of RQ is 5√2 and its slope is -1

So PQ ≅ SR and SP ≅ RQ. By the Perpendicular Bisector theorem, adjacent sides are perpendicular. By the selection of  side, ∠PSR, ∠SRQ, ∠RQP and ∠QPS are right angles. So, the quadrilateral is a rectangle.

To find the lengths and slopes of the sides of the quadrilateral PQRS, we apply the distance formula:

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

and the slope formula:

m = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

1. Length PQ:

Using the distance formula, the length PQ can be calculated as follows:

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

  = √((3 - 0)² + (-4 - 2)²)

  = √(3² + (-6)²)

  = √(9 + 36)

  = √45

  = 3√5

2. Length SR:

Using the distance formula, the length SR can be calculated as follows:

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

  = √((1 - (-2))² + (-5 - 1)²)

  = √((1 + 2)² + (-6)²)

  = √(3² + 36)

  = √(9 + 36)

  = √45

  = 3√5

3. Length SP:

Using the distance formula, the length SP can be calculated as follows:

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

  = √((1 - 0)² + (-5 - 2)²)

  = √(1² + (-7)²)

  = √(1 + 49)

  = √50

  = 5√2

4. Length RQ:

Using the distance formula, the length RQ can be calculated as follows:

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

  = √((-2 - 3)² + (1 - (-4))²)

  = √((-2 - 3)² + (1 + 4)²)

  = √((-5)² + 5²)

  = √(25 + 25)

  = √50

  = 5√2

Now, let's calculate the slopes of the sides:

1. Slope PQ:

The slope of PQ can be calculated using the slope formula:

m = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

  = (-4 - 2) / (3 - 0)

  = -6 / 3

  = -2

2. Slope SR:

The slope of SR can be calculated using the slope formula:

m = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

  = (-5 - 1) / (1 - (-2))

  = -6 / 3

  = -2

3. Slope SP:

The slope of SP can be calculated using the slope formula:

m =[tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

  = (-5 - 2) / (1 - 0)

  = -7 / 1

  = -7

4. Slope RQ:

The slope of RQ can be calculated using the slope formula:

m = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex]

  = (1 - (-4)) / (-2 - 3)

  = 5 / (-5)

  = -1

Therefore, the lengths and slopes of the sides of the quadrilateral PQRS are:

Length PQ: 3√5

Length SR: 3√5

Length SP: 5√2

Length RQ: 5√2

Slope PQ: -2

Slope SR: -2

Slope SP: -7

Slope RQ: -1

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