Solve using the quadratic formula. Show all work. Write each solution in simplest form. No decimals.

The rate exists water being poured into the cup when the water level is 9 cm exists [tex]$$1458 \pi \mathrm{cm}^3 s^{-1}$$[/tex].

To determine the derivative of a composite function in differential calculus, apply the chain rule formula. If y = f(g(x)), then according to the chain rule, the instantaneous rates of change of the functions f and g with respect to x produce the instantaneous rate of change of f with respect to x.

A composition of functions, such as the composition of the functions f and g, f(g(x)), can be used to determine the derivative using the chain rule.

Let us set up the following variables:

R, Radius of conical cup (cm) = 30 cm

H, Height of the conical cup (cm) = 10 cm

Our aim is to find dV/dt when h = 9 and we are given dh/dt = 2

By similar triangles we have

r : h = R : H

r/h = 30/10

⇒ r = 3h

The volume of a cone is 1/3π r²h, So:

V = 1/3π r²h

simplifying the equation, we get

= 1/3π (3h²)h

= 1/3π 9h²h

= 3π h³

Differentiating wrt h we get:

dV/dh = 9π h²

Applying the chain rule we have:

dV/dt = (dV/dh)(dh/dt)

= 9π h² × 2

= 18π h²

If h = 9 then

[tex]$\left[\frac{d V}{d t}\right]_{h=9}=18 \pi\left(9^2\right)=1458 \pi \mathrm{cm}^3 s^{-1}$$[/tex].

Therefore, the rate exists water being poured into the cup when the water level 9 cm exists [tex]$$1458 \pi \mathrm{cm}^3 s^{-1}$$[/tex].

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Answer:

Mr. Hurd received a discount of 57% on the trampoline. The cost of the trampoline was $699.99 and he got it for $429.99, resulting in a discount of 57% when rounded to the nearest percent.

Step-by-step explanation:

The graph of the linear function f(x) = 2/(x + 5) is attached.

The general equation of a straight line is -

y = mx + c

{m} - slope.

{c} - intercept along the y - axis.

Given is the linear function as -

f(x) = 5/(2x + 5)

Refer to the graph of the linear function attached. This graph represents the function f(x).

Therefore, the graph of the linear function f(x) = 2/(x + 5) is attached.

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Answer in explanation:

*change the problem to [tex]\frac{5}{6}[/tex] x [tex]\frac{3}{1}[/tex]

(change 3 into a fraction by putting 1 in the place of a denominator)

next cross multiply

5   x   /1

/2       1

3 can go into 6- 1 time

since anything multiplied by 1 is 1, the answer is  [tex]\frac{5}{2}[/tex]  

Hope this helps ya!

*(disclaimer lol: alternatively, you could cross multiply without changing 3 to a fraction and get the same answer-- however, I wanted to explain as clearly as possible, and figured it was less confusing this way!)

The domain of the function f(x) = log x is (0, ∞). The value of x belongs to (0, ∞).

What is a logarithm?

The logarithm is exponentiation's opposite function in mathematics. This indicates that the exponent to which b must be raised in order to obtain a number x is the logarithm of x to the base b.

The given function is f(x) = log x.

When the value of x is less than or equal to 0, then the value of log x is undefined.

The possible value of x where f(x) exists is the domain of a function.

In the given function the value of x must be greater than zero.

The domain of the function is (0,∞).

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The ratio is 1:4, which means for every 1 butterfly, there are 4 caterpillars.

This ratio represents the relationship between the number of butterflies and the number of caterpillars. In this case, the ratio is saying that for every 1 butterfly, there are 4 caterpillars. This could mean that for every butterfly that is observed, there are 4 caterpillars that are also present or for every butterfly that is caught, there are 4 caterpillars that are also caught.

This ratio may also indicate that for every butterfly that is born, 4 caterpillars are also born. It could also indicate the population ratio of butterflies to caterpillars in a specific habitat.

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The value of x will be; x = 3

An equation is an expression that shows the relationship between two or more numbers and variables.

A mathematical equation is a statement with two equal sides and an equal sign in between. An equation is, for instance, 4 + 6 = 10. Both 4 + 6 and 10 can be seen on the left and right sides of the equal sign, respectively.

We are given that  the equation 5x+7=6x+4

Here, we need to solve for x;

5x+7=6x+4

combine the like terms;

5x - 6x = -7 +4

-x= - 3

x = 3

Therefore, the solution will be as;

x = 3

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The two equation to set up to solve the equation |3x - 2| = 19 are

Question 2

x = 5 OR -9

Question 3

x ≥ 6 OR x ≥ -9

Without taking direction into account, absolute value describes how far away from zero a certain number is on the number line.

A number can never have a negative absolute value.

The equation is written in the form below

|3x - 2| = 19

removing the absolute value sign

3x - 2 = ± 19

the two equations are

3x - 2 = 19

3x - 2 = -19

Question 2

solving for x

3|x + 2| - 7 = 14

3|x + 2| = 14 + 7

3|x + 2| =  21

divide through by 3

|x + 2| =  7

removing the absolute value sign

x + 2 =  ±7

solving for x for positive value

x + 2 = 7

x = 5

solving for the negative sign

x + 2 = -7

x = -9

Question 3

-2|2x+3| + 14 ≥ -16

-2|2x+3| ≥ -30

dividing through by -2

|2x+3| ≥ 15

2x+3 ≥ ±15

solving for the positive sign

2x+3 ≥ 15

x ≥ 6

solving for the negative sign

2x+3 ≤ -15

x ≥ -9

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If the population grew 2.75% each year then each year the population was multiplied by 1+2.75%=1+0.0275=1.0275

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113.778=one hundred thirteen million seven hundred seventy-eight thousand, and 11,664=eleven thousand six hundred sixty-four is the enlarged form.

1.Comprehensive Notation Form:

[tex]$$\begin{aligned}& 11,664= \\& 10,000 \\& +1,000 \\& +600 \\& +\quad 60 \\& +4 \\&\end{aligned}$$[/tex]

Increased Factors Form:

[tex]$$\begin{aligned}& 11,664= \\& 1 \times 10,000 \\& +1 \times 1,000 \\& +6 \times 100 \\& +6 x \quad 10 \\& +4 \times \quad 1 \\&\end{aligned}$$[/tex]

Increased Exponential Form:

[tex]$$\begin{aligned}& 11,664= \\& 1 \times 10^4 \\& +1 \times 10^3 \\& +6 \times 10^2 \\& +6 \times 10^1 \\& +4 \times 10^0 \\&\end{aligned}$$[/tex]

Text Format:

11,664=eleven thousand six hundred sixty-four

2. Comprehensive Notation Form:

[tex]$$\begin{aligned}& 113.778= \\& 100 \\& +10 \\& +3 \\& +\quad 0.7 \\& +\quad 0.07 \\& +\quad 0.008 \\&\end{aligned}$$[/tex]

Comprehensive Notation Form:

[tex]$$\begin{aligned}& 113.778= \\& 1 \times 100 \\& +1 \times \quad 10 \\& +3 \times \quad 1 \\& +7 \times \quad 0.1 \\& +7 \times \quad 0.01 \\& +8 \times \quad 0.001 \\&\end{aligned}$$[/tex]

Increased Exponential Form:

[tex]$$\begin{aligned}& 113.778= \\& 1 \times 10^2 \\+ & 1 \times 10^1 \\+ & 3 \times 10^0 \\+ & 7 \times 10^{-1} \\+ & 7 \times 10^{-2} \\+ & 8 \times 10^{-3}\end{aligned}$$[/tex]

Text  Form:

113.778=one hundred thirteen million seven hundred seventy-eight thousand,

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The equation of the line that passes through (7, 6) and (-1, 2) in slope-intercept form is, [tex]y=\frac{1}{2}x + \frac{5}{2}[/tex].

What is slope intercept form of the line?

Using a straight line's slope and the location of its y-axis intersection, the slope intercept equation can be used to determine the general equation of the line. y = mx + b is the equation in slope intercept form.

Given:

The line passes through the points (7, 6) and (- 1, 2).

We have to find the equation of the line that passes through (7, 6) and (-- 1, 2) in slope-intercept form.

First to find the slope of the line using the formula,

Slope = m = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

Here, [tex](x_1, y_1)= (7, 6), (x_2,y_2)=(-1, 2)[/tex]

Plug the values in the above formula.

[tex]m = \frac{2-6}{-1-7} = \frac{-4}{-8}=\frac{1}{2}[/tex]

Now consider the point slope form of the line,

[tex]y-y_1=m(x-x_1)\\y-6=\frac{1}{2}(x-7) \\y-6=\frac{1}{2}x-\frac{7}{2}\\ y=\frac{1}{2}x -\frac{7}{2}+6 \\y=\frac{1}{2}x+\frac{5}{2}[/tex]

Hence, the equation of the line that passes through (7, 6) and (-1, 2) in slope-intercept form is, [tex]y=\frac{1}{2}x + \frac{5}{2}[/tex].

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Answer:

50x - 200y = 75

3x + y = -5

Step-by-step explanation:

Standard form of a line

Ax + By = C

0.5x +-2y -0.75  Multiply through by 100

50x - 200 - 75 = 0  Add 75 to both sides

50x - 200y = 75

y = -3x - 5  Add 3x to both sides

3x + y = -5

If a value of y relates only to one value of x, then the function does have an inverse function.

A relation between such a collection of inputs and outputs is known as a function.

Thus, if a value of y relates only to one value of x, then the function does have an inverse function.

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Therefore, amu (atomic mass unit) and Avogadro's number are not the same things.

A unit of measurement for atomic and molecular weight is the atomic mass unit (amu).

Molecular weight, commonly known as molecular mass, is the mass of a substance's molecules, calculated using the atomic weight of carbon-12, which is 12.

No, amu and Avogadro's numbers are not the same things because An atomic mass unit (amu) is a unit of measurement for atomic and molecular weight. It is defined as one twelfth of the mass of a neutral atom of carbon-12. It is commonly used in chemistry to express the masses of atoms and molecules While Avogadro's number (N) is a fundamental constant of nature that relates the number of atoms or molecules in a sample to the mass of that sample. It is defined as the number of atoms or molecules in one mole of a substance. It is approximately 6.022 x 10^23 atoms or molecules per mole.

So, amu is a unit of measurement for atomic and molecular weight, while Avogadro's number is a constant that relates the number of atoms or molecules in a sample to the mass of that sample.

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The piecewise function represented by the graph is:

f(x) = 2x when -3 ≤ x < 0

f(x) = 3 When 1/2 < x < 3/2

A function is a relationship between inputs where each input is related to exactly one output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

Let f(x) = y = mx + c

From the graph, when -3 ≤ x < 0.

(-3, -6) and (-1, -2)

m = (-2 + 6) / (-1 + 3) = 4 / 2 = 2

(-1, -2) = (x, y)

-2 = 2 x (-1) + c

-2 = -2 + c

c = 0

f(x) = 2x

From the graph, When 1/2 < x < 3/2.

It is a horizontal line of y = 3.

f(x) = 3

Thus,

f(x) = 2x when -3 ≤ x < 0

f(x) = 3 When 1/2 < x < 3/2

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The equation in option e) is not correct because it would not give correct height of the trees in this problem

Since the Magnolia trees are 3 feet tall and grow at an average of 3/4 foot per year.

If x is the number of years and y is the height. we can write:

y = 3/4(x) + 3

The slope-point equation of a line is given as:

y-y1 = m(x−x1)

y = 3/4(x) + 3 in slope-point  form is:

y = 3/4(x) + 3

y-3 =  3/4(x-0)

This means option a) is correct

Yes, this equation is okay to use for the height of the trees in this problem

b) y - 6 = 3/4(x-4)

y - 6 = 3/4(x-4)

y - 6 = 3/4(x)- 3

y - 6 + 3 = 3/4(x)

y - 3 = 3/4(x)

This is correct

c) y - 4.5 = 3/4(x-2)

y - 4.5 = 3/4(x-2)

y - 4.5 = 3/4(x)- 1.5

y - 4.5 + 1.5 = 3/4(x)

y - 3 = 3/4(x)

This is correct

d) y - 12 = 3/4(x-12)

y - 12 = 3/4(x-12)

y - 12 = 3/4(x) - 9

y - 12 + 9 = 3/4(x)

y - 3 = 3/4(x)

This is correct

(e) y - 15= 3/4(x-15)

y - 15= 3/4(x-15)

y - 15= 3/4(x)-11.25

y - 15 + 11.25= 3/4(x)

y - 3.75 = 3/4(x)

This is not correct because it does not reflect the given information

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The general solution of the given differential equation is y(x) = C₁e^-x + 4xe^-x. The largest interval over which the general solution is defined is (-∞, ∞). There are no transient terms in the general solution.

The differential equation given can be rearranged into the form dy/dx + (x + 2)/(x + 1)y = 8xe^-x, which is a linear first order differential equation. Using the integrating factor method, an integrating factor of e^(x + 1) can be used to obtain the general solution y(x) = C₁e^-x + 4xe^-x.

The integration constant C₁ can be determined by supplying the appropriate initial conditions. As the equation is a linear first order differential equation, the general solution is valid over the entire domain of x, which is (-∞, ∞). As the equation does not involve any terms that provide a particular value for y at any point in the domain, there are no transient terms in the general solution.

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The amount of money in the account after 6 years is $68,847.56

To solve this problem, you can use the formula for compound interest,

where V is the final value of the account, P is the initial principal, r is the annual interest rate (expressed as a decimal), and t is the number of years the money is invested.

The formula for compound interest, V = Pe^(rt) is used to calculate the future value of an investment. It takes into account the effect of compounding, which means that interest is earned not only on the initial investment but also on the accumulated interest over time.

Plugging in the given values, we have

V = 51200e^(0.0546) = 51200e^0.324. e^0.324 is approximately 1.38, so the final value of the account is 512001.38 = $68,847.56

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Answer: The function you provided describes an experiment in which a coin is tossed, a number cube is rolled, and a letter is picked from the word "framer". The outcome of the experiment is represented by the ordered triple (tails, 5, m).

P(tails, 5, m) refers to the probability of the outcome (tails, 5, m) occurring. To find this probability, you would need to know the probability of each individual event:

The probability of getting tails is 1/2,

The probability of getting a 5 on a number cube is 1/6

and probability of getting letter "m" in the word "framer" is 1/6

So P(tails, 5, m) = (1/2) * (1/6) * (1/6) = 1/72

It is to be noted that in order for the inverse function to exist, the function must be one-to-one, meaning each y value has one unique x value, meaning the random variables in question should be independent. But in this case the events are dependent on one another.

Step-by-step explanation:

Following are three steps 1. Isolate the variable: First, identify the variable you are trying to solve for and use the other terms in the equation to isolate that variable. This means using inverse operations to move any terms containing the variable to one side of the equation and any terms without the variable to the other side of the equation.

2. Simplify the equation: Once the variable is isolated, use the properties of equality and inverse operations to simplify the equation, if needed.

3. Solve the equation: Finally, use the inverse operations to solve the equation and find the value of the variable.

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Step-by-step explanation:

(7 - 6)/(4 - p)= 1/9

4 - p = 9

-p= 5

p = -5

The value of 5 divided by 2 is 2.5.

Therefore the answer is 2.5.

When dividing 5 by 2, we are trying to find out how many times 2 can fit into 5. The answer is 2 with a remainder of 1, this is why it can also be written as 2 and 1/2.

When dividing 5 by 2, we can also write the result as a fraction, in this case, it is 2 and 1/2 or 2.5 in decimal form. It is the exact same thing as 2.5, just represented in a different way.

It's important to note that, in general, dividing integers will give a decimal number. If you want the result to be a fraction, you will have to express it as such.

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Answer:

1/ 0.8 kilograms

2/ 800 grams

Step-by-step explanation:

We know

1/

Baseball bat = 28 ounces

1 kilogram = 35 ounces

To find weight in kg we take

28 / 35 = 0.8 kg

2/

We know

1 kg = 1000 g

0.8 kg = 800 grams

So, the baseball bat weights 0.8kg or 800 grams

I'm pretty sure the answer is 72 :))

The spherical coordinates of the point are (r, θ, φ) = (4√2, -60°, 45°).

What is a peripheral coordinate?

A peripheral coordinate is a set of three coordinates (r, θ, φ) that describes the position of a point in space in relation to a fixed origin. r is the distance from the origin to the point, θ is the angle between the positive x-axis and the projection of the point onto the xy-plane, and φ is the angle between the positive z-axis and the line connecting the point to the origin.

To find the spherical coordinates of a point whose rectangular coordinates are (-2, 2√3,4), we can use the following formulas:

[tex]r=\sqrt{(x^2 + y^2 + z^2)} = \sqrt{((-2)^2 + (2\sqrt3)^2 + 4^2)} = \sqrt{(4 + 12 + 16)} = \sqrt32 = 4\sqrt2[/tex]

θ = [tex]tan^{-1}(y/x) = tan^{-1}(2\sqrt3/(-2)) = tan^{-1}((-\sqrt3)/1) = -60^o[/tex]

φ = [tex]cos^{-1}(z/r) = cos^{-1}(4/4\sqrt2) = cos^{-1}(1/\sqrt2) = \degree 45 ^o[/tex]

So, the spherical coordinates of the point are (r, θ, φ) = (4√2, -60°, 45°).

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The spherical coordinates of the point are (r, θ, φ) = (4√2, -60°, 45°).

What is a peripheral coordinate?

A peripheral coordinate is a set of three coordinates (r, θ, φ) that describes the position of a point in space in relation to a fixed origin. r is the distance from the origin to the point, θ is the angle between the positive x-axis and the projection of the point onto the xy-plane, and φ is the angle between the positive z-axis and the line connecting the point to the origin.

To find the spherical coordinates of a point whose rectangular coordinates are (-2, 2√3,4), we can use the following formulas:

[tex]r=\sqrt{x^2+y^2+z^2} = \sqrt{(-2)^2+(2\sqrt{3})^2+4^2}=4\sqrt{2}[/tex]

θ = [tex]tan^{-1}(y/x)=tan^{-1}(2\sqrt{3}/(-2))=60^o[/tex]

φ = [tex]cos^{-1}(z/r)=tan^{-1}(4/4\sqrt{4})=45^o[/tex]

So, the spherical coordinates of the point are (r, θ, φ) = (4√2, -60°, 45°).

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The value of P(X<Y) could be p(1−q) / p+q−pq if  X ∼ Geom(p) and Y ∼ Geom(r) are independent

What is probability ?

Probability could be defined as the ratio of number of favourable outcomes and total number of outcomes.

Given ,

X ∼ Geom(p) and Y ∼ Geom(r) are independent.

P(X<Y)=P(⋃1≤k<n,n∈N{X=k,Y=n})

=∑1≤k<n,n∈NP(X=k,Y=n)

=∑n=1∞∑k=1n−1P(X=k)P(Y=n)

=∑n=1∞∑k=1n−1p(1−p)k−1q(1−q)n−1

=pq∑n=1∞(1−q)n−11−(1−p)n−1p

=q∑n=1∞(1−q)n−1(1−(1−p)n−1)

=1−q / ∑n=1∞[(1−q)(1−p)]n−1

=1−q / 1−(1−p)(1−q)

=1−(1−p)(1−q)−q / 1−(1−p)(1−q)

=p(1−q) / p+q−pq.

Hence, The value of P(X<Y) could be p(1−q) / p+q−pq if  X ∼ Geom(p) and Y ∼ Geom(r) are independent.

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Answer: −2<x<0

Step-by-step explanation: 2<x+4<4 2+−4<x+4+−4<4+−4 −2<x<0

The workers in shift B earns the bonus.

Mean of a set of data is defined as the average of all the values. It gives the exact middle point of the data set.

Mean is found by adding all the data values and then dividing with number of values in the set.

Shift A: {77, 91, 82, 68, 75, 72, 85, 65, 70, 79, 94, 86}

Number of data in shift A = 12

Mean = (77 + 91 + 82 + 68 + 75 + 72 + 85 + 65 + 70 + 79 + 94 + 86) / 12

         = 944 / 12

         = 78.67

Shift B: {68, 93, 53, 100, 77, 86, 91, 88, 72, 74, 66, 82}

Number of data in shift B = 12

Mean = (68 + 93 + 53 + 100 + 77 + 86 + 91 + 88 + 72 + 74 + 66 + 82) / 12

         = 950 / 12

         = 79.17

The workers in the shift with the highest mean will earn a bonus. It is shift B.

Hence the workers in shift B earn the bonus.

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The full question is as follows :

The following are daily outputs from shift A and shift B at a factory.

Shift A: {77, 91, 82, 68, 75, 72, 85, 65, 70, 79, 94, 86}

Shift B: {68, 93, 53, 100, 77, 86, 91, 88, 72, 74, 66, 82}

Q. Compare the means of the shift outputs. The workers in the shift with the highest mean will earn a bonus. Which shift will earn the bonus?

The expression for the sentence that is twice the difference of a number and six is given as follows:

2(n - 6).

The sentence for this problem is defined as follows:

"Twice the difference of a number and six."

The number is unknown, hence the variable that represents the number is given as follows:

n.

The difference between the number and six is given by the subtraction of n by 6, as follows:

n - 6.

(the - symbol represents the subtraction).

The expression twice is equivalent to a multiplication of the expression n - 6 by 2, as follows:

2(n - 6).

(the multiplication symbol is implicit before the parenthesis, meaning that the distributive property is applied).

Which is the expression that represents the sentence given at the beginning of the answer.

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The value of p for the given logarithmic function  -log(2.1x10^12) is, -8.678  

The logarithmic function is an expression which we can write as, y = logₐx, where a>0 & x>0, It is the inverse of exponential function [tex]x = a^y[/tex].

Formulae for logarithmic function are,

LogAB = LogA + LogB         .................(i)

Logₐaᵇ = b                             ................(ii)

Given expression,

p = -log(2.1x10^12)

p = -log2.1 + log10^12  (by applying formula i )

p = -log2.1 + 12             (by applying formula ii )

p =  -(-3.322) + 12        (log2.1 = -3.322)

p =  -8.678

Hence, the value of p is equal to -8.678

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