What are 2 examples of a solution?

The equations which show direct proportion between a and b are:

a=2b and a=b/2.

Direct proportion or direct variation is the relation between two quantities where the ratio of the two is equal to a constant value. It is represented by the proportional symbol, ∝. In fact, the same symbol is used to represent inversely proportional, the matter of the fact that the other quantity is inverted here.

Example: x and y are two quantities or variables which are linked with each other directly, then we can say x ∝ y. When we remove the proportionality symbol, the ratio of x and y becomes equal to a constant, such as x/y = C, where C is a constant.

Given, two variables a and b

If two variables a and b are in direct proportion

then,

a/b = constant.

1. a = b²

a/b = b

which is not constant

⇒ a is not in direct proportion with b.

2. a = 2b

a/b = 2

which is constant

⇒a is in direct proportion with b.

3. a = b/2

a/b = 1/2

which is constant

⇒ a is in direct proportion with b.

4. a=b+2

a/b = 1+2/b

which is not constant

⇒ a is not in direct proportion with b.

5. a = 2/b

ab = 2

⇒a is inversely proportional to b.

Hence, In equation a=2b and a=b/2, a is directly proportional to b.

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A positive second derivative means that the solution curve is concave up. Therefore, all solution curves for the given differential equation in Quadrant II are concave up.

(b) To find the second derivative of y with respect to x, we can use the chain rule. Since y is a function of x, we have [tex]dy/dx = f(x) and d^2y/dx^2 = d/dx(dy/dx) = d/dx(f(x)) = f'(x).[/tex]

To find f'(x), we can use the product rule and the given differential equation:

[tex]f'(x) = d/dx(xy^4) = y^4 + xy^4dy/dx = y^4 + xy^4f(x)[/tex]

Now, to determine the concavity of the solution curves, we need to find the sign of f'(x).

In Quadrant II, x and y are both positive, so [tex]x*y^4[/tex] is positive. Therefore, f'(x) is positive as well.

A positive second derivative means that the solution curve is concave up. Therefore, all solution curves for the given differential equation in Quadrant II are concave up.

(c) To find a particular solution to the differential equation with the given initial condition, we can use separation of variables.

The equation can be rewritten as [tex]dy/y^4 = x dx[/tex]

Integrating both sides gives:

[tex]\int\limits^ {} \,dy/y^4 = \int\limits^ {} \, dx* dx\\-1/y^3 = (x^2)/2 + C\\y^3 = -1/(x^2/2 + C)\\y = (-1/(x^2/2 + C))^(1/3)[/tex]

Using the initial condition f(4) = -1, we can find the value of C:

[tex](-1) = (-1/(4^2/2 + C))^(1/3)\\(-1)^3 = -1/(4^2/2 + C)\\-1 = -1/(16/2 + C)\\C = 15/2[/tex]

So the particular solution is:

[tex]y = (-1/(x^2/2 + 15/2))^(1/3)[/tex]

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Complete question is: Consider the differential equation dy/dx=xy^4.

(b) Find ⅆ2yⅆx2 in terms of x and y. Determine the concavity of all solution curves for the given differential equation in Quadrant II. Give a reason for your answer.

(c) Find the particular solution y=f(x) to the given differential equation with initial condition f(4)=−1.

According to the compound interest formula, the population is 5 years is 422,385.

The term compound interest in math is defined as the interest calculated on the principal and the interest accumulated over the previous period.

Here we have given that the population of 210,000 people is increasing by 15% each year.

And then while looking into the given question, we have identified that the value of

interest rate = 15%

principal count = 210,000

time period = 5

Here we have to convert R as a percent to r as a decimal, then we get

=> r = R/100

=> r = 15/100

=> r = 0.15 rate per year,

Now, we have to solve the equation for A

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

Apply the values on it , then we get,

[tex]A = 210,000.00(1 + \frac{0.15}{1} )^{(1)(5)}[/tex]

When we simplify this one then we get

[tex]A = 210,000.00(1 + 0.15)^5[/tex]

Therefore, the value of A is 422,385.

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The requried inequality to determine the range of body temperature is given as 98.8 ≤ x ≤ 101.2.

Inequality can be defined as the relation of the equation containing the symbol of ( ≤, ≥, <, >) instead of the equal sign in an equation.

Here,
The normal body temperature for an adult horse is 100 degrees F, but it can vary by as much as 1.2 degrees.
Let x be the average temperature of the horse,
100 - 1.2 ≤ x ≤ 100 + 1.2

98.8 ≤ x ≤ 101.2

Thus, the requried inequality determines the range of body temperature is given as 98.8 ≤ x ≤ 101.2.

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Answer:

NEED:

WANT:

Step-by-step explanation:

Right on Plato! <3

Have a wonderful day!

The needs listed are tuition and fees and staple foods, while the wants are a vacation, a Lamborghini, and a lavish wedding.

The listed needs and wants reflect different aspects of a person's life and priorities. "Tuition and Fees" and "Staple Foods" are necessities that are essential for a person's survival and well-being.

On the other hand, "Vacation," "Lamborghini," and "Lavish Wedding" are desires that are often seen as luxury items or experiences. While having wants is normal and human, it is important to understand that not all wants can be fulfilled, and it is equally important to prioritize needs over wants. Having a balanced approach to spending, budgeting, and financial planning can help ensure that both needs and wants can be met in a sustainable manner.

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Let l be the length of the classroom in feet, and w be the width of the classroom in feet. We know that l + 5 = w and l * w = 84.

We can use these two equations to solve for the length and width of the classroom.

First, we can solve the second equation for l:

l = 84 / w

Substituting this expression for l into the first equation gives us:

84 / w + 5 = w

Expanding the left side of the equation gives us:

84 / w + 5 = w

84 / w = w - 5

84 = w^2 - 5w

w^2 - 5w - 84 = 0

This is a quadratic equation that we can solve using the quadratic formula:

w = (5 + sqrt(5^2 - 4 * 1 * -84)) / 2 * 1

= (5 + sqrt(25 + 336)) / 2

= (5 + sqrt(361)) / 2

= (5 + 19) / 2

= 24 / 2

= 12

Now that we know the value of w, we can substitute it back into the equation l = 84 / w to find the value of l:

l = 84 / 12

= 7

Therefore, the dimensions of the classroom are 7 feet by 12 feet.

Answer:

tree = 8

snowman = 1

deer = 4

snowflake = 4

Step-by-step explanation:

say t = tree, s = snowman, d = deer, f = snowflake

1. For 2d + f, since you know d is equal to f, replace the snowflake with a deer, turning the equation into 3d = 12. To find the value of d divide by 3 on both sides. This will result in 4. So, the value of deer = 4.

2. Now, find the value of the snowman using the value of the deer.

4-3 = 1. This means the value of the snowman = 1.

3. Plug in the value of the snowman into 2t + s = 17.

     2t + 1 = 17

  Subtract on both sides to get 2t by itself and then simplify

      2t = 17 -1 --> 2t = 16

  Divide by 2 on both sides to find the value of the tree

      2t/2 = 16/2 --> t = 8

An inequality involving absolute values that describes the range of possible thickness for the laminate.

(a) The range of possible thickness for the laminate is given by the tolerance of 0.004 in., so we can write an inequality involving absolute values as:

|h - 0.016| <= 0.004

(b) To solve the inequality, we need to consider two cases:

Case 1: h - 0.016 <= 0.004

             h <= 0.02

Case 2: h - 0.016 >= -0.004

             h >= 0.012

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Answer:

Step-by-step explanation:

analyze the graph, we found ∠1=∠5,∠2=∠6,

one.because ∠5+∠6=180, so ∠2=∠6=180-∠5=180-55=125

two.∠3=∠2=125

three.∠8=∠5=55

four.∠12=∠9=80

five.∠14=∠10=180-∠9=100

so, the three options are ∠2,∠8,∠14

Yes, a triangle whose sides have lengths 10.2 cm, 5.8 cm and 4.5 cm is possible.

As we know if these sides form a triangle, then

the sum of two sides > the third side

Now, the sum of 10.2 cm and 5.8 cm = 16 cm > 4.5 cm

The sum of 5.8 cm and 4.5 cm = 10.3 cm > 10.2 cm

The sum of 10.2 cm and 4.5 cm = 14.7 cm > 5.8 cm

Here the sum of the two sides is greater than the third side in all three cases.

Therefore a triangle whose sides are 10.2 cm, 5.8 cm and 4.5 cm is possible.

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Answer: y = -1/2x -4

Step-by-step explanation:

To solve for y, subtract 2x on both sides

2x-2x + 4y = -16 - 2x

Simplify

4y = -16 - 2x

Divide by 4 on both sides to get y by itself

4y/4 = -16 - 2x/4

Simplify

y = -4 - 1/2x

Switch the right-side terms to write them in slope-intercept form

y = -1/2x -4

Answer: 12 cups of applesauce

Step-by-step explanation:

Answer:

One way is to make a factor tree. She below for examples.

Step-by-step explanation:

Answer:

You basically just count jumps of that number

Answer:

11√12

Step-by-step explanation:

√12 = 2√3

2√12

= 2 * 2√3

= 4√3

√48 = 4√3

3√48

= 3 * 4√12

= 12√12

√75 = 5√3

2√12 + 3√48 - √75

= 4√3 + 12√12 - 5√12

= 16√12 - 5√12

= 11√12

Answer: My answer is 11√3

Step-by-step explanation:

Pull terms out from under the radical, assuming positive real numbers.

Exact Form:

11√3

Decimal Form:

19.05255888…

Answer:

1. 15

2. 15

3. 13

4. 11

5. 14

6. 11

7. 18

8. 19

9. 18

10. 14

Step-by-step explanation:

The formula for a directed line segment is A = (x2 - x1, y2 - y1). It is a vector equation, meaning that it describes the direction and magnitude of a line segment, rather than its endpoint coordinates.  

A directed line segment is a line segment with a starting point and an ending point, where the direction of the line can be described in terms of its magnitude and direction. To calculate the directed line segment, first calculate the difference between the x-coordinates of the start and end points, and then calculate the difference between the y-coordinates of the start and end points. The result is the vector A, which describes the magnitude and direction of the line segment.

For example, consider a line segment with start point (1, 2) and end point (4, 6). To calculate the directed line segment, first calculate the difference between the x-coordinates of the start and end points (4 - 1 = 3). Then, calculate the difference between the y-coordinates of the start and end points (6 - 2 = 4). The result is the vector A = (3, 4). This vector describes the magnitude and direction of the line segment, with the magnitude being the length of the line segment, and the direction being the direction from the start point to the end point.

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The third side of a triangle, the following formula can be used c² = a² + b² - 2abcos(C)

To calculate the third side of a triangle, first find the lengths of the other two sides and the angle opposite the third side. Then plug these values into the formula above and solve for c.

Where c is the third side of the triangle, a and b are the other two sides, and C is the angle between them.

For example, for a triangle with two sides of length 4 and 6, and an angle between them of 75 degrees, the formula would look like this:

c² = 42 + 62 - 2(4)(6)cos(75)

Solving for c yields a third side length of 8.4.

To solve for c, first square the two side lengths: 42 = 16 and 62 = 36. Then, multiply the two side lengths together and multiply by the cosine of the angle between them: 2(4)(6)cos(75) = 43.9. Finally, add the two squared values together and subtract the result of the multiplication: 16 + 36 - 43.9 = 8.1. Then, take the square root of this result to get the length of the third side: √8.1 = 8.4.

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Answer:

y = - 3

Step-by-step explanation:

Call the line y = a + bx (d)

Since d go through (-7;-3) so -7 = a - 3b (1)

Since d go through (8;-3) so 8 = a - 3b (2)

From (1) and (2), we have equations: [tex]\left \{ {{a - 3b = -7} \atop {a - 3b = 8}} \right.[/tex]

Solve the equations, we have [tex]\left \{ {{a = 0} \atop {b = -3}} \right.[/tex]

The number of miles the taxicab traveled is 12.

The area of mathematics known as algebra aids in the representation of circumstances or problems into mathematical expressions. To create a meaningful mathematical expression, it takes variables like x, y, and z together with mathematical operations like addition, subtraction, multiplication, and division.

Given that,

The fixed charges = $ 2.30

Additional charges per miles = $ 0.80,

Thus, the additional charges for x number of miles = $ 0.80x

So, the total charges for x number of miles = Fixed charges + additional charges

=> 2.30 + 0.80x

According to the question,

2.30 + 0.80x = 11.90 (this is the required equation)

Subtracting 2.30 from both sides,

0.80x = 9.6

Dividing both sides by 0.80

x = 12

Hence, the taxicab traveled 12 miles.

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Answer:

D)  (6.6, 7.4)

Step-by-step explanation:

Begin by finding the margin of error based on the given information.

[tex]\boxed{\begin{minipage}{7.3cm}\underline{Margin of Error formula}\\\\$\textsf{Margin of Error}= Z \times \dfrac{S}{\sqrt{n}}$\\\\where:\\\phantom{ww} $\bullet$ $Z =$ $Z$ score\\\phantom{ww} $\bullet$ $S =$ Standard Deviation of a population\\\phantom{ww} $\bullet$ $n =$ Sample Size\\\end{minipage}}[/tex]

Given:

The z-score for 95% confidence level is 1.96.

Therefore, to find the margin of error, substitute the values into the formula:

[tex]\implies \textsf{Margin of Error}= 1.96 \times \dfrac{1.3}{\sqrt{50}}[/tex]

[tex]\implies \textsf{Margin of Error}=0.360341...[/tex]

To find a reasonable range for the true mean number of hours a teenage spend on their phone, subtract and add the margin of error to the given mean of 7 hours:

[tex]\begin{aligned}\implies \textsf{Lower bound}&=7-0.360341...\\&=6.63965...\\&=6.6\;\; \sf (1\;d.p.)\end{aligned}[/tex]

[tex]\begin{aligned}\implies \textsf{Upper bound}&=7+0.360341...\\&=7.360341...\\&=7.4\;\; \sf (1\;d.p.)\end{aligned}[/tex]

Therefore, the reasonable range for the true mean is:

[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{1}~,~\stackrel{y_1}{-6})\qquad (\stackrel{x_2}{-8}~,~\stackrel{y_2}{-4})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d=\sqrt{(~~-8 - 1~~)^2 + (~~-4 - (-6)~~)^2} \implies d=\sqrt{(-8 -1)^2 + (-4 +6)^2} \\\\\\ d=\sqrt{( -9 )^2 + ( 2 )^2} \implies d=\sqrt{ 81 + 4 } \implies d=\sqrt{ 85 }\implies d\approx 9.22[/tex]

According to a recent study of 35 students,Population-wide, there was a 2.8-hour standard deviation. 16.60 is the population mean's 95% confidence interval.

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The system of equations is inconsistent for any value of k.

If we subtract the first equation from the second one, we get 0 = 12x, which is inconsistent no matter what value k is. This means that the system of equations has no solution and is therefore inconsistent.

The system of equations KX + 3y = K + 2 and 12x + Ky = K is inconsistent for any value of k. This is because when we subtract the first equation from the second one, we get 0 = 12x, which is an inconsistent equation no matter what value k is. This means that the system of equations has no solution and is therefore inconsistent. No matter what value we assign to k, the equations cannot be solved simultaneously and so the system of equations is inconsistent.

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According to the exponential function,

a) The function that models the given situation is f(rabbits) = 65000(√500/13)ˣ

b) The number of rabbits in 1870 is 6.5

c) According to your model, the first pair of rabbits introduced into Australia is 19th century

Here we have given that during the 19th century, here rabbits were brought to Australia.

And here we also know that the rabbits had no natural enemies on that continent, their population increased rapidly.

And we have given that there were 65,000 rabbits in Australia in 1865 and 2,500,000 in 1867.

Then according to the exponential function that could be used to model the rabbit population y in Australia in terms of x, the number of years since 1865 is to be determined.

Then the exponential function can be written as,

=> f(rabbits) = 65000(√500/13)ˣ

When we plot these on the graph then we get the graph like the following.

Through the graph we have identified that the value of rabbits in 1870 is 6.5

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Answer: The answer is $1,317.50

Step-by-step explanation:

First, Find how much they earn in the goods sale.

Make $1,700, their goal to earn, into 1,700/1 and set it to multiply that by 2/5, the fractional amount of money earnt at the goods sale proportional to how much they have to earn.

Simplify that by crossing 1,700 and 5 into a 340 and 1.

Multiply 340/1 by 2/1 to get 680/1.

Make that into a whole number ($680) to show how much money earnt from the goods sale.

Secondly, find how much money they earn by the Bingo Night

Make 1,700, like last time, into 1,700/1 and set it to multiply 3/8, the fractional amount of money earnt from the Bingo Night proportional to how much they have to earn

You can't simplify, so just multiply the amount.

If you multiply, you will get 5,100/8.

Divide 5,100 by 8 (the divisor by the dividend).

Make sure to also add the 0 after the tenths place after you get your quotient, which is supposed to be 637.5 (or $637.50)

Lastly, You have to add them up.

Add $637.50+$680.

You shall get $1,317.50

Answer:

6 tubes of lip balm

Step-by-step explanation:

We can start solving this problem by isolating the variable x.

Given the equation 1.5x + 2.5y = 9, and we know that y = 0 (because we're not buying any bottles of hand sanitizer), we can substitute this into the equation:

1.5x + 2.5(0) = 9

Simplifying this, we get:

1.5x = 9

To solve for x, we can divide both sides of the equation by 1.5:

x = 9/1.5

x = 6

So we can buy 6 tubes of lip balm if we do not buy any bottles of hand sanitizer

1. The Riemann Hypothesis Equation

At the rate of 126cm/s is the area of the square increasing when the area of the square is 81cm2.

If each side of a square is increasing at a rate of 7 cm/s.

The area of the square increasing rate,

Area, A = s2

each side of the square is increasing at the rate of 7cm / s (= dx / dt).

s is the side length

=> dA/dt = 2s. ds/ dt

ds/dt = 7cm/s

Area of the square = 81 cm2

81 cm2 = s2

s = 9 cm

so, dA /dt = 2. 9. 7

               = 126 cm2/s

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