Write a differential equation to represent each situation below. Do NOT solve them. a. A new technology is introduced into a community of 5000 people. If the rate at which the technology is adopted in the community is jointly proportional to the number of people who have adopted the technology and the number of people who have not adopted it, write a differential equation to represent the number of people, x(t), who have adopted the technology by time t. b. A tank with a capacity of 1000 gal originally contains 800 gal of water with 200 lbs of salt in the solution. Water containing 3.5 lbs of salt per gal is entering at a rate of 2 gal/min, and the mixture is allowed to flow out of the tank at a rate of 5 gal/min. Write a differential equation reflecting the information above, clearly stating the requested intermediate results below. Let A(t) represent the amount of salt (in pounds) in the tank after t minutes. Do NOT solve the DE! Show units with each factor in the three setup steps below, and simplify each expression, showing the resulting units as well. *Be sure to include the initial conditions for this DE in your final equation. R_in = Concentration of salt in the tank: c(t) = R_out = Differential equation: A(0) =

The answer is:

Center: (4,3)

Radius: 3 units.

To solve the problem, using the given formula of a circle, we need to find its standard equation form which is equal to:

[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]

Where,

"h" and "k"are the coordinates of the center of the circle and "r" is its radius.

So, we need to complete the square for both variable "x" and "y".

The given equation is:

[tex]x^2+y^2-8x-6y+16=0[/tex]

So, solving we have:

[tex]x^2-8x+y^2-6y=-16[/tex]

[tex](x^2-8x+(\frac{8}{2})^{2} )+(y^2-6y+(\frac{6}{2})^{2})=-16+(\frac{8}{2})^{2}+(\frac{6}{2})^{2}\\\\(x^2-8x+16)+(y^2-6y+9)=-16+16+9\\\\(x^2-4)+(y^2-3)=9[/tex]

Now, we have that:

[tex]h=4\\k=3\\r=\sqrt{9}=3[/tex]

So,

Center: (4,3)

Radius: 3 units.

Have a nice day!

Note: I have attached a picture for better understanding.

By completing the square on the given equation, the center of the circle is found to be (4, 3) and the radius is 3.

This process will transform the equation into a standard form of a circle equation, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius of the circle.

To start, we can rewrite the original equation by adding and subtracting the necessary constants inside the square terms:

Therefore, after completing the square, we get (x - 4)^2 + (y - 3)^2 = 1. Hence, the center of the circle is (4, 3) and the radius is 3.

Answer:

  m∠XQZ = 72°

Step-by-step explanation:

We presume the "if ..." condition is intended to mean that ray QY lies between rays QX and QZ. Then by angle addition, you get

  ∠XQY + ∠YQZ = ∠XQZ

  (4a +8) + (28) = (8a)

  36 = 4a . . . . . . . . . . . . . subtract 4a

  72 = 8a = ∠XQZ . . . . multiply by 2

Answer:

(x-9)^2 + (y+3)^2 = 8^2

or

(x-9)^2 + (y+3)^2 = 64

Step-by-step explanation:

An equation for a circle can be written as

(x-h)^2 + (y-k)^2 = r^2  where (h,k) is the center and r is the radius

(x-9)^2 + (y- -3)^2 = 8^2

(x-9)^2 + (y+3)^2 = 8^2

or

(x-9)^2 + (y+3)^2 = 64

Answer:

\frac{1}{2} \left[\begin{array}{ccc}16&6&2\\20&8&2\\7&3&1\end{array}\right]

Step-by-step explanation:

Given is a matrix 3x3 as

[tex]\left[\begin{array}{ccc}1&0&2\\-3&1&4\\2&-3&4\end{array}\right][/tex]

|A| =2 hence inverse exists.

Cofactors are 16   20   7

                        6     8    3

                         2     2    1

Hence inverse =

[tex]\frac{1}{2} \left[\begin{array}{ccc}16&6&2\\20&8&2\\7&3&1\end{array}\right][/tex]

[tex]\vec F(x,y,z)=e^{xy}\sin z\,\vec\jmath+y\tan^{-1}\dfrac xz\,\vec k[/tex]

Divergence is easier to compute:

[tex]\mathrm{div}\vec F=\dfrac{\partial(e^{xy}\sin z)}{\partial y}+\dfrac{\partial\left(y\tan^{-1}\frac xz\right)}{\partial z}[/tex]

[tex]\mathrm{div}\vec F=xe^{xy}\sin z-\dfrac{xy}{x^2+z^2}[/tex]

Curl is a bit more tedious. Denote by [tex]D_t[/tex] the differential operator, namely the derivative with respect to the variable [tex]t[/tex]. Then

[tex]\mathrm{curl}\vec F=\begin{vmatrix}\vec\imath&\vec\jmath&\vec k\\D_x&D_y&D_z\\0&e^{xy}\sin z&y\tan^{-1}\frac xz\end{vmatrix}[/tex]

[tex]\mathrm{curl}\vec F=\left(D_y\left[y\tan^{-1}\dfrac xz\right]-D_z\left[e^{xy}\sin z\right]\right)\,\vec\imath-D_x\left[y\tan^{-1}\dfrac xz\right]\,\vec\jmath+D_x\left[e^{xy}\sin z}\right]\,\vec k[/tex]

[tex]\mathrm{curl}\vec F=\left(\tan^{-1}\dfrac xz-e^{xy}\cos z\right)\,\vec\imath-\dfrac{yz}{x^2+z^2}\,\vec\jmath+ye^{xy}\sin z\,\vec k[/tex]

To find the curl and divergence of a given vector field, you first identify the vector's components. The curl is calculated using a determinant and the divergence is obtained by computing a dot product of the gradient operator with the vector field.

In order to find the curl and the divergence of the vector field F(x, y, z) = e^(xy) sin z j + y tan^−1(x/z)k, we first need to identify its components. The components are as follows: e^(xy) sin z, and y tan^−1(x/z).  

The Curl of a vector field F in three dimensions is typically denoted as ∇ × F or curl F, where '∇' is the del operator. In Cartesian coordinates, this can be calculated using a determinant that involves the unit vectors î, ĵ, and k, the gradient operator, and the components of F.  

The Divergence of a vector field F in three dimensions, typically denoted as ∇ . F or div F, is obtained by computing a dot product of the gradient operator with the vector field. This can also be calculated using Cartesian coordinates.

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Final answer:

By using the divisibility rules for 9 and 11, we can conclude that the integer 176521221 is divisible by 9 but not by 11, and the integer 149235678 is divisible by 9 but not by 11.

Explanation:

To determine whether the integers 176521221 and 149235678 are divisible by 9 or 11 without performing the division, we can use divisibility rules:

Divisibility by 9: Add up all the digits in the number. If the sum is divisible by 9, then the original number is also divisible by 9.

Divisibility by 11: Take the alternating sum of the digits. If the result is divisible by 11 (including 0), the original number is divisible by 11.

Divisibility of 176521221 by 9

1+7+6+5+2+1+2+2+1 = 27; Since 27 is divisible by 9, 176521221 is also divisible by 9.

Divisibility of 176521221 by 11

(1-7+6-5+2-1+2-2+1) = -3; Since -3 is not divisible by 11, 176521221 is not divisible by 11.

Divisibility of 149235678 by 9

1+4+9+2+3+5+6+7+8 = 45; Since 45 is divisible by 9, 149235678 is also divisible by 9.

Divisibility of 149235678 by 11

(1-4+9-2+3-5+6-7+8) = 9; Since 9 is not divisible by 11, 149235678 is not divisible by 11.

Answer:

Hyperbola.

Step-by-step explanation:

We start from the function:

[tex]x^{2} +y^{2} -z^{2} =36[/tex]

We may get the traces if we cut the surface in planes x=k. This is equivalent to replace x for k:

[tex]k^{2}+y^{2} -z^{2}=36\\y^{2} -z^{2}=36-k^{2}[/tex]

The last equation is the equation of an hyperbola, which varies its characteristics depending on K (the plane cut).

When x=k or y=k is substituted into the equation of the quadric surface x² + y² - z² = 36, the resulting traces are hyperbolas. These traces represent intersections of planes parallel to coordinate planes with the quadric surface.

The trace of a quadric surface can be found by substituting a constant into the equation for one of the variables. The given quadric surface equation is x² + y² - z² = 36. If we substitute x = k into the equation, we get k² + y² - z² = 36. This is an equation of a hyperbola.

Similarly, for the case when y = k, we substitute this into our original equation to get x² + k² - z² = 36. This also represents a hyperbola.

The traces given by these equations represent the intersection of planes parallel to the coordinate planes and the quadric surface. In these particular cases, the traces are hyperbolas.

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Try this suggested solution.

Answer:

The company use 0 ounces of chicken and 100 ounces of grain in each bag of dog food in order to minimize cost.

Step-by-step explanation:

Let x be the number of ounces of chicken

let y be the number of ounces of grains

We are given that Chicken has 10 grams of protein and 5 grams of fat per ounce.

Chicken has protein in 1 ounce = 10

Chicken has protein in x ounces = 10x

Chicken has fat in 1 ounce = 5

Chicken has fat in x ounces = 5x

We are also given that grain has 2 grams of protein and 2 grams of fat per ounce

Grain has protein in 1 ounce = 2

Grain has protein in y ounces = 2y

Grain has fat in 1 ounce = 2

Grain has fat in y ounces =2y

Now we are given that . A bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat.

So, equation becomes:

[tex]10x+2y\geq 200[/tex] ---A

[tex]5x+2y\geq 150[/tex]  ---B

Since x and y represent the ounces .

So, [tex]x\geq 0[/tex] ---C and [tex]y\geq 0[/tex] ----D

Plot A ,B ,C and D on the graph

Refer the attached graph so , the corner points are (0,100),(10,50) and (30,0)

We are also given that chicken costs 10 cents per ounce and grain costs 1 cent per ounce

So, Cost function becomes : [tex]c=10x+y[/tex]

At(0,100)

[tex]c=10x+y[/tex]

[tex]c=10(0)+100[/tex]

[tex]c=100[/tex]

At(10,50)

[tex]c=10x+y[/tex]

[tex]c=10(10)+50[/tex]

[tex]c=150[/tex]

At(30,0)

[tex]c=10x+y[/tex]

[tex]c=10(30)+0[/tex]

[tex]c=300[/tex]

Minimum cost is 100 cents at (0,100)

So,  the company use 0 ounces of chicken and 100 ounces of grain in each bag of dog food in order to minimize cost.

Answer: Option (d) is correct.

Step-by-step explanation:

Regression analysis can be used to determine if there is any linear relationship exists between the two quantitative variables.

There are one dependent variable and one or more than one independent variable in a single regression equation.

After running a simple regression, we get to know the relationship between the dependent variable and explanatory variable.

Various statistical software are used for running regression like STATA, E- Views, SPSS, etc.  

Regression analysis is used primarily to determine if a linear relationship exists between two quantitative variables. While it can identify relationships between variables, it does not imply causation. It does not establish a relationship between two categorical variables or if the difference in the population means may be zero.

Regression analysis is a statistical methodology often used in mathematics and related fields. Primarily, it is used to determine if a linear relationship exists between two quantitative variables.

For example, you could use regression analysis to see if there is a linear relationship between the age of a car (quantitative variable) and the distance it can travel on a tank of gas (another quantitative variable).

It's important to note however that while regression analysis can identify a correlation or relationship between variables, it does not necessarily imply causation, or a cause-and-effect relationship. There could be other related variables causing the effect.

For your options, regression analysis can't be used to determine if a linear relationship exists between two categorical variables or if the difference in the population means may be zero.

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a) After 30 years, the amount in the account will be approximately $499,355.18.

b) The total money deposited over 30 years will be $144,000.

c) The total interest earned over 30 years will be approximately $355355.18.

We have,

Given:

Monthly deposit: $400

Interest rate: 5% (expressed as a decimal, 0.05)

Time: 30 years (in months, 30 * 12 = 360 months)

a.

To calculate the amount in the account after 30 years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = Amount in the account

P = Initial deposit (first deposit)

r = Annual interest rate

n = Number of times interest is compounded per year

t = Time in years

In this case:

P = $400 x 360 = $144000

r = 0.05

n = 12 (compounded monthly)

t = 30

Substituting the values into the formula:

A = 144000(1 + 0.05/12)^(12 * 30)

A ≈ $499,355.18

b.

The total money deposited can be calculated by multiplying the monthly deposit by the number of months:

Total Money Deposited = Monthly deposit * Number of months

Total Money Deposited = $400 * 360

Total Money Deposited = $144,000

c.

The total interest earned can be calculated by subtracting the total money deposited from the amount in the account:

Total Interest Earned = Amount in the account - Total Money Deposited

Total Interest Earned = $499,355.18 - $144,000

Total Interest Earned ≈ $355355.18

Therefore,

After 30 years, the amount in the account will be approximately $499,355.18.

The total money deposited over 30 years will be $144,000.

The total interest earned over 30 years will be approximately $355355.18.

Learn more about compound interest here:

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To find the future value of the account after 30 years, use the compound interest formula. Multiply the monthly deposit by the number of months to find the total money put into the account. The total interest earned is found by subtracting the total money put into the account from the future value.

To calculate the future value of the account, we can use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the future value, P is the principal (initial deposit), r is the interest rate in decimal form, n is the number of times interest is compounded per year, and t is the number of years.

a. Plugging in the values, we have [tex]A = 400(1 + 0.05/12)^(12*30)[/tex]. Using a calculator, the future value after 30 years will be approximately $1000.40.

b. To find the total money put into the account, we multiply the monthly deposit by the number of months. In this case, it will be $[tex]400 * 12 * 30 = $144,000.[/tex]

c. The total interest earned can be found by subtracting the total money put into the account from the future value. In this case, it will be $[tex]1000.40 - $144,000 = -$143,999.60.[/tex]

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Answer:

answer 1

Step-by-step explanation:

Since there's only 1 value in the 90-99 groups it is either 1 or 3.

Furthermore, there are only 2 values starting with 10.

This only fits with answer 1

The answer is A.

A stem plot shows every number with the tens place and above being on the left side while the ones place is on the right side.

For example,

9 is in the tens place while 4 is in the ones place. So, this would look like:

9 | 4

There is only one number with a "9" in the tens place and there is only two numbers with "10" in the tens place. This only sastisfys the first option.

Best of Luck!

Answer:

Yes , S is a basis for [tex]P_3[/tex].

Step-by-step explanation:

Given

S=[tex]\left\{4t-12,5+t^3,5+3t,-3t^2+\frac{2}{3}\righ\}[/tex].

We can make a matrix

Let A=[tex]\begin{bmatrix}-12&4&0&0\\5&0&0&1\\5&3&0&0\\\frac{2}{3}&0&-3&0\end{bmatrix}[/tex]

All rows and columns are linearly indepedent and S span [tex]P_3[/tex].Hence, S is a basis of [tex]P_3[/tex]

Linearly independent means any row or any column should not combination of any rows or columns.

Because  a subset of V with n elements is a basis if and only if it is linearly independent.

Basis:- If B is a subset  of a vector space V over a field F .B is basis of V if satisfied the following conditions:

1.The elements of B are linearly independent.

2.Every element of vector V spanned by the elements of B.

Answer:

11 * 17 = 187.

Step-by-step explanation:

11 * 14 = 154

11 * 15 = 165

11 * 16 =  176

So we have the series 154, 165, 176  which has a common difference of 11.

so the next line is 11*17 = 176 + 11 = 187.

Answer:

There are 19 questions worth 7 points.

There are 17 questions worth 8 points.

Step-by-step explanation:

Assign variables:

Let x = number of questions worth 7 points.

Let y = number of questions worth 8 points.

First we deal with the number of questions.

There are 36 questions, so our first equation is:

x + y = 36

Now we deal with the points.

x questions worth 7 points each are worth 7x points.

y questions worth 8 points each are worth 8y points.

The total worth of all the questions is 7x + 8y.

The total worth of the exam is 269, so our second equation is

7x + 8y = 269

We have a system of 2 equations in 2 variables.

x + y = 36

7x + 8y = 269

Let's use the substitution method to solve the system of equations.

Solve the first equation for x.

x + y = 36

x = 36 - y

Substitute 36 - y in for x in the second equation.

7x + 8y = 269

7(36 - y) + 8y = 269

Distribute the 7.

252 - 7y + 8y = 269

Combine like terms.

252 + y = 269

Subtract 252 from both sides.

y = 17

There are 17 questions worth 8 points.

x + y = 36

x + 17 = 36

x = 19

There are 19 questions worth 7 points.

Check:

What does 19 questions at 7 points each and 17 questions at 8 points each add to?

19 * 7 + 17 * 8 = 133 + 136 = 269

The points add to 269, so our answer is correct.

Answer:

Part 42) Yes, It is possible o form a triangle using the segments with the given measurements

Part 43) Yes, It is possible o form a triangle using the segments with the given measurements

Step-by-step explanation:

we know that

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side

Part 42) we have

8ft, 9ft, 11ft

Applying the triangle inequality theorem

a) 8+9 > 11

17 ft > 11 ft ----> is true

b) 9+11 > 8

20 ft > 8 ft  ----> is true

c) 8+11 >  9

19 ft > 9 ft ----> is true

therefore

Yes, It is possible o form a triangle using the segments with the given measurements

Part 43) we have

7.4cm, 8.1cm, 9.8cm

Applying the triangle inequality theorem

a) 7.4+8.1 > 9.8

15.5 cm > 9.8 cm ft ----> is true

b) 8.1+9.8 > 7.4

17.9 cm > 7.4 cm  ----> is true

c) 7.4+9.8 >  8.1

17.2 cm > 8.1 cm ----> is true

therefore

Yes, It is possible o form a triangle using the segments with the given measurements

6 divided by 4 is 1.5.

1.5 times 20 is 30

it would cost michelle $30 to buy 20 pieces of cookies

Answer:

$30

Step-by-step explanation:

4pcs = $6

20pcs = x

Cross multiplying, we have

x = (20pcs * $6)/4pcs

= $120/4 = $30

[tex]y'+2y=2e^x\Longrightarrow y'=2e^x-2y[/tex]

If [tex]f'(x)=g(x)[/tex] then [tex]y=\int{g(x)dx}[/tex]

So we extract,

[tex]y=\int{2e^x-2x}dx[/tex]

Which becomes,

[tex]y=2e^x-x^2+C[/tex]

Hope this helps.

r3t40

Answer: 0.4052

Step-by-step explanation:

Given : Mean : [tex]\mu=\text{307 ​grams}[/tex]

Standard deviation : [tex]\sigma = \text{4.1 grams}[/tex]

The formula for z -score :

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x=308 ,

[tex]z=\dfrac{308-307}{4.1}=0.24390\approx0.24[/tex]

The p-value = [tex]P(z>0.24)=1-P(z<0.24)[/tex]

[tex]=1-0.5948348= 0.4051652\approx0.4052[/tex]

Hence, the probability that a can will be sold that holds more than 308 ​grams =0.4052.

Answer:

The length of the original rectangle is 14 meters and the width of the original rectangle is 12 meters

Step-by-step explanation:

Let

l ----> the length of the original rectangle

w ----> the width of the original rectangle

we know that

[tex]l=2w-10[/tex] ----> equation A

[tex]196=(w+2)^{2}[/tex] ----> area of a square

Solve for w

square root both sides

[tex](w+2)=(+/-)14[/tex]

[tex]w=14=(+/-)14-2[/tex]

[tex]w=14=14-2=12\ m[/tex]

[tex]w=14=-14-2=16\ m[/tex] -----> this solution not make sense

so

[tex]w=12\ m[/tex]

Find the value of L

[tex]l=2(12)-10=14\ m[/tex]

therefore

The length of the original rectangle is 14 meters and the width is 12 meters

Final answer:

The dimensions of the original rectangle are 12 meters (width) and 14 meters (length).

Explanation:

To find the dimensions of the original rectangle, let's first set up an equation.

Let the width of the rectangle be x meters.

The length of the rectangle is given as 10 m less than twice its width, so the length is (2x - 10) meters.

When 2 m are added to the width, the resulting figure is a square with an area of 196 m2.

The side length of a square with area A is √A, so the side length of the square is √196 = 14 meters.

Since adding 2 m to the width creates a square with side length 14 meters, we have:

x + 2 = 14

Subtracting 2 from both sides gives:

x = 12

Therefore, the dimensions of the original rectangle are 12 meters (width) and

(2*12 - 10)

= 14 meters (length).

Answer:

see below

Step-by-step explanation:

A horizontal line has the same y value and is a constant y value

y = -5

A vertical line has the same x value and is a constant x value

x = -6

A vertical line is a line where all of the [tex]x[/tex] values are the same. In this case, [tex]\boxed{x=-6}[/tex], so that is the equation of the line.

A horizontal line is a line where all of the [tex]y[/tex] values are the same.  Here, [tex]\boxed{y=-5}[/tex], so that is the second line.

Answer:

Yes, the triangles are congruent by Hypotenuse angle congruence

Step-by-step explanation:

we know that

The Hypotenuse-Angle Congruence  states that If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the triangles are congruent.

In this problem

The hypotenuse and an acute angle of the right triangle in the left are congruent to the hypotenuse and corresponding acute angle of the right triangle in the right

therefore

The triangles are congruent by Hypotenuse angle congruence

Answer:

3 cm

Step-by-step explanation:

The perimeter of the square is 24 cm, so the side length is 6 cm.

I assume B is the point between A and C where the tangent line intersects the circle.  If so, B is the midpoint of AC, so it is half the length.  Therefore, BC = 3 cm.

The length of line segment BC is 3 cm.

A square is a polygon with four sides , all the sides of the square are equal.

It is given that the tangents of the circle are forming a square ,

The perimeter of the square is 24 cm.

the length of line segment BC = ?

The perimeter of the square is 4a

where a is the side of the square.

Substituting the values

24 = 4 * a

a = 6 cm

B is the mid point of the tangent length and therefore

BC = 6 /2 = 3 cm

Therefore , the length of line segment BC is 3 cm.

The missing image is attached with the answer.

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Answer:

His total gross pay for the week is $588.25.

Step-by-step explanation:

Consider the provided information.

Jarry has a salary of $43 per day and he worked for 4 days.

Thus, the salary of 4 days is:

4 × $43= $172

The average tips is 18% of his total gross food orders.

Last week he had total food orders of $2,312.5. Therefore, the total tips he received is:

[tex]2312.5 \times(\frac{18}{100})[/tex]

[tex]2312.5 \times(0.18)[/tex]

[tex]416.25[/tex]

Total tips he received is $416.25.

His total gross pay for the week = The salary of 4 days + Total tips he received.

His total gross pay for the week = $172 + $416.25

His total gross pay for the week = $588.25

Hence, his total gross pay for the week is $588.25.

Final answer:

Jerry King's total gross pay for the week, combining his salary and the tips he made on his food orders, is $588.25, after calculating an 18% tip on his total orders and adding his daily salary for the 4 days he worked.

Explanation:

The question asks us to calculate the total gross pay for Jerry King, a server who earns both a salary and tips based on his food orders. First, we need to calculate Jerry's earnings from tips. He averages an 18% tip on his total gross food orders. Since his total food orders for the week were $2,312.5, we calculate the tips as follows:

Tips = (Total Food Orders) × (Tip Percentage)
= $2,312.5 × 18%
= $2,312.5 × 0.18
= $416.25 (rounded to the nearest cent)

Next, we calculate Jerry's salary for the week. Jerry works 4 days per week at a salary of $43 per day:

Salary = (Daily Salary) × (Number of Days Worked)
= $43 × 4
= $172

Finally, to find Jerry's total gross pay for the week, we add the salary and the tips:

Total Gross Pay = Salary + Tips
= $172 + $416.25
= $588.25 (rounded to the nearest cent)

Therefore, Jerry King's total gross pay for the week is $588.25.

Answer: [tex]H_0:p\leq0.72[/tex]

[tex]H_a:p\neq0.72[/tex]

Step-by-step explanation:

Given :  A decade-old study found that the proportion of high school seniors who felt that "getting rich" was an important personal goal was 72% .

Let 'p' be the new proportion of high school seniors who felt that "getting rich" was an important personal goal .

Claim : [tex]p\neq0.72[/tex]

We know that the null hypothesis has equal sign.

Therefore , the null hypothesis for the given situation will be opposite to the given claim will be :-

[tex]H_0:p=0.72[/tex]

And the alternative hypothesis must be :-

[tex]H_a:p\neq0.72[/tex]

Hence,  the null hypothesis and the alternative hypothesis that we would use for this test :

[tex]H_0:p\leq0.72[/tex]

[tex]H_a:p\neq0.72[/tex]

Final answer:

To test the belief that the proportion of high school seniors valuing wealth has changed, we would use H0: p = 0.72 as the null hypothesis and Ha: p

Explanation:

To conduct a hypothesis test regarding the proportion of high school seniors who believe that "getting rich" is an important goal, we state the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis is the statement that the proportion remains the same, while the alternative hypothesis states that the proportion has changed.

For a proportion of high school seniors who believe that getting rich is an important goal, if the decade-old study showed a proportion of 72%, we would have:

H0: p = 0.72

Ha: p ≠ 0.72

In the hypothetical scenario with a disease prevalence of 9.5% in the general population and finding 7% in a local survey, we would determine the null and alternative hypotheses regarding whether the local proportion is less than the national average as follows:

H0: p ≥ 0.095

Ha: p < 0.095

When conducting a hypothesis test, the null hypothesis typically represents no change or no effect, while the alternative hypothesis represents a change, difference, or effect that we are trying to detect.

This ODE is separable:

[tex]yy'-\cot t=0\implies y\,\mathrm dy=\cot t\,\mathrm dt[/tex]

Integrate both sides to get

[tex]\dfrac12y^2=\ln|\sin t|+C[/tex]

Given that [tex]y\left(\frac\pi2\right)=-1[/tex], we get

[tex]\dfrac12(-1)^2=\ln|\sin\dfrac\pi2\right|+C\implies C=\dfrac12[/tex]

Then

[tex]\dfrac12y^2=\ln|\sin t|+\dfrac12[/tex]

[tex]y^2=2\ln|\sin t|+1[/tex]

[tex]y^2=\ln\sin^2t+1[/tex]

[tex]\implies\boxed{y(t)=\pm\sqrt{\ln\sin^2t+1}[/tex]

Answer:

The annual rate of return is 13.14%.

Step-by-step explanation:

As it is not mentioned whether the amount was compounded, so we will assume this to be simple interest.

Given is - A $3500 loan was settled ten years later with a payment of $8100.

Means total amount paid back was = $8100

And original principle was = $3500

So, interest paid = [tex]8100-3500=4600[/tex] dollars

Now simple interest formula is :

[tex]I=p\times r\times t[/tex]

Where p = 3500

I = 4600

r = ?

t = 10

Now putting these values in formula we get;

[tex]4600=3500\times r\times10[/tex]

[tex]r=4600/35000[/tex]

r=  0.1314

So, rate of interest = 13.14%

Answer:

The constant of variation is -4

Step-by-step explanation:

If y is directly proprtional to x, then we write:

[tex]y \propto \: x[/tex]

If we introduce the constant of proportionality or variation k, then we obtain:

[tex]y = kx[/tex]

When x=1, y=-4 (We can choose any ordered pair from the table)

[tex] - 4 = k(1)[/tex]

[tex]k = - 4[/tex]

Therefore the constant of variation is -4.

Answer:

-4

Step-by-step explanation:

Answer:

The proposition is false if p is true.

Step-by-step explanation:

I made the truth table in the picture, to fill it you need to know:

v : only is false when both propositions are false.

^ : only is true when both propositions are true.

⇒: only is false when the left proposition is true and the right proposition is false.

So, the answer is the third option.

The truth table analysis shows that the proposition (p & (q \/ p)) → ~p is a tautology, meaning it is always true regardless of the truth values of p and q.

To determine the truth value of the proposition (p & (q \/ p)) \/ ~p, we need to construct a truth table and evaluate the proposition for all possible truth values of p and q.

Truth Table Construction

Let's go step by step:

Construct two initial columns for the variables p and q with their possible truth values.

Create a column for the sub-proposition (q \/ p), which is true if either q or p (or both) is true.

Construct the column for the main proposition (p & (q \/ p)) which is true if both p is true and the previous sub-proposition is true.

Determine the negation of p (~p), which is true when p is false.

Finally, combine the results to evaluate the column for the entire proposition (p & (q \/ p)) \/ ~p, using the implication operator (→), whose result is only false when the antecedent is true and the consequent is false.

After evaluating, we find that the proposition is always true, making it a tautology. Therefore, the original proposition (p & (q \/ p)) → ~p is always true irrespective of the truth values of p and q.