Which one of the following will result in movement along the demand curve for clothing as opposed to a shift in the demand curve itself?

Answer:

Option 2) [tex]a_n = -6(a_{n-1})[/tex]            

Step-by-step explanation:

We are given the following sequence in the question:

[tex]1, -6, 36, -216, ...[/tex]

We have to find the recursive relation for the sequence.

[tex]a_1 =1\\a_2 = -6 = -6(1) = -6(a_1)\\a_3 = 36 = -6(-6) = -6(a_2)\\a_4 = -216 = -6(36) = -6(a_3)[/tex]

Thus, continuing in the following manner, we get,

[tex]a_n = -6(a_{n-1})[/tex]

Thus, the recursive rule is given by

Option 2) [tex]a_n = -6(a_{n-1})[/tex]

Answer:

Step-by-step explanation:

Answer:

The weight of grapefruit is now 80 pound.

Step-by-step explanation:

Consider the provided information.

Let the x is the weight loss. The weight of grapefruit is 100 pounds and water is 92%. After evaporation water is 90%.

Thus the weight loss is:

[tex]0.92\times100-0.90(100 - x) = x[/tex]

[tex]92-90+0.90x=x[/tex]

[tex]2=x-0.90x[/tex]

[tex]2=0.1x[/tex]

[tex]x=20[/tex]

Hence, the weight loss is 80 pounds.

Therefore,  New weight is 100 - 20 = 80 pounds

The weight of grapefruit is now 80 pound.

The variable is the average miles per gallon (mpg), which is a quantitative measure. The implied population is all new cars.

(a) The variable in this situation is the average miles per gallon (mpg) for all new cars.

(b) The variable is quantitative, as it deals with a numerical measure, i.e., the number of miles a car can travel per a gallon of fuel.

(c) The implied population would be all new cars in general - though it's specified that this is based on a sample from Consumer Reports, which may not cover every single new car in existence.

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Answer:

k = ln (6/5)

Step-by-step explanation:

for

f(x)=A*exp(kx)+B

since f(0)=1, f(1)=2

f(0)= A*exp(k*0)+B = A+B = 1

f(1) = A*exp(k*1)+B =  A*e^k + B = 2

assuming k>0 , the horizontal asymptote H of f(x) is

H= limit f(x) , when x→ (-∞)

when x→ (-∞) , limit f(x) =  limit (A*exp(kx)+B) = A* limit [exp(kx)]+B* limit = A*0 + B = B

since

H= B = (-4)

then

A+B = 1 → A=1-B = 1 -(-4) = 5

then

A*e^k + B = 2

5*e^k + (-4) = 2

k = ln (6/5)    ,

then our assumption is right and k = ln (6/5)  

The value of k is [tex]k=ln(\frac{6}{5} )[/tex].

Given function is,

                           [tex]f(x)=Ae^{kx} +B[/tex]

Substitute [tex]f(0)=1,f(1)=2[/tex] in above equation.

We get,

                       [tex]A+B=1\\\\Ae^{k}+B=2[/tex]

Given that horizontal asymptote of f is -4.

               [tex]\lim_{x \to -\infty} Ae^{kx}+B=-4\\ \\ B=-4[/tex]

So,  [tex]A=1-B=1-(-4)=5[/tex]

Substitute value of A and B.

                [tex]5e^{k}-4=2\\ \\e^{k} =\frac{6}{5}\\ \\k=ln(\frac{6}{5} )[/tex]

Hence, the value of k is [tex]k=ln(\frac{6}{5} )[/tex].

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Answer:

The answer to your question is  25 and 28

Step-by-step explanation:

Number 1 = x

number 2 = y

Conditions

1)                    x + y = 53         ------------- l

2)                   x - y =  3            ------------ ll

Solve this system of equations by elimination

                     x + y = 53

                     x - y =  3

                   2x      = 56

Solve for x

                      x = 56/2

                      x = 28

Substitute x in equation 2

                    28 - y = 3

                         - y = -28 + 3

                         - y = -25

                           y = 25

Answer:

A

Step-by-step explanation:

The linear model can be assessed by the checking the independent variables having power 1 which shows the linear relationship between x and y. For example, as in the option B, C and D, the power of Xi's is one. Whereas the non linear model has the power for independent variables greater than 1. For example, as in option  A the model is a quadratic model because X associated with β2 has a power of a 2.

Thus the nonlinear model can be expressed as

Y = β0 + β1X + (β2X)2 + ε.

Answer:

School collected 273 coats in second month.

Step-by-step explanation:

Given

Total number of coats collected in 2 months = 542

Number of coats collected in first month = 269

We need to find the number of coat school collected in second month.

Solution:

Now we can say that;

Total number of coats collected in 2 months is sum of Number of coats collected in first month and Number of coats collected in Second month.

Also We can say that;

Number of coats collected in Second month is equal to Total number of coats collected in 2 months minus Number of coats collected in first month.

framing in equation form we get;

Number of coats collected in Second month = [tex]542-269 =273[/tex]

Hence School collected 273 coats in second month.

Answer:the difference between the cost of one evening ticket and one day time ticket s $1.8

Step-by-step explanation:

The cost of four evening movie tickets is $33.40. This means that the cost of one evening ticket would be

33.4/4 = $8.35

The cost of 6 daytime tickets is 39.30. This means that the cost of one daytime ticket would be

39.30/6 = $6.55

Therefore, the difference between the cost of one evening ticket and one day time ticket would be

8.35 - 6.55 = $1.8

Answer: The other dimension can be expressed as

(50 - 2L)/2

Step-by-step explanation: First and foremost, we would let the other dimension be represented by B. Then, the perimeter of a rectangle is measured as L+L+B+B or better put;

Perimeter = 2L + 2B

Where L is the measurement of the longer side and B is the measurement of the shorter side.

In this case the perimeter of the rectangle measures 50m, and this can now be written as

50 = 2L + 2B

Subtract 2L from both sides of the equation

50 - 2L = 2L - 2L + 2B

50 -2L = 2B

Divide both sides of the equation by 2

(50 - 2L)/2 = B

Answer:

  25-L

Step-by-step explanation:

Let W represent the other side length. The perimeter (P) of the rectangle is ...

  P = 2(W+L)

Solving for W, we get ...

  P/2 = W+L

  P/2 -L = W

Filling in the given value for P, we find ...

  W = 50/2 -L = 25 -L

The other dimension is (25-L) meters.

The distance of a dog from a post and the cost of buying apples can be represented by functions. We can find the values of these functions by substituting specific values and calculating the corresponding outputs.

In this question, we are given a function that represents the distance of a dog from a post as a function of time. To find the value of f(20), we substitute 20 into the function and calculate the corresponding distance. To find the value of f(140), we do the same thing, substituting 140 into the function.

f(20) = 20 feet

f(140) = 140 feet

In the second part of the question, we are given a function C that represents the cost of buying n apples. To find the meaning of each expression or equation, we substitute the given value of n and calculate the corresponding cost.

C(8) = $4.50

C(2) represents the cost of buying 2 apples.

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Answer: 5 seconds

Step-by-step explanation:

Given : A ball is thrown into the air from the top of a building.

The height, h(t), of the ball above the ground t seconds after it is thrown can be modeled by [tex]h(t) = -16t^2 + 64t + 80[/tex] .

When ball reaches the ground , then its height from ground become zero.

i.e. [tex]h(t) = -16t^2 + 64t + 80=0[/tex]

Divide equation by 16 , we get

[tex]-t^2 + 4t + 5=0\\\\\Rightarrow\ t^2-4t-5=0\\\\\Rightarrow\ t^2+t-5t-5=0\\\\\Rightarrow\ t(t+1)-5(t+1)=0\\\\\RIghtarrow\ (t+1)(t-5)=0 \\\\\Rightarrow\ t= -1 , 5[/tex]

Since time cannot be negative , therefore t= 5

Hence, the ball will take 5 seconds ( after being thrown) to hit the ground.

The ball will hit the ground 5 seconds after being thrown.

Settingh(t) equal to zero gives us the equation:

[tex]\[ -16t^2 + 64t + 80 = 0 \][/tex]

 To solve this quadratic equation, we can factor out the common factor of -16:

[tex]\[ -16(t^2 - 4t - 5) = 0 \][/tex]

 Now we need to factor the quadratic expression inside the parentheses.

We are looking for two numbers that multiply to -5 and add up to -4. These numbers are -5 and +1. So we can write:

[tex]\[ -16(t - 5)(t + 1) = 0 \][/tex]

 Setting each factor equal to zero gives us two possible solutions for t

[tex]\[ t - 5 = 0 \quad \text{or} \quad t + 1 = 0 \][/tex]

 Solving these, we get:

[tex]\[ t = 5 \quad \text{or} \quad t = -1 \][/tex]

Since time cannot be negative, we discard the solution  t = -1

Therefore, the ball will hit the ground 5 seconds after being thrown."

y=2.50 units

Step-by-step explanation:

Given that angle ∠Y=75°, ∠Z=50°, side XY=2 units, and side XZ is y then applying the sine rule for this case,

x/sin ∠x =y/sin y =z/sin z

2/sin 50°=y/sin 75°

2 sin 75° =y sin 50°

y= 2 sin 75°/sin 50°

y=2.50 units

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Answer:

C

Step-by-step explanation:

I just finished the test :)

Final answer:

To solve for q in the equation 3(q + 4/3) = 23, we distribute the 3, subtract 4 from both sides, and then divide by 3 to find that q is approximately 6.33.

Explanation:

To solve for q in the equation 3(q + \dfrac{4}{3}) = 23, we need to apply some basic algebra principles. First, we distribute the 3 into the parentheses.

3q + 3 \times \dfrac{4}{3} = 23

3q + 4 = 23

Now, subtract 4 from both sides to get 3q alone on one side.

3q = 23 - 4

3q = 19

Last, divide both sides by 3 to solve for q.

q = \dfrac{19}{3}

q = 6.333...

Thus, q is approximately equal to 6.33 when rounded to two decimal places.

The value of q is q= 19 / 3.

Let's solve for q in the equation:

3(q+ 3 / 4)=23

We can solve the equation by distributing the terms, adding/subtracting to both sides, dividing both sides by the same factor, and simplifying.

Steps to solve:

1. Distribute the terms:

3q+4=23

2. Add/subtract to both sides:

3q+4−4=23−4

3q=19

3. Divide both sides by the same factor:

3q / 3 = 19 / 3

4. Simplify:

q= 19 / 3

Therefore, the value of q is q= 19 / 3.

Answer:

a. is symmetric but not reflexive and transitive

b. is reflexive and transitive but not symmetric

c. is reflexive, symmetric and transitive

Step-by-step explanation:

The cases are missing in the question.

Let the cases be as follows:

a. R = {(1, 3), (3, 1), (2, 2)}

b. R = {(1, 1), (2, 2), (3, 3), (1, 2)}

c. R = ∅

R is defined on the set {1, 2, 3}

a.  R = {(1, 3), (3, 1), (2, 2)} is

b. R = {(1, 1), (2, 2), (3, 3), (1, 2)} is

is reflexive because (1, 1), (2, 2), (3, 3) is in R

is not symmetric because for (1,2) (2,1) is not in R

is transitive becaue for (1,1) and (1,2) we have (1,2) in R

c. R = ∅ is

reflexive, symmetric and transitive because it satisfies the definitions since there is no counter example.

Subtracting a value from the input x shifts the graph that number of units to the right.

The answer would be C.

Answer: Zoey needs 168 square feet if carpet squares.

Step-by-step explanation:

Zoey wants to cover her bedroom floor with carpet squares. Each square has an area of 1 square foot.

The formula for determining the area of a rectangle is expressed as

Area = length × width

Her bedroom measures 12 feet by 14 feet. Therefore, the area of her bedroom would be

12 × 14 = 168 square feet.

Therefore, the number of carpet squares that Zoey needs would be

168/1 = 168 square feet

Answer:

  168 squares

Step-by-step explanation:

Each square is 1 foot on a side, so along the 14-foot wall, Zoey will need 14 squares. Altogether, Zoey will need 12 rows of 14 squares, so 12×14 = 168 squares.

Answer:

The fourth root is 3

If the 4th root is not a real therefore it must be a  complex number (a+ib),and its conjugate will be also a root ,therefore there would be 5 roots instead of 4  roots.

Therefore the fourth root is real.

The roots are -1 with multiplicity 2 and 3 with multiplicity 2

Therefore it has four roots

Step-by-step explanation:

Given polynomial equation is [tex]X^4-4X^3-2X^2+12X+9=0[/tex]

And also given that 3,-1 and -1 are  the roots of the given polynomial equation

To find the fourth root of the polynomial equation and to solve the fourth root is real :

By synthetic division

_3|   1    -4    -2    12    9

       0    3    -3   -15    -9

___________________

_-1|       1     -1     -5    -3     0

            0     -1      2     3

___________________

            1      -2     -3     0

Therefore x-3 and x+1 is a factor

Therefore 3 and -1 are roots

Now we have the quadratic equation [tex]x^2-2x-3=0[/tex]

[tex](x+1)(x-3)=0[/tex]

Therefore x=-1,3 are the roots

Therefore the fourth root is 3

If the 4th root is not a real therefore it must be a  complex number (a+ib),and its conjugate will be also a root ,therefore there would be 5 roots instead of 4  roots.

Therefore the fourth root is real.

The roots are -1 with multiplicity 2 and 3 with multiplicity 2

Therefore it has four roots.

Final answer:

The fourth root of the polynomial equation X⁴-4X³-2X²+12X+9=0 must be real because a polynomial of degree n has n roots, and since we already have real roots, the remaining root must also be real to have a pair. Upon analyzing, the fourth root is found to be 3.

Explanation:

The student has provided three roots of the fourth-degree polynomial equation X⁴-4X³-2X²+12X+9=0: 3, -1, and -1 (the latter being a repeated root). To determine why the fourth root must also be a real number, we can invoke the fundamental theorem of algebra, which states that a polynomial of degree n will have exactly n roots in the complex number system (including real and complex roots). Given that a polynomial with real coefficients will have complex roots that come in conjugate pairs, and since the known roots are all real, the unknown fourth root must also be real to satisfy the theorem.

Let's find the fourth root. The polynomial can be factored using the known roots:

Therefore, we have the equation (X-3)(X+1)²(X-a)=0, where 'a' is the unknown root. The product of the roots taken one at a time equals the constant term (9) of the polynomial with an alternate sign. This gives us the equation: 3 × -1 × -1 × a = 9. Solving for 'a' yields a=3, which is the fourth root.

Answer:

D = A < D < B < C

Step-by-step explanation:

A = 4.6 x 10-4

Can be written as,

= 0.00046

B = 2.4 x 10-3

Can be written in this form,

= 0.0024

C = 3.5 x 105

Is written as,

350000

D = 6.3 x 10-4

Is also written as,

= 0.00063

A and D are in 4 decimal places and therefore from the 3th decimal place 46 is less than 63 so therefore 0.00046 is less than 0.00063.

B is greater than A and D because B which is 0.0024 is in 3 decimal places. C which is 35000 is the greatest because there are no decimal places and it is in tenth thousand.

So therefore,

A < D < B < C

Answer:option 1 and option 2 represents a direct variation.

Step-by-step explanation:

A direct variation is one in which, as one variable increases in value, the other variable increases in value. Also as one variable decreases in value, the other variable decreases in value.

Looking at the options,

1) y = 2х

When x = 2, y = 2×2 = 4

When x = 3, y = 2×3 = 6

Therefore, y = 2x is a direct variation.

2) y=x+4

When x = 2, y = 2 + 4 = 6

When x = 3, y = 3 + 4 = 7

Therefore, y = x + 4 is a direct variation.

3) y= 1/2x

When x = 2, y = 1/2×2 = 1/4 = 0.25

When x = 3, y = 1/2×3 = 1/6 = 0.17

Therefore, y = 1/2x is not a direct variation.

4) y = 3/x

When x = 2, y = 3/2 = 1.5

When x = 3, y = 3/3 = 1

Therefore, y = 3/x is not a direct variation.

Answer:

4m width and 9m length

Step-by-step explanation:

Let the width of the rectangle be x

Length is 1 more than twice width= 1 + 2x

Area of rectangle is L * B

x(2x + 1) = 36

2x^2 + x = 36

2x^2 + x -36 = 0

2x^2 + 9x - 8x -36 = 0

Solving this:

(2x+9)(x - 4) = 0

X = 4 or -4.5

Distance cannot be negative, so x = 4m

The length is thus 2(4) + 1 = 9m

Final answer:

The width of the plot is 4 meters and the length is 9 meters.

Explanation:

To solve this problem, we can let the width of the plot be x meters. According to the problem, the length of the plot is one more than twice its width, so the length would be 2x + 1 meters. The area of a rectangle is given by the formula A = length * width. So we have the equation (2x + 1) * x = 36. Expanding and rearranging, we get 2x² + x - 36 = 0.

Factoring this quadratic equation, we get (2x + 9)(x - 4) = 0. Setting each factor equal to zero and solving for x, we find x = -4/2 and x = 4. Since the width cannot be negative, we discard x = -4/2 and conclude that the width of the plot is 4 meters. Substituting this value back into the equation for the length, we find the length is 2(4) + 1 = 9 meters.

Using the process of Gaussian elimination, the system of linear equations is rewritten in the form of a matrix. It is then transformed into the Row-Echelon form, which helps determine possible solutions. The solution for this particular system of equations is x1 = 2, x2 = 2, and x3 = 1.

To solve this system of linear equations, you can use a process called

Gaussian elimination

. You start by rewriting the system in augmented matrix. Thus, the system

2x1 + x2 − 2x3 = 4

4x1 + 2x3 = 10

−4x1 + 5x2 − 17x3 = −15

becomes the matrix

[2 1 -2 4]

[4 0 2 10]

[-4 5 -17 -15]

The next step is to convert this matrix into the Row-Echelon form. Once you have a matrix in Row-Echelon form, you can easily see if there are any solutions by looking at the location of the zeros. If there is a row with all zeros on the left and non-zero terms on the right, then there is no solution. If there are infinite many solutions, its row will end with zeros. In this case, the solution is x1 = 2, x2 = 2, and x3 = 1.

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Answer:the first equation can be multiplied by 5 and the second equation by 3

Step-by-step explanation:

The operation that would create an equivalent system of equations with opposite like terms is that the first equation can be multiplied by 5 and the second equation by 3, the correct option is A.

The system of equation is set of equations which have a common solution.

The equations are

3x-3y = 3

4x+5y = 13

The value of x and y can be determined using Elimination Method.

In elimination method like terms have to be created to make an equivalent system,

The first equation can be multiplied by 5 and the second equation by 3 to solve the equations.

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Answer:   2+i

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

Work Shown:

-3 + 6i - (-5 - 3i) - 8i

-3 + 6i + 5 + 3i - 8i

(-3+5) + (6i+3i-8i)

2+1i

2+i

The simplified expression is in the form a+bi with a = 2, b = 1.

Answer:

2 + i.

Step-by-step explanation:

-3 + 6i - (-5 - 3i) - 8i       Distribute the negative over the parentheses:

= -3 + 6i + 5 + 3i - 8i

= - 3 + 5 + 6i + 3i - 8i     Now simplify like terms:

= 2 + i.

Answer:

a) [tex]P(\bar X >4)=P(Z>\frac{4-3}{\frac{3}{\sqrt{50}}}=2.357)[/tex]

[tex]P(Z>2.357)=1-P(Z<2.357) =1-0.9908=0.0092[/tex]

b) [tex]P(\bar X <2.5)=P(Z>\frac{2.5-3}{\frac{3}{\sqrt{50}}}=-1.179)[/tex]

[tex]P(Z<-1.179)=0.1192[/tex]

c)  

[tex] P(2.5 < \bar X< 3.5) = P(\frac{2.5-3}{\frac{3}{\sqrt{50}}} <Z<\frac{3.5-3}{\frac{3}{\sqrt{50}}})[/tex]

[tex]P(-1.179<Z<1.179)=P(Z<1.179)-P(Z<-1.179)=0.8808-0.1192=0.7616 [/tex]Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the number of people of a population, and for this case we know that:

Where [tex]\mu=3[/tex] and [tex]\sigma=\sqrt{9}=3[/tex]

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:

[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]

And we want to find this probability:

[tex] P(\bar X >4)= P(z> \frac{4-3}{\frac{3}{\sqrt{50}}})[/tex]

And using a calculator, excel or the normal standard table we have that:

[tex]P(Z>2.357)=1-P(Z<2.357) =1-0.9908=0.0092[/tex]

Part b

[tex] P(\bar X <2.5)= P(z> \frac{2.5-3}{\frac{3}{\sqrt{50}}})[/tex]

And using a calculator, excel or the normal standard table we have that:

[tex]P(Z<-1.179)=0.1192[/tex]

Part c

For this case we want this probability:

[tex] P(2.5 < \bar X< 3.5) = P(\frac{2.5-3}{\frac{3}{\sqrt{50}}} <Z<\frac{3.5-3}{\frac{3}{\sqrt{50}}})[/tex]

And using a calculator, excel or the normal standard table we have that:

[tex]P(-1.179<Z<1.179)=P(Z<1.179)-P(Z<-1.179)=0.8808-0.1192=0.7616 [/tex]

The number of orange cans of paint is 6

Solution:

Given that,

[tex]\text{Number of cans of red paint } = 3\frac{1}{4}\\\\\text{Number of cans of yellow paint } = 3\frac{2}{12}[/tex]

Let us convert the mixed fractions to improper fractions

Multiply the whole number part by the fraction's denominator.

Add that to the numerator.

Then write the result on top of the denominator

[tex]\rightarrow 3\frac{1}{4} = \frac{4 \times 3 + 1}{4} = \frac{13}{4}\\\\\rightarrow 3\frac{2}{12} = \frac{12 \times 3 + 2}{12} = \frac{38}{12}[/tex]

Now we have to add red cans of paint and yellow cans of paint to get orange cans of paint

[tex]\text{Number of cans of orange paint } = \frac{13}{4} + \frac{38}{12}[/tex]

Take L.C.M for denominators

The prime factors of 4 = 2 x 2

The prime factors of 12 = 2 x 2 x 3

For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.

The new superset list is

2, 2, 3

Multiply these factors together to find the LCM.

LCM = 2 x 2 x 3 = 12

[tex]\text{Number of cans of orange paint } = \frac{13}{4} + \frac{38}{12}[/tex]

[tex]\text{Number of cans of orange paint } = \frac{13 \times 3}{4 \times 3} + \frac{38}{12}\\\\\text{Number of cans of orange paint } = \frac{39}{12} + \frac{38}{12}\\\\\text{Number of cans of orange paint } = \frac{77}{12} = 6.4 \approx 6[/tex]

Thus the number of orange cans of paint is 6

Answer:12 baskets must be sold in order to break even

Step-by-step explanation:

The materials needed by the fundraiser team to make each basket cost $3.75.

Let x represent the number of baskets that the team made and also sold. if they spent $75 to advertise their site, then the total cost for x baskets would be

3.75x + 75

The baskets are being sold for $10 each. It means that the total revenue would be

10x

In order to break even, the revenue must be equal to total cost. Therefore,

10x = 3.75x + 75

10x - 3.75x = 75

6.25x = 75

x = 75/6/25 = 12

Answer:

[tex]m = \dfrac{a^2-99}{2a}[/tex]

Step-by-step explanation:

on a number line, m is the point that is the midpoint of two other points.

the distance between each of the points to the midpoint is 10 units..

if a is the point 10 units less than m

and b is the point 10 units greater than m,

then,

[tex]m = a+10[/tex]

[tex]m = b-10[/tex]

we can add the two equations to form the midpoint formula.

[tex]2m = a+b[/tex]

we also know that the product of both numbers equal -99.

[tex]ab = -99[/tex]

we can substitute either 'a' or 'b' to the equation of m.

[tex]2m = a-\dfrac{99}{a}[/tex]

[tex]m = \dfrac{a^2-99}{2a}[/tex]

and this is the equation for the midpoint of the two numbers.

Answer:

c

Step-by-step explanation:

Answer: None of the above

Step-by-step explanation:

Each of the presented points helps to describe how to collect and summarize data and how to make appropriate scientific inferences.

It provides a guide on how to use biostatistical principles with grounded mathematical and probability theory. It aims is to help understand and to interpret biostatistical analysis generally.

The answer is

[tex]f(x) = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]

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

Here's how I got that answer:

Start with the piecewise definition for y = |x|.

[tex]g(x) = \begin{cases}x \text{ if }x \ge 0 \\ -x \text{ if }x < 0\end{cases}[/tex]

Everywhere you see an 'x', replace it with x+2

[tex]g(x+2) = \begin{cases}x+2  \text{ if }x+2 \ge 0 \\ -(x+2) \text{ if }x+2 < 0\end{cases}[/tex]

[tex]g(x+2) = \begin{cases}x+2  \text{ if }x \ge -2 \\ -x-2 \text{ if }x < -2\end{cases}[/tex]

Now tack on "-4" at the end of each piece so that we shift the function down 4 units

[tex]g(x+2)-4 = \begin{cases}x+2-4  \text{ if }x \ge -2 \\ -x-2-4 \text{ if }x < -2\end{cases}[/tex]

[tex]g(x+2)-4 = \begin{cases}x-2  \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]

[tex]f(x) = \begin{cases}x-2  \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]

Check out the attached images below. In figure 1, I graph y = x-2 and y = -x-6 as separate equations on the same xy coordinate system. Then in figure 2, I combine them to form the familiar V shape you see with any absolute value graph.

The minimum force required to open the jar using the wrench is 41.5 N, calculated based on the given torque of 8.9 N∙m and the effective radius of 0.2145 m.

Calculate the minimum force required to open the jar using the jar wrench:

1. Identify the torque required:

The problem states that the torque required to open the jar is 8.9 N∙m. This means that you need to apply a force that creates a twisting moment of 8.9 N∙m to overcome the friction between the lid and the jar.

2. Determine the effective radius:

The effective radius is the distance from the center of rotation (the center of the lid) to the point where the force is applied (the end of the wrench handle).

In this case, the effective radius is the sum of:

The length of the wrench handle (17 cm = 0.17 m)

Half the diameter of the lid (4.45 cm = 0.089 m / 2, assuming a circular lid)

So, the effective radius is 0.17 m + 0.0445 m = 0.2145 m.

3. Apply the torque formula:

The formula for torque is: τ = rF

τ = torque (in N∙m)

r = effective radius (in meters)

F = force (in Newtons)

You can rearrange this formula to solve for force: F = τ / r

4. Calculate the force:

Plug in the values: F = 8.9 N∙m / 0.2145 m

Calculate: F = 41.5 N

Therefore, the minimum force required to open the jar using the wrench is 41.5 N.