Write the equations of functions satisfying the given properties, in expanded form. a. Cubic polynomial, x-intercepts at - and -2, y-intercept at 10. 14 b. Rational function, x-intercepts at -2,-2, 1; y-intercept at - %; vertical asymptotes at 2, 3, -4; horizontal asymptote at 1.

Answer:

216 cm³

Step-by-step explanation:

large prism volume = 6 x 6 x 8 = 288 cm³

small cutout volume = 3 x 3 x 8 = 72 cm³

288- 72 = 216 cm³

Answer:

6x

Step-by-step explanation:

Answer:

d. -2/3x+5

Step-by-step explanation:

Because the line is perpendicular, the slope must be the inverse of the original slope. The inverse of 3/2 is -2/3. To find the b value, you plug (6,1) into the equation y=-2/3x+b

1=-2/3(6)+b

1=-4+b

5=b

The final equation is y=-2/3x+5

The true statements are:a. Angle R is congruent to angle R'

b. (P' * Q')/(PQ) = 4

e. (C * Q')/(CQ) = 4

To determine which statements are true about triangle PQR and its image P' * Q' * R' after dilation, let's analyze each statement:

a. Angle R is congruent to angle R': This statement is true. When a triangle is dilated, the corresponding angles remain congruent.

b. (P' * Q')/(PQ) = 4: This statement is true. The scale factor of dilation is 4, which means the corresponding side lengths are multiplied by 4. Therefore, (P' * Q')/(PQ) = 4.

c. (QR)/(Q' * R') = 4: This statement is false. The scale factor of dilation applies to individual side lengths, not ratios of side lengths. Therefore, (QR)/(Q' * R') will not necessarily be equal to 4.

d. (C * P')/(CP) = 5: This statement is false. The scale factor of dilation is 4, not 5. Therefore, (C * P')/(CP) will not be equal to 5.

e. (C * Q')/(CQ) = 4: This statement is true. The scale factor of dilation is 4, so the corresponding side lengths are multiplied by 4. Therefore, (C * Q')/(CQ) = 4.

f. (C * P')/(CP) = (C * R')/(CR): This statement is false. The dilation does not guarantee that the ratios of the distances from the center C to the vertices will be equal.

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The correct GCD of 57 and 17 is 1, obtained through the Euclidean algorithm.

To find the correct GCD (Greatest Common Divisor) of 57 and 17 using the Euclidean algorithm, we follow these steps:

1.) Divide the larger number (57) by the smaller number (17) and find the remainder:

57 ÷ 17 = 3 remainder 6

2.) Replace the larger number with the smaller number and the smaller number with the remainder:

17 ÷ 6 = 2 remainder 5

3.)  Repeat step 2 until the remainder is 0:

6 ÷ 5 = 1 remainder 1

5 ÷ 1 = 5 remainder 0

4.) The GCD is the last nonzero remainder, which is 1.

Therefore, the correct GCD of 57 and 17 is 1.

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Answer:

Step-by-step explanation:

20km east - 12km south= 8km east

Answer:

its 120.96

Step-by-step explanation:

i dont have any but i know that is the answer

The given expression is $\frac{6}{t}+\frac{8}{u}-\frac{9}{v}$ and we need to rewrite this expression without any denominator in it. Step-by-step explanation: We can use the concept of the Least Common Multiple (LCM) of the denominators to remove the fractions in the expression. By taking the LCM of the denominators of the given expression, we have,$LCM\text{ of }t, u, v = t \cdot u \cdot v$ Now, multiplying each term of the given expression with the LCM $t \cdot u \cdot v$, we get,$\frac{6}{t}\cdot t \cdot u \cdot v+\frac{8}{u}\cdot t \cdot u \cdot v-\frac{9}{v}\cdot t \cdot u \cdot v$$6uv + 8tv - 9tu$$\therefore \text{The given expression without any denominator is } 6uv + 8tv - 9tu.$Thus, we can rewrite the given expression $\frac{6}{t}+\frac{8}{u}-\frac{9}{v}$ without any denominator in it as $6uv + 8tv - 9tu$.

LCM (a,b) in mathematics stands for the least common multiple, or LCM, of two numbers, such as a and b. The smallest or least positive integer that is divisible by both a and b is known as the LCM. Take the positive integers 4 and 6 as an illustration.

There are four multiples: 4,8,12,16,20,24.

6, 12, 18, and 24 are multiples of 6.

12, 24, 36, 48, and so on are frequent multiples for the numbers 4 and 6. In that lot, 12 would be the least frequent number. Now let's attempt to get the LCM of 24 and 15.

LCM of 24 and 15 is equal to 222235 = 120.

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Cov(X, Y) = -1/3 and ρ(X, Y) = -1/18.

Given data:X and Y are two random variables,

σ² of X=4,σ² of Y=9.Z=2X − Y and W = X + Y are independent

To find:

Cov(X, Y) and ρ(X, Y)

Solution:

We know that:

Cov(X, Y) = E(XY) - E(X)E(Y)ρ(X, Y) = Cov(X, Y) / σX σY

Let's find E(X), E(Y), E(XY)E(X) = E(W - Y) = E(W) - E(Y)E(W) = E(X + Y) = E(X) + E(Y)

From this equation, E(X) = E(W)/2 ------- (1)

Similarly, E(Y) = E(W)/2 ------- (2)

To find E(XY), we will use the following equation:

E(XY) = Cov(X, Y) + E(X)E(Y)Using equations (1) and (2) in the above equation:

E(XY) = Cov(X, Y) + E(W)²/4

Now, we will use the independence of Z and W to find Cov(X, Y).Cov(X, Y) = Cov((W - Z)/2, (W + Z)/3)= 1/6[Cov(W, W) - Cov(W, Z) + Cov(Z, W) - Cov(Z, Z)]= 1/6[Var(W) - Var(Z)]

Here,Var(W) = Var(X + Y) = Var(X) + Var(Y) [using independence]= 4 + 9 = 13Var(Z) = Var(2X - Y) = 4Var(X) + Var(Y) - 2 Cov(X, Y)= 4 + 9 - 2 Cov(X, Y)

Now, putting these values in Cov(X, Y),Cov(X, Y) = -1/3

Also,σX = 2 and σY = 3ρ(X, Y) = Cov(X, Y) / σX σY= -1/18

Hence, Cov(X, Y) = -1/3 and ρ(X, Y) = -1/18.

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Answer:

3x - 24

Step-by-step explanation:

this is probably wrong

Answer:

1116

Step-by-step explanation:

Hey!

We can use the algebraic expression, 3x - 24, to solve.

Just substitute 380 in for x.

⇒3(380) - 24

⇒1140 - 24

⇒ 1116

--------------------------------------------------------------------------------------------------------------

Hope I Helped, Feel free to ask any questions to clarify :)

Have a great day!

More Love, More Peace, Less Hate.

       -Aadi x

The solution to the equation [x³ - 1] = 0 is x = 1

From the question, we have the following parameters that can be used in our computation:

[x³ - 1] = 0

Remove the square bracket in the equation

So, we have

x³ - 1 = 0

Add 1 to both sides

This gives

x³ = 1

Take the cube root of both sides

x = 1

Hence, the solution to the equation [x³ - 1] = 0 is x = 1

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Given: A team of 3 employees is preparing 20 reports. Mary takes 30 minutes to complete a report. Matt takes 45 minutes to complete a report.

All reports are completed in 4 1/2 hours. To Find: How long does it take the third team member to complete a report?Solution: Let the third employee takes x minutes to complete a report work done by Mary in 1 minute = 1/30Work done by Matt in 1 minute = 1/45Work done by the third employee in 1 minute = 1/x Total work done by all three in 1 minute = 1/30 + 1/45 + 1/x (As all are working together) a Total number of reports to be prepared = 20Therefore, total work = 20Now,

we know that all reports are completed in 4 1/2 hours = 9/2 hours∴ Total time = 9/2 x 60 = 270 minutes according to the problem statement, Total work = Total time x Total work done by all three in 1 minute20 = 270 (1/30 + 1/45 + 1/x)Solving the above equation for x, we get :x = 90 minutes therefore, it takes the third team member 90 minutes to complete a report.

Answer: 90 minutes.

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Let's represent the third team member as t. Mary takes 30 minutes to complete a report, while Matt takes 45 minutes to complete a report.

Thus, it takes the third team member 3 hours to complete a report.

Therefore, we can use the information given to form an equation. We are given that the team is preparing 20 reports, so:

30 minutes/report × M reports + 45 minutes/report × N reports + T minutes/report × O reports = 4.5 hours

To make the equation simpler, let the unit conversion 4.5 hours to minutes:

4.5 hours × 60 minutes/hour = 270 minutes

Thus:

30M + 45N + TO = 270

O= 20 - M - N

From the third team member: TO = T × 20

Therefore:

30M + 45N + T × 20 = 270

Solving for T:

30M + 45N + 20T = 270

T = (270 - 30M - 45N)/20

We know that there are only three members in the team, and that M and N have already been defined, so we can substitute these values:

T = (270 - 30(20) - 45(0))/20

T = 3

Thus, the time taken by the third team member is 3 hours to complete a report.

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Answer:

4x-5y=10

Step-by-step explanation:

A possible value for g(6) is 27. The only option greater than 18 is:

e. 27

To determine a possible value for g(6), we can make use of the given information and the properties of the function g(x).

Since g'(x) > 0 for all real numbers x, we know that g(x) is strictly increasing. This means that as x increases, g(x) will also increase.

Furthermore, since g''(x) > 0 for all real numbers x, we know that g(x) is a concave up function. This implies that the rate at which g(x) increases is increasing as well.

Given that g(4) = 12 and g(5) = 18, we can conclude that between x = 4 and x = 5, the function g(x) increased from 12 to 18.

Considering the properties of g(x), we can deduce that g(6) must be greater than 18. Since the function is strictly increasing and concave up, the increase from g(5) to g(6) will be even greater than the increase from g(4) to g(5).

Among the given answer choices, the only option greater than 18 is:

e. 27

Therefore, a possible value for g(6) is 27.

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(A intersect B) is a subset of A, A is a subset of (A union B), A is a subset of (D union E), and |sin(nx)| <= n|sin(x)| for any natural number and real number x.

To prove that (A intersect B) is a subset of A, we need to show that every element in (A intersect B) is also in A. Let x be an arbitrary element in (A intersect B). This means x is in both A and B. Since x is in A, it follows that x is also in the union of A and B, which means x is in A. Therefore, (A intersect B) is a subset of A.

To prove that A is a subset of (A union B), we need to show that every element in A is also in (A union B). Let x be an arbitrary element in A. Since x is in A, it follows that x is in the union of A and B, which means x is in (A union B). Therefore, A is a subset of (A union B).

Given A is a subset of (B union C), B is a subset of D, and C is a subset of E, we want to prove that A is a subset of (D union E). Let x be an arbitrary element in A. Since A is a subset of (B union C), it means x is in (B union C). Since B is a subset of D and C is a subset of E, we can conclude that x is in (D union E). Therefore, A is a subset of (D union E).

To prove |sin(nx)| <= n |sin(x)| for any natural number n and real number x, we can use mathematical induction. For the base case, when n = 1, the inequality reduces to |sin(x)| <= |sin(x)|, which is true. Assuming the inequality holds for some positive integer k, we need to show that it holds for k+1. By using the double-angle formula for sin, we can rewrite sin((k+1)x) as 2sin(x)cos(kx) - sin(x). By the induction hypothesis, |sin(kx)| <= k|sin(x)|, and since |cos(kx)| <= 1, we have |sin((k+1)x)| = |2sin(x)cos(kx) - sin(x)| <= 2|sin(x)||cos(kx)| + |sin(x)| <= 2k|sin(x)| + |sin(x)| = (2k+1)|sin(x)| <= (k+1)|sin(x)|. Therefore, the inequality holds for all natural numbers n and real numbers x.

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Answer:

12(x−2.57)=0.36

Step-by-step explanation:

Let x represent the initial amount of money Mrs. Sorenstam had to spend on each student.

The cost of the 3 items is:

1.49+0.59+0.49=2.57

The change left for each student will be:

x−2.57

For 12 students, the change left will be

12(x−2.57) which equals 36 cents, according to the problem

So, the equation to represent this situation will be:

12(x−2.57)=0.36

The minimum value of the objective function 12x + 14y is 156 at point C(6, 9).Answer: 156.

Given:

Minimize and maximize objective function = 12x + 14y–2x + y ≥ 6x + y ≤ 15x ≥ 0, y ≥ 0.

The graphical method is a simple and easy method of solving a linear programming problem (LP).

LP issues are represented on a graphical scale using graphical method.

Let's plot the given inequalities on the graph. The graph of all inequalities must be in the first quadrant since x, y ≥ 0.Initially, let us consider x = 0 and y = 0 for (2) and (3) respectively.

(2) y ≤ 15 - x On plotting the line y = 15 - x in first quadrant, we get the following graph:

(3) x ≤ 15 - y On plotting the line x = 15 - y in first quadrant, we get the following graph:Now let's check for the first inequality, -2x + y ≥ 6.It can be written as y ≥ 2x + 6.

On plotting the line y = 2x + 6 in first quadrant, we get the following graph:The region containing common feasible points for all the three inequalities is shown in the figure below:Thus, the feasible region is OACD.The corner points of the feasible region are A(2, 13), B(3.8, 11.2), C(6, 9) and D(15, 0).

We need to determine the minimum and maximum values of the objective function 12x + 14y at each corner point as follows:At point A, 12x + 14y = 12(2) + 14(13) = 194At point B, 12x + 14y = 12(3.8) + 14(11.2) = 184.8At point C, 12x + 14y = 12(6) + 14(9) = 156At point D, 12x + 14y = 12(15) + 14(0) = 180.

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To find the minimum and maximum values of the objective function 12x + 14y subject to the given constraints using graphical method.

Therefore, the minimum value of the objective function is 210 at (10.5, 3) and the maximum value of the objective function is not bounded.

We can follow these steps:

Step 1: Convert the inequality constraints into equation form by replacing the inequality signs with equality signs. So, -2x + y = 6 and

x + y = 15

Step 2: We find the values of x and y for each equation.

Step 3: Plot the two lines on the coordinate axis formed by the values obtained in Step 2.

Step 4: Determine the feasible region by identifying the portion of the plane where the solution satisfies all the constraints. In the present case, it is the region

above the line -2x + y = 6 and

below the line x + y = 15 and

to the right of the y-axis.

Step 5: Plot the objective function 12x + 14y on the same graph.

Step 6: Move the objective function line either up or down until it just touches the highest or lowest point of the feasible region. The point of contact is the solution to the linear programming problem. The graph of the feasible region and the objective function is shown below:

graph

y = 15 - x [-10, 20, -5, 25]

y = 2x + 6 [-10, 20, -5, 25]

y = -(6/7)x + 180/7 [-10, 20, -5, 25](-1/2)x+(1/14)

y = 0.5[0, 20, 0, 20](-1/2)x+(1/7)

y = 1[0, 20, 0, 20]12x + 14

y = 210[0, 20, 0, 20]

Therefore, the minimum value of the objective function is 210 at (10.5, 3) and the maximum value of the objective function is not bounded.

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Answer:

30000 times

Step-by-step explanation:

3 x 10^3 x 30000 = 90000000

Answer:

30000

Step-by-step explanation:

9×10^7 =90000000

also

3×10^3 =3000

divide 90000000 by 3000

=30000

The vector equation r(t) = t^2i + t^4j + t^6k represents a parametric curve in three-dimensional space. To sketch the curve, we can substitute values of t and plot corresponding points in the coordinate system.

By examining the components of the vector equation, we can observe that x = t^2, y = t^4, and z = t^6. This implies that the curve lies in the x-y-z coordinate system, where the x-coordinate is determined by t^2, the y-coordinate is determined by t^4, and the z-coordinate is determined by t^6.

To start sketching, we can choose a range of values for t and substitute them into the equations. For example, for t = -1, 0, 1, we can calculate the corresponding x, y, and z values.

By plotting these points and connecting them, we can obtain an approximate shape of the curve. Additionally, we can observe that as t increases, the curve moves in the direction of increasing t, which can be indicated by an arrow along the curve.

Note that without specific values for t or a specific range, the sketch will be a general representation of the curve.

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The function for this problem is given as follows:

y = 0.25(x + 5)²(x - 4)²

We are given the roots for each function, hence the factor theorem is used to define the functions.

The function is defined as a product of it's linear factors, if x = a is a root, then x - a is a linear factor of the function.

The roots of the function in this problem are given as follows:

Hence the linear factors are given as follows:

The function is:

y = a(x + 5)²(x - 4)²

In which a is the leading coefficient.

When x = 0, y = 100, hence the leading coefficient a is given as follows:

100 = a(5²)(-4)²

400a = 100

a = 0.25.

Hence the function is:

y = 0.25(x + 5)²(x - 4)²

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The is specified initial approximation X1  x3 is equal to 5.

We absolutely need to accentuate using the recipe in order to find x3 using Newton's method:

In this particular instance, we are informed that x5 is equal to x2 minus 4 and that X1 equals 1. Because we need to find x3, let's use the given equation to find x2.

We can solve for x2 because we have x5: x2 + 4

As of now we have x2 = x5 - 4 from x2 = x5 - 4. This ought to be added to the Newton's system recipe, and afterward we can find x3:

We ought to portray our ability f(x) and its subordinate f'(x) as Xn+1 = Xn - f(Xn)/f'(Xn).

We can now calculate x3 by using X1 = 1 as our underlying estimate: X2 = X1 - f(X1)/f'(X1) = 1 - ((1)2 + 4 - 1)/(- 1) = 1 - (1 + 4 - 1)/(- 1) = 1 + 4 = 5 In this way, x3 is the same as 5.

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Answer:

y = -1/4x + 16

Step-by-step explanation:

the slope is -1/4 and the y-intercept is 16

The five-number summary for the given data set is: Minimum = -7, First Quartile = 2, Median = 6, Third Quartile = 9, Maximum = 24.

To calculate the five-number summary for the given data set, we need to arrange the data in ascending order and then determine the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum values.

The given data set: -7, -5, -2, 0, 4, 6, 8, 8, 10, 22, 24

Arranged in ascending order: -7, -5, -2, 0, 4, 6, 8, 8, 10, 22, 24

Now, let's calculate the values for the five-number summary:

Minimum: The smallest value in the data set is -7.

First Quartile (Q1): This represents the median of the lower half of the data set. Since we have 11 data points, Q1 is the median of the first 5 data points. Q1 = (0 + 4) / 2 = 2.

Median (Q2): The median is the middle value of the data set. Since we have an odd number of data points, the median is the 6th value, which is 6.

Third Quartile (Q3): This represents the median of the upper half of the data set. Q3 is the median of the last 5 data points. Q3 = (8 + 10) / 2 = 9.

Maximum: The largest value in the data set is 24.

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The probability that an employed person earns a living through agriculture is ≈ 17%.

The frequency is the number of times the data appear within each category.

A percentage frequency distribution is used to summarize data and report on the proportion or percentage of observations that fall within a specified category.

It is the process of showing how often a particular value or category occurs in a set of data.

In order to create a percentage frequency distribution, we will first add all the values together:

Total number of people employed = 6,678 + 1,492 + 653 + 2,865 + 914

= 12,602

Now we can calculate the percentage of people employed in each sector:

Formal business sector =

(6,678 / 12,602) x 100% = 53.0%

Commercial agricultural sector =

(1,492 / 12,602) x 100% = 11.8%

Subsistence agricultural sector =

(653 / 12,602) x 100% = 5.2%

Informal business sector =

(2,865 / 12,602) x 100% = 22.7%

Domestic service sector =

(914 / 12,602) x 100% = 7.3%

The likelihood that an employed person earns a living through agriculture can be calculated by adding the number of people employed in the commercial agricultural sector and the number of people employed in subsistence agriculture.

This gives a total of 2,145 people employed in agriculture.

Therefore, the probability that a person earns a living through agriculture is:

Probability = (2,145 / 12,602) x 100% ≈ 17%

The probability that an employed person earns a living through agriculture is ≈ 17%.

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The differential equation y'' - 12y' + 36y = 36[tex]x^4[/tex] is solved using the method of undetermined coefficients. The particular solution is found to be y_p = (1/72)[tex]x^6[/tex] - (1/12)[tex]x^4[/tex]+ (1/6)[tex]x^{2}[/tex].

To solve the given differential equation using the method of undetermined coefficients, we assume a particular solution of the form y_p = A[tex]x^6[/tex] + B[tex]x^4[/tex] + C[tex]x^{2}[/tex], where A, B, and C are constants to be determined. We differentiate y_p twice to find its derivatives: y_p' = 6A[tex]x^5[/tex] + 4B[tex]x^3[/tex]+ 2Cx and y_p'' = 30A[tex]x^4[/tex] + 12B[tex]x^{2}[/tex] + 2C.

Substituting these derivatives into the original differential equation, we have:

30A[tex]x^4[/tex] + 12B[tex]x^{2}[/tex] + 2C - 12(6A[tex]x^5[/tex] + 4B[tex]x^3[/tex] + 2Cx) + 36(A[tex]x^6[/tex] + B[tex]x^4[/tex] + C[tex]x^{2}[/tex]) = 36[tex]x^4[/tex].

Simplifying and equating the coefficients of like powers of x, we obtain the following equations:

36A = 0 (coefficient of x^6 term),

-72A + 36B = 0 (coefficient of x^4 term),

-36B + 36C = 36 (coefficient of x^2 term).

Solving these equations, we find A = 0, B = -1/12, and C = 1/6. Therefore, the particular solution is y_p = (1/72)[tex]x^6[/tex]- (1/12)[tex]x^4[/tex]+ (1/6)[tex]x^{2}[/tex].

The general solution of the given differential equation is the sum of the particular solution and the homogeneous solution. However, since the equation does not specify any initial conditions, we only provide the particular solution in this case.

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In part (A) of the problem, it is required to prove that if the initial conditions f(x) and g(x) are even functions and the forcing function F(x, t) is even for every t > 0, then the solution u(x, t) is also even with respect to x for every t > 0. In part (B), the task is to prove that if f(x) and g(x) are periodic functions and the forcing function F(x, t) is periodic for every t ≥ 0, then the solution u(x, t) is also periodic for every t ≥ 0.

To prove part (A), we can use the principle of superposition, which states that if the initial conditions and forcing function are even, then the solution will also possess the property of evenness.

To prove part (B), we can use the fact that if the initial conditions and forcing function are periodic, the solution will be a linear combination of periodic functions. The sum of periodic functions is also periodic, thus making the solution u(x, t) periodic for every t ≥ 0.

By leveraging these principles and the given assumptions about the initial conditions and forcing function, it can be shown that the solutions u(x, t) will also possess the specified properties of evenness or periodicity, depending on the case.

Note: The explanation provided is a general overview of the approach without delving into the mathematical details and formal proofs.

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Answer:

54

Step-by-step explanation:

First of all,

Cross multiplication

4 = 12

18 = x

Let the value of the number of eggs be represented by x

4x = 12×18

4x = 216

4. 4

(Same as 216÷4)

x = 54