Help Please................

If Mr. and Mrs. Smith choose the hire purchase plan, the total cost of the washing machine will be $2320.00.

If Mr. and Mrs. Smith decide to purchase the washing machine on a hire purchase plan, they have two options: making a cash payment or choosing the hire purchase plan with a down payment and monthly installments.

Cash Payment:

If they choose to make a cash payment, they will pay the full price of $2000.00 upfront, and they will own the washing machine immediately.

Hire Purchase Plan:

If they opt for the hire purchase plan, they need to make a down payment and pay equal monthly installments. Here are the details:

Down Payment:

The down payment is 20% of the total price, which is $2000.00. So, 20% of $2000.00 is:

Down payment = 20/100 ×$2000.00 = $400.00

Monthly Installments:

The remaining amount after the down payment is $2000.00 - $400.00 = $1600.00.

They will pay this remaining amount in 12 equal monthly installments of $160.00 each.

Total Cost:

To calculate the total cost, we need to add the down payment to the sum of the monthly installments:

Total Cost = Down Payment + (Monthly Installments x Number of Months)

Total Cost = $400.00 + ($160.00 x 12) = $400.00 + $1920.00 = $2320.00

Therefore, if Mr. and Mrs. Smith choose the hire purchase plan, the total cost of the washing machine will be $2320.00.

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The answer is True.

The logistic regression model is designed to describe the probability of a certain outcome (in this case, disease development) based on a given set of independent variables. It models the relationship between the independent variables and the probability of the outcome, which is the risk for the disease.

Logistic regression models the probability of the dependent variable being 1 (i.e., having the disease) as a function of the independent variables, using the logistic function. The logistic function maps any real-valued input to a value between 0 and 1, which can be interpreted as the probability of the dependent variable being 1.

Therefore, the logistic regression model can be used to estimate the risk of disease development based on a given set of independent variables.

By examining the coefficients of the independent variables in the logistic regression equation, we can identify which variables are associated with an increased or decreased risk of disease development.

This information can be used to develop strategies for preventing or treating the disease.

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Circle parametric equations are equations that define the coordinates of points on a circle in terms of a parameter, such as the angle of rotation. The equations are often written in the form x = r cos(theta) and y = r sin(theta), where r is the radius of the circle and theta is the parameter.

These equations can be used to graph circles and to solve problems involving circles, such as finding the intersection of two circles or the area of a sector of a circle. Circle parametric equations are commonly used in mathematics, physics, and engineering.

Given the circle equation (x−9)²+(y−6)²=4, we can find the parametric equations to represent the circle being traced clockwise as the parameter increases.

Step 1: Rewrite the circle equation in terms of radius
The circle equation can be written as (x−h)²+(y−k)²=r², where (h, k) is the center of the circle and r is the radius. In this case, h=9, k=6, and r=√4 = 2.

Step 2: Write the parametric equations for x and y
Since the circle is traced clockwise, we use negative sine for the y-coordinate. The parametric equations for the circle are:
x = h + rcos(t) = 9 + 2cos(t)
y = k - rsin(t) = 6 - 2sin(t)

As given, x = 9 + 2cos(t). The parametric equations representing the circle being traced clockwise are:
x = 9 + 2cos(t)
y = 6 - 2sin(t)

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The average temperature between time 0 and time 10 is 40°F.

To find the average temperature, you need to integrate the temperature function over the interval [0, 10] and then divide by the length of the interval. The given temperature function is T(t) = -0.6t² + 2t + 70. First, integrate T(t) with respect to t from 0 to 10:

∫(-0.6t² + 2t + 70) dt from 0 to 10 = [-0.2t³ + t² + 70t] evaluated from 0 to 10.

Next, substitute the limits of integration and subtract:

[-0.2(10³) + (10²) + 70(10)] - [-0.2(0³) + (0²) + 70(0)] = 400.

Finally, divide the result by the length of the interval (10 - 0 = 10):

Average temperature = 400/10 = 40°F.

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Leo multiplied all the numbers from 1 to 11 and wrote the answer on the board. The erased digits on the board are 3, 9, and 6.

The product of these numbers is calculated as 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11. During the break, three digits were erased: 39, 9, and 6.

To find the erased digits, we can divide the remaining product on the board by the product of the non-erased digits. The remaining product is equal to 1 x 2 x 4 x 5 x 7 x 8 x 10 x 11. By dividing the original product by the remaining product, we can determine the missing digits.

Calculating (1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11) / (1 x 2 x 4 x 5 x 7 x 8 x 10 x 11), we find that the result is 3 x 9 x 6.

Therefore, the erased digits on the board are 3, 9, and 6.

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Answer:

  286.51 cm

Step-by-step explanation:

You want the circumference of a circle with radius 45.6 cm.

The circumference of a circle is given by the formula ...

  C = 2πr

For the given radius, the circumference is ...

  C = 2π(45.6 cm) = 286.51 cm

The circumference is about 286.51 cm.

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The matrix A is:
A = [-2 1 0; 0 -5 0; 4 0 0; 0 0 -2; 0 0 0; 0 0 6]
T(-2, 5, 4) = (-18, -25, -8, 4, 0, 24).

(a) To find the matrix A, we need to find the image of each basis vector under T and write them as columns of a matrix. Therefore, we have:

T(e1) = (-2, 0, 4, 0, 0, 0)T
T(e2) = (1, -5, 0, 0, 0, 0)T
T(e3) = (0, 0, 0, -2, 0, 6)T


(b) To find T(-2, 5, 4), we simply need to multiply the matrix A by the vector (-2, 5, 4, 0, 0, 0)T, i.e.,

T(-2, 5, 4) = [-2 1 0; 0 -5 0; 4 0 0; 0 0 -2; 0 0 0; 0 0 6] * [-2; 5; 4] = [-18; -25; -8; 4; 0; 24]



(c) To determine whether T is one-to-one or not, we need to check if the nullspace of A is trivial or not. The nullspace of A is the set of all vectors x such that Ax = 0. We can find the nullspace of A by row reducing the augmented matrix [A|0].

However, we can see that the rank of A is 3, which means that the nullspace of A is non-trivial, and hence, T is not one-to-one.

(d) To determine whether T is onto or not, we need to check if the range of T is equal to R3 or not. Since the columns of A span R3,

we can conclude that the range of T is equal to the column space of A, which is a subspace of R3. Therefore, T is not onto.

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The derivative of the function h(x) = ∫[3+√(x)]^3 n(r) dr can be found using Part 1 of the Fundamental Theorem of Calculus. The result is h'(x) = n([3+√(x)]) * [3+√(x)]^2.

According to Part 1 of the Fundamental Theorem of Calculus, if a function h(x) is defined as the integral of another function n(r) with respect to r over a certain interval, then the derivative of h(x) with respect to x can be found by evaluating the integrand at the upper limit of integration and multiplying it by the derivative of the upper limit with respect to x.

In this case, the function h(x) is defined as the integral of n(r) with respect to r, where the lower limit is a constant 3 and the upper limit is 3+√(x). To find h'(x), we evaluate n(r) at the upper limit of integration, which is [3+√(x)], and multiply it by the derivative of the upper limit with respect to x, which is 2√(x).

Therefore, h'(x) = n([3+√(x)]) * 2√(x) = 2√(x) * n([3+√(x)]) = n([3+√(x)]) * [3+√(x)]^2.

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To determine μx and σx, we can use the formula:

μx = μ
σx = σ / √n

Plugging in the values we get:

μx = 68
σx = 20 / √29 ≈ 3.71

Therefore, the sample mean is 68 and the sample standard deviation is approximately 3.71.


μx represents the mean of the sample and σx represents the standard deviation of the sample. We can calculate these values using the formula provided above, which involves the population mean (μ), population standard deviation (σ), and sample size (n).

In this case, the population mean is 68, the population standard deviation is 20, and the sample size is 29. By plugging in these values into the formula, we can calculate the sample mean and sample standard deviation.


By calculating the sample mean and sample standard deviation, we have a better understanding of the distribution of the sample data. These values can be used to make inferences about the population, such as estimating population parameters or testing hypotheses.
 Let's determine μx (the mean of the sample) and σx (the standard deviation of the sample) using the given population parameters and sample size.


μx = μ = 68
σx = σ / √n = 20 / √29

Explanation:
1. The mean of the sample (μx) is equal to the mean of the population (μ), so μx = 68.
2. To find the standard deviation of the sample (σx), you need to divide the population standard deviation (σ) by the square root of the sample size (n). In this case, σ = 20 and n = 29, so σx = 20 / √29.


For the given population parameters and sample size, the mean of the sample (μx) is 68, and the standard deviation of the sample (σx) is approximately 3.71 (20 / √29 ≈ 3.71).

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The power set in roster notation requires listing all the possible subsets of a set, including the empty set and the set itself. The number of subsets in a power set can be calculated using the formula 2^n, where n is the number of elements in the original set.

The power set of a set is the set of all its subsets, including the empty set and the set itself. To write out the power set in roster notation, we need to list all the possible subsets of a given set.
(a) The set {a} has two subsets: {a} and {}. Therefore, the power set of {a} in roster notation is {{}, {a}}.
(b) The set {1,2} has four subsets: {1,2}, {1}, {2}, and {}. Therefore, the power set of {1,2} in roster notation is {{}, {1}, {2}, {1,2}}.
It is important to note that the cardinality (number of elements) of the power set of a set with n elements is 2^n. For example, the set {1,2} has two elements, so its power set has 2^2 = 4 subsets. Similarly, the set {a} has one element, so its power set has 2^1 = 2 subsets.
In conclusion, writing out the power set in roster notation requires listing all the possible subsets of a set, including the empty set and the set itself. The number of subsets in a power set can be calculated using the formula 2^n, where n is the number of elements in the original set.

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Answer:

4 according to the numbers you provided integer x the = 4

Step-by-step explanation:

The student population at Chapman Elementary School decreased by approximately 10% after rezoning. This corresponds to a decrease of 67 students from the previous year's population of 670.

In order to calculate the percent decrease in the student population, we can use the following formula:

Percent decrease = ((Initial population - Final population) / Initial population) * 100

Substituting the given values into the formula, we get:

Percent decrease = ((670 - 603) / 670) * 100

= (67 / 670) * 100

= 0.1 * 100

= 10%

Therefore, the percent decrease in the student population at Chapman Elementary School after rezoning is 10%. This indicates that the student population decreased by 10% from the previous year's count of 670 students, resulting in a current population of 603 students.

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The perimeter of a regular octagon with a side length of 2.4mm can be calculated by multiplying the length of one side by the number of sides, which is 8.

A regular octagon is a polygon with eight equal sides and angles. To find the perimeter, we need to calculate the total distance around the octagon.

Since all sides of a regular octagon are equal, we can simply multiply the length of one side by the number of sides to find the perimeter. In this case, the side length is given as 2.4mm, and the octagon has 8 sides.

Perimeter = Side length * Number of sides = 2.4mm * 8 = 19.2mm.

Therefore, the perimeter of the regular octagon with a side length of 2.4mm is 19.2mm.

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Therefore, the work done by the force field F is 10π given by the line integral.

The work done by the force field F along the arch of the cycloid is given by the line integral of F·dr over the curve r(t), i.e.,

W = ∫C F · dr = ∫0^2π F(r(t)) · r'(t) dt

Using the given values of F(x,y) and r(t), we can compute F(r(t)) · r'(t) as follows:

F(r(t)) · r'(t) = (t - sin(t))i + (5 - cos(t))j · (cos(t)i + sin(t)j)

= (t - sin(t))cos(t) + (5 - cos(t))sin(t)

Hence, we have:

W = ∫0^2π [(t - sin(t))cos(t) + (5 - cos(t))sin(t)] dt

integration by parts, we can evaluate this integral to get:

W = [t sin(t) + (5 - cos(t))cos(t)]|0^2π

= 10π

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Using Cramer's rule the values are:

a) b12 = -15/22

b) b22 = 1/22

c) bs2 = 5/22

d) b23 = -3/22

To find the entries of A-1, we can use Cramer's rule and the formula for computing inverse matrices. First, we need to compute the determinant of A using row reduction:

|1 0 5 3|

|0 1 3 2| = det(A)

|1 0 1 1|

|1 0 0 1|

We can reduce the matrix to upper triangular form by subtracting the first row from the third and fourth rows:

|1 0 5 3|

|0 1 3 2|

|0 0 -4 -2|

|0 0 -5 -2|

Now, the determinant of A is the product of the diagonal entries, which is (-4)(-2)(1)(1) = 8.

To find b12, we replace the second column of A with the column vector [0 1 0 0] and compute the determinant of the resulting matrix. We get:

|-15 0 5 3|

| 8 1 3 2| = b12 det(A)

|-11 0 1 1|

| 4 0 0 1|

Using the formula for 4x4 determinants, we can expand along the first column to get:

b12 = (-15)(-2)(1) + (8)(1)(1) + (-11)(0)(-2) + (4)(0)(5) = -15/22

Similarly, we can find b22, bs2, and b23 by replacing the corresponding columns of A with [0 1 0 0], [0 0 1 0], and [0 0 0 1], respectively, and computing the determinants of the resulting matrices using Cramer's rule. We get:

b22 = 1/22

bs2 = 5/22

b23 = -3/22

Therefore, the entries of A-1 are:

| -15/22 1/22 5/22 |

| 7/22 1/22 -3/22 |

| 1/22 -2/22 1/22 |

Note that we can also find A-1 directly using the formula A-1 = (1/det(A)) adj(A), where adj(A) is the adjugate matrix of A. The adjugate matrix is obtained by taking the transpose of the matrix of cofactors of A, where the (i,j)-cofactor of A is (-1)^(i+j) times the determinant of the submatrix obtained by deleting the i-th row and j-th column of A.

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(a) The function f = 1A is f-measurable.

(b) For every m ∈ N (m > 1), the set Em is f-measurable.

(a) To show that f = 1A is f-measurable, we need to show that the preimage of any Borel set B in R is f-measurable. Since f can only take values 0 or 1, the preimage of any Borel set B is either the empty set, X, A or X \ A, all of which are f-measurable. Therefore, f is f-measurable.

(b) To show that Em is f-measurable, we need to show that its complement E^c_m is f-measurable. Let E^c_m be the set of points that belong to less than m sets Ak.

Then E^c_m is the union of all intersections of at most m-1 sets Ak. Since each Ak is f-measurable, any finite intersection of at most m-1 sets Ak is also f-measurable. Hence, E^c_m is f-measurable, and therefore Em is also f-measurable.

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The probability of getting a green ball and purple ball is 4/27

A probability is a number that reflects the chance or likelihood that a particular event will occur. The certainty of an event is 1 and the equivalent in percentage is 100%.

Probability = sample space /Total outcome

total outcome = 4+3+2 = 9

For the first draw,

probability of picking a green = 4/9

for the second draw;

probability of picking a purple = 3/9 = 1/3

The probability of getting a green and a purple = 1/3 × 4/9

= 4/27

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Answer: None of the above are irrational numbers

Step-by-step explanation:

Answer:

2 and 3 only

Step-by-step explanation:

1 ) 10n = 103

    n = 103/10 = 10.3  

2) 5n = 15

    n  =  15/5 = 3    

3)  

[tex]\frac{1}{4}+n = \frac{13}{4}\\ n = \frac{13}{4}-\frac{1}{4}\\ n = \frac{13-1}{4}\\ n = \frac{12}{4} = 3[/tex]

4) n/2 = 6

    n = 12

5) n/3 = 3

   n = 9

The rate of change of the banking angle when the cars are accelerating at 2 m/s², banking angle is at 45 degrees, velocity is 80 km/h, and the radius of the arc is 20 m is approximately 0.454 radians/s.

The chain rule of differentiation to calculate the rate of change of the banking angle.

Let v be the speed, r be the radius, and x be the banking angle.

Next, we have

v2 = rg(tan x)

r[g(sec2 x)(dx/dt)] + g(tan x)(dr/dt) = 2v(dv/dt) is the result of differentiating both sides with regard to time t.

Using the values supplied, we can reduce the equation as follows:

v = 80 km/h

= 22.22 m/s dv/dt

= 2 m/s2 r

= 20 m g

= 9.8 m/s2 x

= 45 degrees

= /4 radians

When we enter these numbers into the equation, we obtain:

20(9.8(sec2 /4)(dx/dt) plus 9.8(tan /4)(dr/dt) equals 2(22.22).(2)

To put it simply, we obtain 196(dx/dt) plus 98(dr/dt) = 88.88.

We must provide a solution for the banking angle change rate (dx/dt) using the radius change rate (dr/dt).

Rearranging

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Let the number of Eagles albums sold be x, therefore number of Thriller albums sold would be `(x/2)+15`.

We know that Together Eagles – "Their Greatest Hits (1971-1975)" album and Michael Jackson’s Thriller album sold 72 million copies.Hence, we can form the equation:x + (x/2 + 15) = 72 million

2x + x + 30 = 144 million

3x = 144 million - 30 million

3x = 114 million

x = 38 million

Therefore, the number of Eagles albums sold was 38 million.

The number of Thriller albums sold would be `(x/2)+15

= (38/2)+15

= 19+15

= 34`.

Thus, 38 million copies of Eagles album and 34 million copies of Michael Jackson's Thriller album were sold.

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In this random graph, we expect to find approximately 14 unordered cycles of length three.

In an undirected random graph of eight vertices, where the probability of an edge existing between any pair of vertices is 1/2, we can calculate the expected number of unordered cycles of length three.

To determine the expected number, we need to analyze the probability of forming a cycle of length three through any three vertices.

To form a cycle of length three, we must select three distinct vertices. The probability of selecting a particular vertex is 1, and the probability of not selecting it is (1 - 1/2) = 1/2. Hence, the probability of selecting three distinct vertices is (1)(1/2)(1/2) = 1/4.

Since we have eight vertices, the number of ways to choose three distinct vertices is given by the combination formula C(8, 3) = 8! / (3! * (8 - 3)!) = 56.

Therefore, the expected number of unordered cycles of length three is the product of the probability and the number of ways to choose the vertices: (1/4) * 56 = 14.

Therefore, in this random graph, we expect to find approximately 14 unordered cycles of length three.

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Parametric equations:

(a) x(t) = 3cos(-t) = 3cos(t), y(t) = 1 + 3sin(-t) = 1 - 3sin(t)

(b) x(t) = 3cos(4t), y(t) = 1 + 3sin(4t)

(c) x(t) = 3cos(t + π), y(t) = 4 + 3sin(t + π)

(a) Once around clockwise, starting at (3, 1):

We can parameterize the circle by using the cosine and sine functions:

x(t) = 3cos(t)

y(t) = 1 + 3sin(t)

where 0 ≤ t ≤ 2π. To move around the circle clockwise, we can use a negative value of t:

x(t) = 3cos(-t) = 3cos(t)

y(t) = 1 + 3sin(-t) = 1 - 3sin(t)

where 0 ≤ t ≤ 2π.

(b) Four times around counterclockwise, starting at (3, 1):

We can use the same parameterization as in part (a), but use a larger range for t:

x(t) = 3cos(4t)

y(t) = 1 + 3sin(4t)

where 0 ≤ t ≤ 2π/4.

(c) Halfway around counterclockwise, starting at (0, 4):

We can use a similar parameterization as in part (a), but shift the starting point and adjust the range of t:

x(t) = 3cos(t + π)

y(t) = 4 + 3sin(t + π)

where 0 ≤ t ≤ π.

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The set of all 2×3 matrices with entries from ℝ, where each row of entries sums to zero, is indeed a vector space.

To determine if the set of 2×3 matrices with entries from ℝ, where each row sums to zero, forms a vector space, we need to verify if it satisfies the necessary properties of a vector space. These properties include closure under addition and scalar multiplication, associativity, commutativity, existence of a zero vector, existence of additive inverses, and distributive properties.

To check closure under addition, we need to ensure that the sum of any two matrices from the given set is also a matrix in the set. Let's take two arbitrary matrices A and B from the set. Each row of A and B sums to zero. Now, when we add corresponding entries of A and B, the resulting matrix C will also have rows that sum to zero. Thus, the set is closed under addition.

For closure under scalar multiplication, we need to verify that multiplying any matrix from the set by a scalar also produces a matrix within the set. Let's consider an arbitrary matrix A from the set and a scalar c from ℝ. When we multiply each entry of A by c, the resulting matrix cA will also have rows that sum to zero. Therefore, the set is closed under scalar multiplication.

Matrix addition is associative, meaning that for any matrices A, B, and C in the set, (A + B) + C = A + (B + C). This property holds true for matrices in this set since addition of matrices follows the same rules regardless of their row sums.

Matrix addition is commutative, meaning that for any matrices A and B in the set, A + B = B + A. This property also holds true for matrices in this set because the order of addition does not affect the row sums of the resulting matrix.

A zero vector is an element of the set that when added to any other matrix in the set, leaves the other matrix unchanged. In this case, the zero vector is a 2×3 matrix with all entries equal to zero. When we add this zero matrix to any other matrix in the set, the resulting matrix still has rows that sum to zero. Hence, the set contains a zero vector.

For every matrix A in the set, there must exist an additive inverse -A in the set such that A + (-A) = 0. Since each row of A sums to zero, the additive inverse -A will also have rows that sum to zero. Therefore, the set contains additive inverses.

The set needs to satisfy the distributive properties of scalar multiplication over addition and scalar multiplication over scalar addition. These properties hold true for matrices in this set, as the row sums are preserved when performing these operations.

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The inequality to describe the relationship between the number of floors (f) and the maximum number of offices (o) is:

f * o ≤ 195.

Let's assume that the number of floors in the building is represented by the variable "f" and the maximum number of offices on each floor is represented by the variable "o". To write an inequality describing the relationship between the number of floors and the maximum number of offices, we can use the following inequality:

f * o ≤ 195

In this inequality, the product of "f" and "o" represents the total number of offices in the building. We multiply the number of floors by the maximum number of offices per floor to obtain the total number of offices. The inequality states that the total number of offices must be less than or equal to 195.

This inequality ensures that the building does not exceed the maximum limit of 195 offices. It allows for flexibility in the distribution of offices across the floors, as long as the total number of offices does not exceed the given limit.

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Desiree is from Mill City and wants to see "Chili Revenge".

Based on the given information, we can determine the friend from Mill City who wants to see "Chili Revenge". Let's analyze the clues:

There are three friends from Mill City.

Four friends want to see "Out of Asparagus".

Three friends want to see "Chili Revenge".

Paul, Aaron, and Desiree are from the same city.

Lavinia and Jennifer are from different cities.

Xavier, Lavinia, and Sparkly want to see the same movie.

From these clues, we can deduce that Xavier, Lavinia, and Sparkly want to see "Chili Revenge" since they all want to see the same movie. This means that the friend from Mill City who wants to see "Chili Revenge" is Sparkly. Therefore, Sparkly is the friend from Mill City who wants to see "Chili Revenge".

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The function reaches a. n reaches 30 first. b. m reaches 100 first.

We are given that;

Function=m(x) = (1.05) and n(x) = x

Now,

To find the value of x that makes m(x) = 30, we need to solve the equation

m(x) = 30 (1.05)^x = 30 x = log(30)/log(1.05) x ≈ 23.44

n(x) = 30 x = 30

To compare these values, we see that n(x) reaches 30 first, when x = 30, while m(x) reaches 30 later, when x ≈ 23.44.

Similarly, to find the value of x that makes m(x) = 100, we need to solve the equation:

m(x) = 100 (1.05)^x = 100 x = log(100)/log(1.05) x ≈ 46.89

n(x) = 100 x = 100

To compare these values, we see that m(x) reaches 100 first, when x ≈ 46.89, while n(x) reaches 100 later, when x = 100.

Therefore, by the function answer will be a. n reaches 30 first. b. m reaches 100 first.

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A enclose numerators and denominators in parentheses.  f(x)=x 2+4x and g(x)=1−x² is fg(x) = x² - x⁴ + 4x - 4x³ ,gf(x) = x² - x⁴ + 4x - 4x²

To find the values of (f+g)(x), (f-g)(x), fg(x), and gf(x), the respective operations on the given functions f(x) and g(x).

Given:

f(x) = x² + 4x

g(x) = 1 - x²

(f+g)(x):

To find (f+g)(x), the two functions f(x) and g(x):

(f+g)(x) = f(x) + g(x) = (x² + 4x) + (1 - x²)

= x² + 4x + 1 - x²

= (x² - x²) + 4x + 1

= 4x + 1

Therefore, (f+g)(x) = 4x + 1.

(f-g)(x):

To find (f-g)(x), subtract the function g(x) from f(x):

(f-g)(x) = f(x) - g(x) = (x² + 4x) - (1 - x²)

= x² + 4x - 1 + x²

= (x² + x²) + 4x - 1

= 2x² + 4x - 1

Therefore, (f-g)(x) = 2x² + 4x - 1.

fg(x):

fg(x), multiply the two functions f(x) and g(x):

fg(x) = f(x) × g(x) = (x² + 4x) × (1 - x²)

= x² - x⁴ + 4x - 4x³

Therefore, fg(x) = x² - x⁴ + 4x - 4x³.

gf(x):

gf(x), multiply the two functions g(x) and f(x):

gf(x) = g(x) × f(x) = (1 - x²) × (x² + 4x)

= x² - x⁴ + 4x - 4x³

Therefore, gf(x) = x² - x⁴ + 4x - 4x³.

[tex](f+g)(x) = 4x + 1\\\\(f-g)(x) = 2x^2 + 4x - 1\\\\fg(x) = x^2 - x^4 + 4x - 4x^3\\\\gf(x) = x^2 - x^4 + 4x - 4x^3\\[/tex]

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The Null hypothesis would be rejected in favor of an alternative hypothesis, indicating that the type of water used does have an effect on plant growth.

The gardener is testing whether using rainwater instead of tap water would lead to faster plant growth, the null hypothesis (H₀) is a statement that assumes no significant difference or effect between the two variables being compared. In this case, the null hypothesis would state that there is no difference in plant growth between using rainwater and tap water.

The null hypothesis for this scenario can be formulated as follows:

H₀: There is no significant difference in the growth rate of house plants when using rainwater compared to tap water.

This null hypothesis assumes that the type of water used (rainwater or tap water) has no impact on the growth rate of the house plants. It suggests that any observed differences in growth between the two groups (rainwater and tap water) are due to chance or random variation.

When conducting an experiment or study, the purpose is to gather evidence to either support or reject the null hypothesis. If the evidence suggests a significant difference in plant growth between using rainwater and tap water, the null hypothesis would be rejected in favor of an alternative hypothesis, indicating that the type of water used does have an effect on plant growth.

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h'(1) is equal to (g'(1) - 2g(1)). To find the specific value of h'(1), you would need to know the explicit form or additional information about the function g(x) and evaluate it at x = 1.

To find h'(1), we will differentiate h(x) using the quotient rule and then substitute x = 1 into the derivative expression.

Using the quotient rule, the derivative of h(x) = g(x)/[tex]x^{2}[/tex] is given by:

h'(x) = (g'(x) × [tex]x^{2}[/tex] - g(x) × 2x) / [tex](x^{2})^{2}[/tex]

= (g'(x) × x^2 - 2g(x) × x) / [tex]x^{4}[/tex]

= ([tex]x^{2}[/tex] × g'(x) - 2x × g(x)) / [tex]x^{4}[/tex]

= (x × (x × g'(x) - 2g(x))) / x^4

= (x × (x × g'(x) - 2g(x))) / ([tex]x^{2}[/tex] × [tex]x^{2}[/tex])

= (x × (x × g'(x) - 2g(x))) / ([tex]x^{2}[/tex])

Now, substitute x = 1 into the derivative expression:

h'(1) = (1 × (1 × g'(1) - 2g(1))) / (1)

= (g'(1) - 2g(1))

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