Solve the following recurrence relation: remarks: ∑i=1 i = n(n + 1) / 2
∑i=1 i^2 = n(n + 1) (2n +1) / 6

Let's begin by using the formulas for the volume and surface area of a cube:

Volume = edge^3

Surface Area = 6 * edge^2

1. To find how fast the volume is shrinking when each edge is 13 cm, we need to take the derivative of the volume formula with respect to time:

dV/dt = 3(edge)^2 * (d(edge)/dt)

We know that the rate of change of the edge is -3 cm/sec (since it is shrinking), and we are given that the edge length is 13 cm. Substituting these values into the derivative formula, we get:

dV/dt = 3(13)^2 * (-3) = -1521 cm^3/sec

Therefore, the volume is shrinking at a rate of 1521 cm^3/sec when each edge is 13 cm.

2. Similarly, to find how fast the surface area is decreasing when each edge is 13 cm, we need to take the derivative of the surface area formula with respect to time:

dS/dt = 12(edge) * (d(edge)/dt)

Using the same values as before, we get:

dS/dt = 12(13) * (-3) = -468 cm^2/sec

Therefore, the surface area is decreasing at a rate of 468 cm^2/sec when each edge is 13 cm.
1. To find the rate at which the volume is shrinking, we can use the formula for the volume of a cube (V = a³) and differentiate with respect to time (t):

dV/dt = d(a³)/dt = 3a²(da/dt)

Given that the edges are shrinking at a rate of 3 cm/sec (da/dt = -3 cm/sec), and each edge is 13 cm:

dV/dt = 3(13²)(-3) = -1521 cm³/sec

The volume is shrinking at a rate of 1521 cm³/sec.

2. To find the rate at which the surface area is decreasing, we can use the formula for the surface area of a cube (S = 6a²) and differentiate with respect to time (t):

dS/dt = d(6a²)/dt = 12a(da/dt)

Given that the edges are shrinking at a rate of 3 cm/sec (da/dt = -3 cm/sec), and each edge is 13 cm:

dS/dt = 12(13)(-3) = -468 cm²/sec

The surface area is decreasing at a rate of 468 cm²/sec.

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The function represents the area of the rectangle as a varying quadratic function of the length. It is also worth noting that the function is defined for values of l within the range of 0 to 2, as a rectangle cannot have negative or greater than 2 lengths in this scenario.

To express the area A of rectangle R as a function of length l, we can use the formula for the area of a rectangle, which is A = l * w, where l represents the length and w represents the width.

Since we are given that the perimeter of the rectangle is constant at 4ft, we can write an equation using the perimeter formula: 2l + 2w = 4. Simplifying this equation gives us l + w = 2. By solving for w, we have w = 2 - l.

Now, substituting this value of w into the area formula, we get A = l * (2 - l).

The function for the area of the rectangle as a function of length l is A = l(2 - l).

Regarding this function, we know that it is a quadratic function because of the squared term (l^2) present in the expression. The function represents the area of the rectangle as a varying quadratic function of the length. It is also worth noting that the function is defined for values of l within the range of 0 to 2, as a rectangle cannot have negative or greater than 2 lengths in this scenario.

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To write the equation of the straight line parallel to the straight line 2y = 4x + 5 which passes through the point (0, 2), we will use the following steps.

Step 1: We first find the slope of the straight line 2y = 4x + 5.

We can write the equation 2y = 4x + 5 in the slope-intercept form of a straight line y = mx + b by dividing both sides by 2.2y / 2 = 4x / 2 + 5 / 2y = 2x + 5 / 2

The slope m of the straight line 2y = 4x + 5 is the coefficient of x, which is 2.

Thus, the slope m of the straight line parallel to the straight line 2y = 4x + 5 is also 2.

Step 2: We use the point-slope form of a straight line to write the equation of the straight line parallel to the straight line 2y = 4x + 5 which passes through the point (0, 2).

The point-slope form of a straight line is y - y1 = m(x - x1), where (x1, y1) is a given point on the straight line and m is its slope.Substituting m = 2 and (x1, y1) = (0, 2) in the above equation, we get:

y - 2 = 2(x - 0)y - 2 = 2x The required equation of the straight line parallel to the straight line 2y = 4x + 5 which passes through the point (0, 2) is y = 2x + 2.

Note: The equation of the straight line 2y = 4x + 5 is equivalent to the equation y = 2x + 5 / 2 in the slope-intercept form of a straight line.

It is better to use the exact coefficients of x and y in the point-slope form of a straight line to avoid possible errors.

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In order to match the expressions on the left to their simplified versions on the right, specific expressions and their simplified forms need to be provided. Without the expressions to work with, it is not possible to generate a specific answer.

To match expressions to their simplified versions, we need the actual expressions. It could involve algebraic expressions, equations, mathematical operations, or any other mathematical representation. Without knowing the expressions, it is not possible to provide a specific answer.

If you can provide the expressions, I will be more than happy to help match them to their simplified forms and provide a detailed explanation for each match.

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Answer:

they are supplementary set the equal to 180 so angle 1 would be 112

We can conclude that the possible values of the third side of the triangle in the figure are between 4 cm and 20 cm inclusive (i.e. 4 cm ≤ third side ≤ 20 cm).

When finding the possible values of the third side of a triangle, it is important to consider the triangle inequality theorem.

This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.In the figure given, we see that the two sides of the triangle are 8 cm and 12 cm in length.

Therefore, we can use the triangle inequality theorem to find the possible values of the third side.

We know that the sum of the lengths of the two sides must be greater than the length of the third side.Thus, we can write:8 + 12 > third side20 > third side

Therefore, the length of the third side must be less than 20 cm. Additionally, we know that the length of the third side must be greater than the difference between the lengths of the other two sides.

Thus, we can write:12 - 8 < third side4 < third side

Therefore, the length of the third side must be greater than 4 cm.

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Answer:

Answers

Step-by-step explanation:

Answer:

Yes they can.

The only time they would have to be different lengths is if it was a square

The first four terms of the binomial series are 1-  1/2 x+  3/16 x^2-  1/16 x^3. (Option C)

In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like ^n for a nonnegative integer n. For a binomial series, the formula for the expansion of any terms is given by

(1+x)^m=1+mx+  m(m-1)/2! x^2+  m(m-1)(m-2)/3! x^3+⋯.

For the given function (1+ x/4)^(-2)

m = -2

x = x/4

Hence, using the above expansion formula

The first term = 1

The second term = mx = (-2)(x/4) = -x/2

The third term =m(m-1)/2! (x/4)^2=  (-2(-2-1))/(2*1*16) x^2=3/16 x^2

The fourth term = m(m-1)(m-2)/3! (x/4)^3=(-2(-2-1)(-2-2))/(3*2*1*64) x^3=-1/16 x^3

Hence, the first four terms of the binomial series are 1-  1/2 x+  3/16 x^2-  1/16 x^3.

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Answer:

the slope of the line would be 3/4

Step-by-step explanation:

to find this use your equation for slope: y2-y1/x2-x1

this will give you 5-2/4-0

simplified it is 3/4

another way to know the slope is that the slope will be rise/run, so since the line is going up 3 places and over 4 places, your slope would be 3/4

\(f(x) = \log_b(x)\) is a decreasing function when \(0 < b < 1\)

This is because we can use the change of base formula to say

\(\log_b(x) = \frac{\log(x)}{\log(b)}\)

If b is between 0 and 1, not including either endpoint, notice how the log(b) term is negative.

For example, if b = 0.5, then log(b) = log(0.5) = -0.301 approximately. I'm using log base 10 to get log(0.5) = -0.301

So for b = 0.5, we have,

\(\log_{0.5}(x) = \frac{\log(x)}{\log(0.5)} \approx \frac{\log(x)}{-0.301} \approx -3.322\log(x)\)

The log(x) part on its own is always increasing. The negative coefficient out front flips it to always decreasing.

By applying the behavior rules we notice that the expression \(F(x) = \log_{b} x\)decreasing function if the base of the logarithm is 0 < b < 1. (Correct choice: D) #SPJ5

Logarithms are trascendent functions whose form is defined by the following expression:

\(\log _{b} x\) such that \(x = b^{a}\), where b > 0.

Where b is the base of the power.

Whose rules are described below:

By applying the behavior rules we notice that the expression \(F(x) = \log_{b} x\) is a decreasing function if the base of the logarithm is 0 < b < 1. (Correct choice: D)

The statement is poorly formatted and reports several mistakes. Correct form is shown below:

For what values of b will \(F(x) = \log_{b} x\) be a decreasing function?

A. b > 0

B. 0

C. b < 0

D. 0 < b < 1

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Answer:

9 1/2

Step-by-step explanation:

Add 2 1/2 plus 3 1/2 plus 3 1/2 to get 9 1/2.

The entry in the check register should show a debit of $30.00 in the savings account and a credit of $30.00 in the checking account, indicating that the money has been transferred from one account to the other.

A check register is a piece of writing that lists all check payments' payment dates, check numbers, payment amounts, and payee names. The report is a crucial component of the accounts payable process because it is used to identify the precise payments that are included in a check run.

For this transaction, the entry process is explained below:

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Answer:

x = 105.41

Step-by-step explanation:

24•log(3x) = 60

Divide by 24

log (3x) = 60/24

log (3x) = 5/2

Raise each side to the power of 10

10 ^ log (3x) = 10^5/2

3x= 10 ^5/2

Divide by 3

3x/3 = 1/3   * 10 ^ 5/2

x = 1/3 (100 sqrt(10))

x=105.40925

To two decimal places

x = 105.41

The mean of the sampling distribution of means always equals the mean of the underlying raw score population when the sample size is large. This is known as the central limit theorem, which is a fundamental principle in statistics that describes the behavior of sample means.

The central limit theorem states that as the sample size increases, the distribution of sample means approaches a normal distribution with a mean equal to the population means and a standard deviation equal to the population standard deviation divided by the square root of the sample size. This means that the larger the sample size, the more representative the sample mean is of the population mean.

The central limit theorem is important in statistical analysis because it allows us to make inferences about population parameters based on sample data. By calculating the mean and standard deviation of the sampling distribution of means, we can estimate the population means and assess the probability of obtaining certain sample means.

However, it is important to note that the central limit theorem applies only to random samples from a population with finite variance. It may not hold for non-random samples or populations with infinite variances. Additionally, the theorem assumes that the sample means are independent and identically distributed and that the sample size is sufficiently large.

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Answer: 4.25ft

Step-by-step explanation:

As a result, 51 inches equals 4.25 feet.

Division is one of the four fundamental arithmetic operations that allow integers to be joined to create new numbers. Addition, subtraction, and multiplication are the other operations. Division is the process of dividing a certain amount into equal pieces in mathematics. For example, a group of 20 people can be divided into four groups of five people each, or five groups of four people each, and so on. To split is to divide into two or more equal pieces, areas, classes, categories, groups, or divisions. Divide simply means to cut something into equal parts and give it to a group.

Here,

1 foot is equal to 12 inches, so to convert from inches to feet, divide the number of inches by 12.

So, 51 inches = 51 ÷ 12 = 4.25 feet

Therefore, 51 inches is equal to 4.25 feet.

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The scale factor is 2 and center dilation is reduced.

The scale factor is a ratio that describes how much a figure has been enlarged or reduced. It is calculated by dividing the length of the corresponding sides of the original and dilated figures. If the scale factor is greater than 1, then the figure is enlarged, and if it is less than 1, then the figure is reduced.

To find the scale factor in an IXL dilations problem, you need to compare the corresponding sides of the original and dilated figures.

If the original figure has a side length of 4 units, and the dilated figure has a corresponding side length of 8 units, then the scale factor is 8/4=2. This means that the dilated figure is twice as large as the original figure.

The center of dilation is the point about which the figure is enlarged or reduced. It is the fixed point that remains unchanged during the dilation process.

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The perimeter of the triangle whose coordinates are A (2, 2), B (3, 3), and C (4, 2) is 4.828

We can determine the perimeter of a triangle with vertices by utilizing the distance formula.

Given that, we have the vertices (x₁,y₁), (x₂, y₂), (x₃, y₃). Then the length of each sides can be expressed as follows:

SIde AB = \(\mathbf{\sqrt{(x_1 -x_2)^2 +(y_1 -y_2)^2}}\)

Side AC = \(\mathbf{\sqrt{(x_1 -x_3)^2 +(y_1 -y_3)^2}}\)

Side BC = \(\mathbf{\sqrt{(x_2 -x_3)^2 +(y_2 -y_3)^2}}\)

Replacing the values of the coordinates into the given formula above the perimeter of the triangle is 4.828

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The given expression √10 and √26  are irrational numbers.

The sum of √10 and √26 cannot be simplified because the two numbers are not like terms. However, we can estimate the value of the sum by using the fact that the square root of a number is between two consecutive perfect squares.

For √10, we know that 9 < 10 < 16, so √10 is between 3 and 4. For √26, we know that 25 < 26 < 36, so √26 is between 5 and 6. Therefore, we can estimate the sum of √10 and √26 to be between 8 and 10.

To get a more precise value, we can use a calculator to evaluate the sum of √10 and √26. Using a calculator, we find that the sum of √10 and √26 is approximately 7.051. Therefore, we can describe the sum of √10 and √26 as a decimal number that is approximately 7.051.

Alternatively, we can write the sum of √10 and √26 as a simplified radical expression. To do this, we need to combine the two radical terms into a single term using the distributive property of multiplication. We have:

√10 + √26 = √(10 × 1) + √(26 × 1) = √10 × √1 + √26 × √1 = √10 + √26

Therefore, we cannot simplify the sum of √10 and √26 any further, and we can describe it as the sum of two irrational numbers.

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The equation of line passing through the point (4,5) and has a slope of 0 is y = 5

Slope is the ratio of the change in the y-coordinate to the change in the x-coordinate.

It used to know how steep a line is.

Slope can be negative or positive or zero depending on how the line or curve appears.

if the line passes through a point with a slope of 0, it means the line is either vertical or horizontal.

equation of line passing through a point with a slope is

y - \(y_{1}\) / x - \(x_{1}\) = slope

y - 5/x --4 = 0

y - 5/x + 4 = 0

by cross multiplying,

y - 5 = 0( x + 4)

y - 5 = 0

y = 5

In conclusion, the equation of line is y = 5

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Based on the fact that Brielle mixed 8 cups of blue and 6 cups of yellow paint, the complete table to show a new mixture that matches this paint color is:

The original paint color is in the ratio:
8 : 6

This can be simplified to:

(8 : 6 ) / 2

4 : 3

For every 4 blue cups, there will be 3 yellow cups.

If there are 9 yellow cups then the number of blue cups would be:

= (9 x 4) / 3

= 12 blue cups

If there are 16 blue cups, there will be:
= (16  x 3) / 4

= 12 yellow cups

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Pi written as 3.l4 and π in mathematics is the ratio for the diameter of a circle to the circumference of that circle. The value is unending

Pi, which is represented by the Greek letter p, or, is a shorthand for the ratio of a circle's diameter to its circumference. No matter how big the circle is, this ratio will always be equal to pi. The value of pi is about 3.14 in decimal notation.

We are to multiply Pi to any equation

let the equation be (20.5 + 3x)\(\pi\)

pi is represented in numbers by 3.14

Hence we would have the equation to be of the form

(20.5 + 3x)3.14

multiply through the bracket

64.37 + 9.42x

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A van can carry 12 students, and a bus can carry 36 students. The senior class at NAHS rented and filled 6 vans and 12 buses with 504 students.

To determine how many students a van can carry and how many students a bus can carry, we can use a system of equations based on the given information.

Let's assume that the number of students a van can carry is "v" and the number of students a bus can carry is "b".

According to the information provided:

The senior class at NAHS rented and filled 6 vans and 12 buses with a total of 504 students.

The senior class at FCHS rented and filled 2 vans and 5 buses with a total of 204 students.

Based on these conditions, we can form the following equations:

Equation 1: 6v + 12b = 504

Equation 2: 2v + 5b = 204

We now have a system of two equations with two unknowns (v and b). We can solve this system to find the values of v and b.

Let's solve the system using any preferred method, such as substitution or elimination.

Multiplying Equation 2 by 6, we get:

12v + 30b = 1224

Subtracting this equation from Equation 1, we can eliminate v:

6v + 12b - (12v + 30b) = 504 - 1224

-6v - 18b = -720

Simplifying this equation, we get:

-6v - 18b = -720

Dividing this equation by -6, we obtain:

v + 3b = 120

Now we have a new equation:

v + 3b = 120 -------------- Equation 3

We can now solve Equations 2 and 3 as a new system of equations.

Multiplying Equation 3 by 2, we get:

2v + 6b = 240

Subtracting this equation from Equation 2, we can eliminate v:

2v + 5b - (2v + 6b) = 204 - 240

b = -36

Dividing both sides of the equation by -1, we find:

b = 36

Now, substituting the value of b back into Equation 3:

v + 3(36) = 120

v + 108 = 120

v = 12

Therefore, a van can carry 12 students, and a bus can carry 36 students.

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Answer:

-x

Step-by-step explanation:

I'm not sure if every teacher does this but my teacher does.

The regular price of the poster is $ 15.6.

It is a question of linear equation in one variable.

It is given in the question that:-

Number of posters Sunita wants to buy = 5

Total money paid by Sunita = $ 63.

Discount on each poster = $ 3.

We have to find the regular price of the poster.

Let the regular price of the poster be p.

Hence, we can write,

Number of posters*(regular price of the poster - discount on each poster) = total money paid by Sunita

5*(p-3) = 63

p - 3 = 63/5

p - 3 = 12.6

p = 12.6 + 3

p = $ 15.6

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Answer:  (1) D. BE ≡ AD

               (2) B. AAS Postulate

               (3) B. RF

Step-by-step explanation:

1) Place the letters underneath each other and match them up

                                 B A E

                                 D E A

∠B = ∠D                 ∠A = ∠E                ∠E = ∠A

BA ≅ DE                  AE ≅ EA              BE ≅ DA

2)

      ∠V = ∠Y              Angles are congruent (given)

∠VWZ = ∠YWZ        Angles are congruent (def of ∠ bisector)

     WZ = WZ            Sides are the same (reflexive property)          

ΔWZV ≡ ΔWZY      Angle-Angle-Side (AAS) Postulate

3)   F ∴__________________∴ R

     Segment FR = RF is between ∠F and ∠R

Answer:

x=12

Step-by-step explanation:

Try using m a t h w a y, it helps with this kind of math.

Answer: 12

Step-by-step explanation:

Seems like your in a rush so I'll edit the explanation after- goodluck!

The range would be the most appropriate measure of spread in this case because it takes into account the extreme values of $300 and $1,300 and provides a clear measure of the difference between them.

To measure the amount of money college students spend on rent per month, the most appropriate measure of spread would be the range. The range is the simplest measure of spread and is calculated by subtracting the lowest value from the highest value in a data set. In this case, the range would be $1,300 - $300 = $1,000.

The median would not be the best choice in this scenario because it only represents the middle value in a data set. It does not take into account extreme values like the $1,300 rent expense.

Standard deviation would not be the most appropriate measure of spread in this case because it calculates the average deviation of each data point from the mean. However, it may not accurately represent the spread when extreme values like the $1,300 rent expense are present.

The interquartile range (IQR) would not be the best choice either because it measures the spread of the middle 50% of the data set. It does not consider extreme values and would not accurately represent the range of rent expenses in this scenario.

In summary, the range would be the most appropriate measure of spread in this case because it takes into account the extreme values of $300 and $1,300 and provides a clear measure of the difference between them.

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