when you have an exponent outside of parenthesis, do you do the parenthesis first?

The average rate of change of a function f(x) on the interval [4, 9] is -67.

What is the average rate of change?

The average rate of change of a function between two points is the same that the slope of the line passes through these points (secant line).

The given table:

The average rate of change of a function f(x) over the interval [a, b] is

= f(b) - f(a)/b-a.

Interval : [4, 9]

f(9) = -419,

f(4) = -84

The average rate of change of a function f(x) on the interval [4, 9] is

=  f(9) - f(4)/9 - 4.

= -419 - (-84) / 5

= -297/ 5

= -67

Hence, The average rate of change of a function f(x) on the interval [4, 9] is -67.

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Step-by-step explanation:

(a)

If m is an odd integer, then m + 1 is an even integer.

m (odd integer) m + 1 (even integer)

1 2

3 4

5 6

... ...

Proof:

Suppose m is an odd integer. We can write m as 2n + 1 for some integer n. Then,

m + 1 = (2n + 1) + 1 = 2n + 2

Since 2n + 2 is clearly an even integer, it follows that m + 1 is an even integer if m is an odd integer.

(b)

If x is an even integer and y is an odd integer, then x + y is an odd integer

x (even integer) y (odd integer) x + y (odd integer)

0 1 1

2 3 5

4 5 9

... ... ...

Proof:

Suppose x is an even integer and y is an odd integer. We can write x as 2n and y as 2m + 1 for some integers n and m. Then,

x + y = 2n + (2m + 1) = 2(n + m) + 1

Since n + m is clearly an integer, it follows that x + y is an odd integer if x is an even integer and y is an odd integer.

(c)

If m is an even integer, then 3m^2 + 2m + 3 is an odd integer.

m (even integer) 3m^2 + 2m + 3 (odd integer)

0 3

2 27

4 99

... ...

Proof:

Suppose m is an even integer. We can write m as 2n for some integer n. Then,

3m^2 + 2m + 3 = 3(2n)^2 + 2(2n) + 3 = 12n^2 + 4n + 3

Since 12n^2 + 4n + 3 is clearly an odd integer, it follows that 3m^2 + 2m + 3 is an odd integer if m is an even integer.

The distance Luciano walked more on Sunday is given by the equation A = ( 1/6 ) of a mile

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the distance Luciano walked more on Sunday be A

Now , the equation will be

The distance walked by Luciano on Sunday = ( 5/9 ) mile

The distance walked by Luciano on Saturday = ( 7/18 ) mile

So , the distance Luciano walked more on Sunday A = distance walked by Luciano on Sunday - distance walked by Luciano on Saturday

Substituting the values in the equation , we get

The distance Luciano walked more on Sunday A = ( 5/9 ) - ( 7/18 )

On simplifying the equation , we get

The distance Luciano walked more on Sunday A = ( 10 - 7 ) / 18

The distance Luciano walked more on Sunday A = 3/18 miles

The distance Luciano walked more on Sunday A = 1/6 miles

Hence , the equation is A = 1/6 of a miles

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√(a² + b²)²  - 4a²b²cos²θ is the product of the lengths of the two diagonals is given by the expression.

What is a mathematical expression?

A mathematical expression is a phrase that includes at least two numbers or variables, at least one arithmetic operation, and the expression itself. This mathematical operation may be addition, subtraction, multiplication, or division.

                                An expression's structure is as follows: Number/variable, Math Operator, Number/Variable is an expression.

we have AB as a, AD as b and the angle between them is theta.

So using the cosine rule, we have

  BD = √a² + b² - 2abcosθ

So now consider the triangle ABC

Here AB is a, BC is b and the angle is 180-theta

So using cosine rule, we get AC as

   AC = √a² + b² - 2abcosθ( 180 - θ )

    AC = √a² + b² - 2ab(-cosθ )

    AC = √a² + b² - 2abcosθ

Now we have the two diagonals AC and BD. So multiplying, we get

 AC × BD = √a² + b² + 2abcosθ × √a² + b² - 2abcosθ

Simplifying, we get

AC × BD = √(a² + b² + 2abcosθ) × (√a² + b² - 2abcosθ)

AC × BD = √(a² + b²)² - (2abcosθ)²

AC × BD = √(a² + b²)²  - 4a²b²cos²θ

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Answer:

[tex]y = \frac{3}{2}x + 6[/tex]

Step-by-step explanation:

Rearranging the equation:

[tex]4y = 6x + 24[/tex]

The coefficient of y has to be '+1' for the above equation to be considered as the slope-intercept form.

Dividing both sides of the equation by '4':

[tex]\frac{4}{4}y = \frac{1}{4}(6x + 24)[/tex]

Expand the brackets by applying the Distributive Law:

[tex]\frac{4}{4}y = \frac{6}{4}x + \frac{24}{4}[/tex]

[tex]y = \frac{6}{4}x + 6[/tex]

Divide the numerator and the denominator present in the coefficient of x by the Highest Common Factor '2':

[tex]y = \frac{3}{2}x + 6[/tex]

The square matrix A is invertible, i.e., inverse of A is exist, if and only if the inverse of matrices B and C exists.

We have a matrix A such that

[tex]A = \begin{bmatrix} B & 0 \\0 & C \\ \end{bmatrix}[/tex]

where, B and C are square matrices. We have to prove or show that matrix A is invertible iff B and C are invertible. We shall prove it by using the inverse and determinant of matrices. The determinant is a scalar value which is associated with the square matrix. If X is a matrix, then the determinant of a matrix is denoted by |X|. Inverse of matrix A is calculated by using formula as below,

A = adj(A)/|A|. As we see a matrix A is invertible if and only if it has non-zero determinant. Also for a diagonal matrix its determinant is product of diagonal entries here diagonal entries are again matrices so |A|=|B|×|C|. From here we can clearly says |A| ≠ 0 if and only if |B| ≠ 0 and |C| ≠ 0. Hence, the matrix A is invertible iff B and C are invertible.

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Complete question :

Let A = [ B, 0 ; 0, C], where B and C are square. show that A is invertible if and only if both B and C are invertible.

Hamilton's method of allocating can be used to assign the 21 salespeople in the table in the following order:

2, 5, 6, 8.

Alexander Hamilton was the one who first suggested the strategy that now carries his name. In 1791, Congress approved of his strategy, but President Washington disapproved of it. It was commonly used from 1852 through 1911. He begins by figuring out exactly how many objects each group requires.

We'll use the terminology of shift and hours since the relevant question is about the allocated salesperson in order to calculate the appropriate number of salespeople for each shift.

Now total no. of customers given = 125 + 305 + 439 + 515 = 1375

Now total no of salesperson available = 21.

So, the divisor = 1375/21 = 65.47

Hamilton's approach is now used to determine the number of salespeople assigned by dividing the average number of customers per shift by the divisor.

So, the salesperson assigned are:

Morning = 125/65.47 = 2

Midday = 305/65.47 = 5

Afternoon = 430/65.47 = 6

Evening = 515/65.47 = 8 (Decimals rounded according to Hamilton's principle).

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Answer: To calculate the amount of pasta needed for eight servings, we need to multiply the original amount of pasta in the recipe by 8/6.

So, 1.75 cups * 8/6 = 2.3 cups of pasta.

Therefore, Rosa will use 2.3 cups or four and one quarter cups of pasta if she wants to make eight servings.

Step-by-step explanation:

Answer:

A = 81 units²

Step-by-step explanation:

the sides of a square (s) are congruent

given perimeter = 36 , then

s = 36 ÷ 4 = 9

the area (A) of a square is calculated as

A = s²

then

A = 9² = 81 units²

The value of m in the limit is (-13 + sqrt(169 + 4ε)) / 2, where -84.875 < ε < 1.

To prove that lim as x approaches 6 of (x^2+x) = 42, we need to find a δ > 0 such that if 0 < |x - 6| < δ, then |(x^2+x) - 42| < ε, where ε > 0 is arbitrary.

We can start by working with the expression |(x^2+x) - 42| and trying to bound it by ε. We have:

|(x^2+x) - 42| = |x^2 + x - 6^2 - 6 + 36| = |(x-6)(x+7)|.

To get a bound on |(x-6)(x+7)|, we can assume that 0 < |x - 6| < δ, where δ > 0 is some quantity we need to determine. This means that:

|x-6| < δ.

We can then use the triangle inequality to bound |x+7| as follows:

|x+7| <= |x-6| + |13| < δ + 13.

Thus, we have:

|(x-6)(x+7)| <= |x-6|*|x+7| < δ(δ+13).

Now, we want to choose δ such that δ(δ+13) < ε. Let's choose:

δ = min(ε/m, 1),

where m is some constant we need to determine. We want to make sure that this choice of δ satisfies δ(δ+13) < ε.

Substituting δ = min(ε/m, 1) into the inequality δ(δ+13) < ε, we get:

min(ε/m, 1)(min(ε/m, 1) + 13) < ε.

Simplifying this inequality gives:

min(ε/m, 1)^2 + 13min(ε/m, 1) - ε < 0.

Since we want to choose the smallest possible value of δ that satisfies this inequality, we want to choose m to be the largest value that satisfies this inequality.

The discriminant of the quadratic equation is:

b^2 - 4ac = 13^2 + 4ε.

Since ε > 0, we know that the discriminant is positive, so there are two roots to the quadratic equation. The largest root is:

(-13 + sqrt(169 + 4ε)) / 2.

So, we can choose:

m = (-13 + sqrt(169 + 4ε)) / 2.

Then, we have:

min(ε/m, 1) = ε/((-13 + sqrt(169 + 4ε)) / 2).

Substituting this value of δ into the inequality δ(δ+13) < ε, we get:

ε^2 / (13 - sqrt(169 + 4ε)) < ε.

This simplifies to:

ε < 13 - sqrt(169 + 4ε).

Squaring both sides and rearranging, we get:

4ε^2 + 338ε - 1444 < 0.

This quadratic inequality holds if and only if:

(-338 - sqrt(338^2 + 441444)) / (24) < ε < (-338 + sqrt(338^2 + 441444)) / (24).

Simplifying this gives:

-84.875 < ε < 1.375.

Thus, we have found that if we choose:

m = (-13 + sqrt(169 + 4ε)) / 2,

where:

-84.875 < ε < 1.

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4966.5

Step-by-step explanation:

Starting at finding out how the population will increase in 3 years we take 3 and divide it by 4. This produces an increase of 75% every 3 years. If we multiply 2838 by 75% we get 2128.5. If we add it back to 2838, we get 4966.5

The volume of the spherical boulder is 7234.56 cubic feet.

What is the diameter?

A line connecting the center and the circumference at its opposite ends is called the diameter. Its length is double that of the circle's radius.

The formula for the volume of a sphere is [tex]V=\frac{4}{3}\pi r^{3}[/tex].

Given the diameter of the sphere is 24 feet.

therefore radius is equal to 12 feet.

The volume of the sphere is equal to

[tex]V=\frac{4}{3}\pi (12)^{3} \\V=\frac{4}{3} *3.14*1728\\V=7234.56[/tex]

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Answer:

1.5322e+30

Step-by-step explanation:

calculator, there were too many numbers!

The probabilities of the random sample of 223 students are solved

P ( male ) = 0.475

P ( female ) = 0.525

P ( male | pet ) = 0.538

P ( female | no pet ) = 0.547

The probability that an event will occur is measured by the ratio of favorable examples to the total number of situations possible

Probability = number of desirable outcomes / total number of possible outcomes

The value of probability lies between 0 and 1

Given data ,

Let the total number of students be = 223 students

Let the total number of male students be = 49 + 57 = 106 students

Let the total number of female students be = 64 + 53 = 117 students

Now , the equation will be

Let the number of male students who own a pet = 57 students

Let the number of male students who does not own a pet = 49 students

And ,

Let the number of female students who own a pet = 53 students

Let the number of female students who does not own a pet = 64 students

The probability of choosing a male student P ( male ) = number of male students / total number of students

The probability of choosing a male student P ( male ) = 106 / 223

The probability of choosing a male student P ( male ) = 0.475

And ,

The probability of choosing a female student P ( female ) = number of male students / total number of students

The probability of choosing a female student P ( female ) = 117 / 223

The probability of choosing a female student P ( female ) = 0.525

And ,

Probability of choosing a male student who owns a pet P ( male | pet ) = number of male students who own a pet / number of male students

P ( male | pet )  = 57 / 106

P ( male | pet ) = 0.538

The probability of choosing a female student who does not own a pet is P ( female | no pet ) = number of female students who does not own a pet / number of female students

P ( female | no pet ) = 64 / 117

P ( female | no pet ) = 0.547

Hence , the probabilities are solved

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P(A) = 1/2, P(B) = 3/4, and P(A and B) = 3/8, where A is the event that the first student is a boy and B is the event that the second student is a girl.

The total number of possible outcomes in this scenario is 4C2, which is equal to 6. These outcomes are AB, AC, AD, BC, BD, and CD, where A represents Franklin and B, C, and D represent Sandra, Marta, and Jane, respectively.

The probability that the first student is a boy is P(A) = 1/2, since there are two boys and four students total.

The probability that the second student is a girl is P(B) = 3/4, since there are three girls and four students total.

The probability that the first student is a boy and the second student is a girl is P(A and B) = 1/2 x 3/3 = 3/8, since the probability of the first student being a boy is 1/2 and the probability of the second student being a girl is 3/4 (after one girl has already been chosen as the first student).

Therefore, the probability that the first student is a boy is 1/2, the probability that the second student is a girl is 3/4, and the probability that the first student is a boy and the second student is a girl is 3/8.

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The benefits of using quadratic models in real-world applications include: -They are versatile and can be used for a variety of problems.

Researchers may find the quadratic model to be a useful data analytic approach for helping them identify the combined impacts of achievement goals on academic accomplishment.

We anticipate that future studies on the impact of academic achievement goals will frequently use the quadratic model to analyze their data.

Situations that can be approximated by quadratic functions include tossing a ball, firing a cannon, jumping off a platform, and hitting a golf ball.

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The five-number summary of the given data 18, 15, 1, 14, 18, 11, 12, 20, 13, 19, 14, 8, 24, 18, 17 is ,

Your data set's five-number summary offers you a general notion of how it is organised. You will, for instance, have your best value and lowest value (the maximum). The main reason you'll want to locate a five-number summary is to find other valuable statistics, like the interquartile range, also known as the middle fifty, even if it's important in and of itself.

The five number summary includes 5 items:

Step 1: Put your numbers in ascending order (from smallest to largest). For this particular data set, the order is:

1, 8, 11, 12, 13, 14, 14, 15, 17, 18, 18, 18, 19, 20, 24.

Step 2: For your data set, determine the lowest and maximum. This ought to be obvious now that your math is correct.

In the example in step 1, the minimum (the smallest number) is 1 and the maximum (the largest number) is 24.

Step 3: Find the median. The median is the middle number.

median = 17

Step 4: (This is not technically necessary, but it makes Q1 and Q3 easier to find).

(1, 8, 11, 12, 13, 14, 14, 15), 17, (18, 18, 18, 19, 20, 24).

Step 5: Find Q1 and Q3. Q1 can be thought of as a median in the lower half of the data, and Q3 can be thought of as a median for the upper half of data.

(1, 8, 11, 12, 13, 14, 14, 15), 17, (18, 18, 18, 19, 20, 24).

Step 6: Write down your summary found in the above steps.

minimum = 1, Q1 = 12, median = 17, Q3 = 19, and maximum = 24.

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[tex]\begin{array}{llll} \textit{Logarithm of rationals} \\\\ \log_a\left( \frac{x}{y}\right)\implies \log_a(x)-\log_a(y) \end{array} \\\\[-0.35em] ~\dotfill\\\\ \log_8(x)-\log_8(7)=\log_8\left( \cfrac{5}{7} \right)\implies \log_8(\stackrel{x }{5})-\log_8(7)=\log_8\left( \cfrac{5}{7} \right)[/tex]

Step-by-step explanation:

To find the amount of money represented by C, we need to find the value of A and B first.

Since the ratio A:B is 2:5, we can assume that B is 5 times the amount of A.

Let's assume the amount of money represented by A is x.

Then B = 5x and C = 8x.

We know that the total amount of money split is £370, so we can write an equation:

x + 5x + 8x = £370

14x = £370

x = £26.43

So, the amount of money represented by C is 8x = 8 * £26.43 = £211.44

Answer: 4/1 is 1/4 but what are the 4 fractions your talking about?

Step-by-step explanation:

The side length of the field after two years (24 months) would be 46368.

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

To determine the side length of the field after two years, we can calculate the number of rabbit pairs after two years.

At the end of the first month, there is 1 pair of rabbits.

At the end of the second month, there are 1 + 1 = 2 pairs of rabbits.

At the end of the third month, there are 1 + 1 = 2 pairs of rabbits.

At the end of the fourth month, there are 2 + 1 = 3 pairs of rabbits.

At the end of the fifth month, there are 3 + 2 = 5 pairs of rabbits.

At the end of the sixth month, there are 5 + 3 = 8 pairs of rabbits.

At the end of the seventh month, there are 8 + 5 = 13 pairs of rabbits.

At the end of the eighth month, there are 13 + 8 = 21 pairs of rabbits.

At the end of the 24th month, there are fibonacci(24) = 46368 pairs of rabbits.

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Answer:

6

Step-by-step explanation:

30 divided by 5 = 6

6 is the maximum possible number of groups possible.

The best interpretation for this interval is that "She is 99% confident that the population parameter is within the interval" (option C).

In statistics and related fields, the confidence interval refers to a percentage that determines the population fits the interval set. Due to this, a high confidence rate is considered to be positive.

In this case, the confidence interval implies that the researcher is 99% confident that the population parameter is within the interval (option C)

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Answer:

24 min

Step-by-step explanation:

1 hr 36 min = 96 min

96 min / 4 = 24 min

Answer:

4604 (when rounded to the nearest whole number)

Step by step explanations:

Using the exponential growth formula, we can determine Amuel's population at the time the factory shut down: N = N0 * e^(rt) (rt) Where r is the growth rate, t is the number of years the growth takes place, N is the ultimate population, N0 is the starting population, and r represents growth.

We may use the population data from 1995 and 1996 to get the growth rate: r = ln(1,381 / 1,256) / (1996 - 1995) (1996 - 1995) 

Putting the numbers in: r = 0.117 

Next, we determine how many years the growth took place: t = 13 years (2007 - 1994) Lastly, we enter the values into the formula as follows: N = 1,256 * e^(0.117 * 13) 

Calculating the answer, we discover: 4,604 persons (N = 1,256 * 3.63) Consequently, there were about 4,604 residents there when the facility shut down in 2007.

The integral evaluates to 11, which is the sum of the areas of the three regions (R1 + R2 + R3 = 4 + 5 + 6 = 11).

R1 = 4, R2 = 5, R3 = 6

The integral evaluates to 11, which is the sum of the areas of the three regions (R1 + R2 + R3 = 4 + 5 + 6 = 11).

The integral is given by:

∫ (R1 + R2 + R3) dA

where R1, R2, and R3 are the areas of the labeled regions in the figure.

By interpreting the integral in terms of areas, we can calculate the value of the integral. The integral evaluates to 11, which is the sum of the areas of the three regions (R1 + R2 + R3 = 4 + 5 + 6 = 11).

The complete question is :

Evaluate the integral below by interpreting it in terms of areas in the figure. The areas of the labeled regions are A = 3, B = 4, C = 5, and D = 6.

∫DBCA x dA

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[tex]y=\frac{2x\pm\sqrt{5x^2-(4x^2-16x+4\bar{c})} }{2(1)}[/tex]Solve the given DE, [tex](y-x)\frac{dy}{dx} =y-x+2[/tex].

Rewriting,

=> [tex](y-x)\frac{dy}{dx} =y-x+2[/tex]

=> [tex](y-x)dy =(y-x+2)dx[/tex]

=> [tex]-(y-x+2)dx+(y-x)dy =0[/tex]

=> [tex](-y+x-2)dx+(y-x)dy =0[/tex]

Check to see if this is an exact DE by taking the partial derivative of M with respect to y and N with respect to x.

[tex]M=(-y+x-2)dx[/tex]

=> [tex]M_{y} =-1[/tex]

[tex]N=(y-x)dy[/tex]

=> [tex]N_{x}=-1[/tex]

[tex]M_{y} =N_{x}[/tex], so this is an exact DE. Now integrate M with respect to x and N with respect to y.

[tex]\int\ ({-y+x-2)} \, dx[/tex]

=>[tex]-xy+\frac{x^2}{2}-2x[/tex]

[tex]\int\ ({y-x)} \, dy[/tex]

=> [tex]=\frac{y^2}{2} -xy[/tex]

So we can say the solution to the given DE is, [tex]\frac{x^2}{2}+\frac{y^2}{2}-xy-2x=c[/tex].