Ide whole numbers
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franny spent 35 minutes walking around a track. she made 7 laps around the track.
it took franny the same amount of time to walk each lap. how many minutes did it take her to walk each lap?
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The theoretical probability is: 12.5%. After 100 trials, the experimental probability is of: 20%. After 400 trials, the experimental probability is of: 11%. After more trials, the experimental probability is closer to the theoretical probability.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The dice has eight sides, hence the theoretical probability of rolling a six is given as follows:

1/8 = 0.125 = 12.5%.

(eight sides, each of them is equally as likely).

The experimental probabilities are obtained considering the trials, hence:

The more trials, the closer the experimental probability should be to the theoretical probability.

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Answer: B, C, E

Step-by-step explanation: Other dude posted wrong answer.

Answer:

1) a = 14

2) -4 (x - 2)² - 5

Step-by-step explanation:

To obtain a vertex, you take h and k in a equation.

So a(x-h)²+k = a(x-2)² -5

For the point (1, - 9),

a[(1)-2]² - 5 = - 9

a(1) = -9+5

a = -4

so the final equation is

-4(x-2)² - 5

I'm not 100% sure about this but I tried. Let me know if it makes sense

Lizzie's divisibility test states that a number n is divisible by a certain number m if and only if the alternating sum of its two-digit chunks is divisible by m.

Lizzie's divisibility test involves breaking a positive integer into two-digit chunks, finding the alternating sum of these chunks, and then determining if the result is divisible by a certain number m.

To apply the test:

To prove that this is a divisibility test for m, you need to show that n is divisible by m if and only if the result of the alternating sum is divisible by m.

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The equation that can be used to decode the secret code is m = c - 2

For you to decode the secret message, you need to turn the the encoding process around. Find the inverse.

Since the encoding process uses the equation c = m + 2, to decode the message, all that need to be found is the value of m. This can be done by rearranging the encoding equation to solve for m

move 2 to c side. it becomes m = c-2

The above answer is in response to the full question below;

Suppose M and C each represent the position number of a letter in the alphabet, but M represents the letters in the original message and C represents the letters in a secret code. The equation c=m+2 is used to encode a message.

Write an equation that can be used to decode the secret code into the original message.

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Value of p is approximately 0.987.

To find the p-value for the given claim that the mean body temperature at 12 AM is μ > 98.6°F with a sample size of n=8 and a test statistic of t=-2.687, follow these steps:

1. Identify the degrees of freedom: Since the sample size is n=8, the degrees of freedom (df) are calculated as n-1, which is 8-1=7.

2. Determine the tail of the test: The claim states that the mean body temperature is greater than 98.6°F (μ > 98.6), which indicates a right-tailed test.

3. Find the p-value using the t-distribution table or a calculator: With a test statistic of t=-2.687 and df=7, you can look up the corresponding p-value using a t-distribution table or an online calculator. Since it's a right-tailed test, the p-value will be the area to the right of the test statistic in the t-distribution.

After completing these steps, the p-value is found to be approximately 0.987.

Therefore, your answer is: The p-value for the claim that the mean body temperature at 12 AM is μ > 98.6°F, given a sample size of n=8 and a test statistic of t=-2.687, is approximately 0.987.

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Answer:

48.91

Step-by-step explanation:

r=cos^-1(.92)

r=23.07

cos(23.07)=45/y

y=45/cos(23.07)

48.91

The two algebraic expressions for the square root of x are x^(1/2) and √x.

An algebraic expression is an expression built up from constant algebraic numbers, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number).

The square root function involves the square root symbol √ (which is read as "square root of"). The square root of a number 'x' is a number 'y' such that y2 = x. i.e., if y2 = x ⇒ y = √x. i.e., if 'x' is the square of 'y' then 'y' is the square root of 'x'.

There are two common ways to write the square root of x as an algebraic expression. The first way is to use fractional exponent notation, which is x^(1/2). The second way is to use radical notation, which is √x. Both of these expressions represent the square root of x in algebraic form.

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There is no one definitive list of steps to solving a word problem, as different types of problems may require different approaches.

However, a general set of steps that can be useful in solving many word problems is:

1. Read the problem carefully to understand what it is asking.

2. Identify the relevant information and the unknown quantity you need to find.

3. Translate the problem into an equation or set of equations that relate the given information to the unknown quantity.

4. Solve the equation(s) to find the value of the unknown quantity.

5. Check your answer to make sure it makes sense in the context of the problem.

Additional steps or variations on these steps may be necessary depending on the specific problem, but this general framework can be a useful starting point.

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For the g(x) functions provided, here are their key features:

g(x) = (x + 2)(x − 1)(x − 2)

End behavior: As x approaches negative or positive infinity, g(x) approaches positive infinity.

Y-intercept: g(0) = -4

Zeros: x = -2, 1, 2

Ray and Kelsey could both be accurate, all depending on the stated third-degree polynomial function.

It is conceivable for a third-degree polynomial to present up to three zeros, thus corroborating Kelsey's point that the function can have up to three intersection points with the x-axis maximum. Moreover, it can even occur that this function possesses a repeatable zero, causing a fourth interception.

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Both situations can be represented by the number line as they both involve values either greater than or less than 75.

The number line the teacher drew can represent both Brooke's and Eileen's situations.

In Brooke's situation, the number line can represent the amount of money Christopher needs to earn to buy a new bicycle. If he needs to earn more than $75, any point on the number line greater than 75 would represent the amount of money he has earned that is sufficient for purchasing the bicycle.

In Eileen's situation, the number line can represent the time left in Paul's flight. If Paul has less than 75 minutes left, any point on the number line less than 75 would represent the time remaining in his flight.

Both situations can be represented by the number line as they both involve values either greater than or less than 75.

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194 children, 58 students, and 97 adults attended the movie.

Let's use algebra to solve this problem.

Let's assume the number of children who attended the movie is C, the number of students is S, and the number of adults is A.

From the problem, we know that:

The seating capacity of the theater is 349:

C + S + A = 349

The theater charges $5 for children, $7 for students, and $12 for adults:

5C + 7S + 12A = $2540

There are half as many adults as there are children:

A = 1/2C

Now we can substitute A = 1/2C from the third equation into the first and second equations:

C + S + 1/2C = 349

3/2C + S = 349

5C + 7S + 12(1/2C) = $2540

5C + 7S + 6C = $2540

11C + 7S = $2540

Now we have two equations with two variables, C and S.

We can solve for S in the first equation:

3/2C + S = 349

S = 349 - 3/2C

Now we can substitute S = 349 - 3/2C into the second equation:

11C + 7S = $2540

11C + 7(349 - 3/2C) = $2540

11C + 2443 - 10.5C = $2540

0.5C = 97

C = 194

Therefore, 194 children attended the movie of total sales.

We can use A = 1/2C from the third equation to find the number of adults:

A = 1/2C

A = 1/2(194)

A = 97

Therefore, 97 adults attended the movie.

We can use C + S + A = 349 to find the number of students:

C + S + A = 349

194 + S + 97 = 349

S = 58

Therefore, 58 students attended the movie.

In summary, 194 children, 58 students, and 97 adults attended the movie.

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To compute Bu(1.-3) - 88 - (1, -3), we need to substitute the values of u and v into the expression for w.

First, we need to find the values of u and v. Since u = 1.-3 and v = (1, -3), we have:

u = 1.-3 = 1 - 0.3 = 0.7
v = (1, -3)

Next, we can substitute these values into the expression for w:

w = 2xy + y^2 - 4x^2
 = 2(1)(-3) + (-3)^2 - 4(1)^2  (substituting x = 1 and y = -3)
 = -6 + 9 - 4
 = -1

Finally, we can compute Bu(1.-3) - 88 - (1, -3) by multiplying the gradient of w by the vector (1, -3) and subtracting 88:

Bu(1.-3) - 88 - (1, -3) = (-8x + 2y, 2x + 2y)  (1, -3) - 88
                        = (-8(1) + 2(-3), 2(1) + 2(-3))  (1, -3) - 88
                        = (-14, -4)  (1, -3) - 88
                        = (-14)(1) + (-4)(-3) - 88
                        = -14 + 12 - 88
                        = -90

Therefore, Bu(1.-3) - 88 - (1, -3) = -90.

Since the question seems to have some typos or missing information, I'll assume you want to find the partial derivatives of w with respect to x and y, and evaluate them at the point (1, -3).

Given w = 2xy + y² - 4x², let's compute the partial derivatives:

∂w/∂x = 2y - 8x
∂w/∂y = 2x + 2y

Now, let's evaluate these partial derivatives at the point (1, -3):

∂w/∂x(1, -3) = 2(-3) - 8(1) = -6 - 8 = -14
∂w/∂y(1, -3) = 2(1) + 2(-3) = 2 - 6 = -4

Thus, the evaluated partial derivatives are ∂w/∂x(1, -3) = -14 and ∂w/∂y(1, -3) = -4.

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The circumference of the circle with a diameter of 10m is 31.4m, using 3.14 as the value of pi and including the correct unit in the answer.

The formula for the circumference of a circle is C = πd, where C is the circumference and d is the diameter. Substituting the given value of the diameter, we get C = 3.14 x 10m = 31.4m as the circumference of the circle.

Since the value of pi is irrational, it cannot be expressed as a finite decimal or fraction, so we use an approximation, such as 3.14, to calculate the circumference. It is important to include the correct unit, which is meters in this case, in the answer to indicate the quantity being measured. Therefore, the circumference of the circle is 31.4m.

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The mode of the number of push-ups is 40.

A stem-and-leaf plot is a way of organizing data where the stems (the tens digit) and leaves (the ones digit) are separated. Each row represents a stem and the leaves represent the values that belong to that stem.

Here's the stem-and-leaf plot for the number of push-ups:

3 | 5 6 8

4 | 0 0 1 2 2 3 5 6 8 9

5 | 0 1 3 4 5 5 7 8 9

6 | 0 1 2 2 3 4 5 7 8 9

7 | 0 2 5 8

8 | 1 2 4

9 | 0

To find the mode, we look for the value that appears most frequently. In this case, the number 40 appears three times, which is more than any other value. Therefore, the mode of the number of push-ups is 40.

Note that the stem-and-leaf plot makes it easy to see the distribution of the data. For example, we can see that there are a lot of values between 40 and 49, and relatively few values above 60.

We can also see that there are no values between 90 and 99.

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The extremum of the function is a minimum at x = -2.5

The given function is k(x)(300+10x)(5-0.2x).

To check for symmetry about the y-axis, we replace x with -x in the given function and simplify as follows:

k(-x)(300-10x)(5+0.2x)

To check for symmetry about the x-axis, we replace y with -y in the given function and simplify as follows:

k(x)(300+10x)(5-0.2x) = -k(x)(-300-10x)(5+0.2x)

To find these points, we set the function equal to zero and solve for x:

k(x)(300+10x)(5-0.2x) = 0

This equation has three solutions:

x = 0

x = -30

x = 25.

The midpoint of the line segment connecting these points is

(x1+x2) ÷ 2 = (-30+25) ÷ 2 = -2.5.

To determine the type of extremum at this point, we need to check the sign of the second derivative. The second derivative of the function is:

k(x)(-1200+x)(0.2x+15)

Since the function is symmetric about the x-axis, the second derivative will be negative at the extremum if it is maximum and positive if it is a minimum.

When x = -2.5, the second derivative is positive, which means that the function has a minimum at x = -2.5.

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In both cases, the inequality holds true. The inequality is 415 + d ≥ 500.  

Part A:
To write an inequality that represents the situation, we can use the following format: money raised so far + additional money needed ≥ goal. In this case, the money raised so far is $415, and the goal is $500. Let d represent the additional money needed. So the inequality would be:

415 + d ≥ 500

Part B:

The inequality 415 + d ≥ 500 has infinitely many solutions, as there are countless values of d that can satisfy the inequality. This is because as long as the total amount raised is equal to or greater than $500, the student council meets its goal. For example, if d is 85, then the council would exactly meet its goal (415 + 85 = 500). If d is 100, the council would exceed its goal (415 + 100 = 515). In both cases, the inequality holds true.

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The technician will bill a total of 150 hours to the job. To find the total hours, we need to determine how many hours the technician will work during the first three weeks and the last three weeks, and then add them together.

1. First three weeks:

The technician will be 25% billable during these weeks. They work 40 hours per week, so we need to calculate 25% of 40 hours for each week:

25% of 40 hours = 0.25 * 40 = 10 hours per week

Since there are three weeks, we'll multiply these hours by 3:
10 hours/week * 3 weeks = 30 hours

2. Last three weeks:

The technician will be 100% billable during these weeks. They work 40 hours per week, so they'll bill 40 hours for each of these weeks:
40 hours/week * 3 weeks = 120 hours

3. Finally,

we need to add the hours from the first and last three weeks together to find the total hours the technician will bill to the job:
30 hours (first three weeks) + 120 hours (last three weeks) = 150 hours

The technician will bill a total of 150 hours to the job.

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We use the fundamental property of proportions where:

[tex] \space [/tex]

[tex] \bf \frac{4x + 2}{39} = \frac{5x - 2}{42} \\ \\ \bf 39 \cdot (5x - 2) = 42 \cdot (4x + 2) \\ \\ \bf 195x - 78 = 168x + 84 \\ \\ \bf 195x - 168x = 84 + 78 \\ \\ \bf 27x = 162 \\ \\ \bf x = \frac{162}{27} \implies \bf \red{ \boxed{ \bf x = 6} } [/tex]

The number is 6.

Hope that helps! Good luck! :)

To create a story context for the given expressions, which are (5 1/4 - 2 1/8) divided by 4 and 4 x (4.8/0.8).

Imagine there is a fruit store where you have to prepare fruit baskets for a local charity event. The first expression (5 1/4 - 2 1/8) divided by 4 can be a story about the number of apples to be distributed equally among four baskets.

You initially have 5 1/4 dozen apples, but you realize that 2 1/8 dozen of them are not suitable for the baskets.

To find out how many dozens of apples should be put into each basket, you need to subtract the unsuitable apples and divide the result by 4:

(5 1/4 - 2 1/8) / 4

Now, let's move on to the second expression, 4 x (4.8/0.8). This can be a story about the number of oranges you need to purchase for the fruit baskets. You already have 4.8 dozen oranges, but you need to add more to reach the desired ratio of oranges to apples.

Your friend suggests that for every 0.8 dozen oranges you currently have, you should add 4 more dozen oranges. To find out how many dozens of oranges you need to buy, you can use this formula:

4 x (4.8/0.8)

By creating these story contexts, you can use the given expressions to solve real-life problems, such as distributing fruits among charity baskets.

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Answer: ln|y/8| + C
Explanation:

First, we need to recognize that the derivative of arctan(x) is 1/(1+x^2). Therefore, the derivative of arctan(y/8) is 8/(64+y^2).

Now, using the substitution u = y/8, we can rewrite the integral as:

∫(1/u)(64+64u^2)(8/(64+64u^2))du

Simplifying, we get:

∫(1/u)du = ln|u| = ln|y/8|

Therefore, the final answer is:

ln|y/8| + C

where C is the constant of integration.

a) The probability that exactly five men will still be alive in 20 years is approximately 0.024.

b) The probability that more than eight men will still be alive in 20 years is approximately 0.057.

c) The probability that at least two men will still be alive in 20 years is approximately 0.999.

To calculate the probabilities, we can use the binomial distribution formula, where n is the number of trials, p is the probability of success, and x is the number of successes. Therefore,

a) P(X = 5) = (10 choose 5) * (0.69)⁵ * (0.31)⁵ ≈ 0.024

b) P(X > 8) = P(X = 9) + P(X = 10) = [(10 choose 9) * (0.69)⁹ * (0.31)¹] + [(10 choose 10) * (0.69)¹⁰ * (0.31)⁰] ≈ 0.057

c) P(X ≥ 2) = 1 - P(X = 0) - P(X = 1) = 1 - [(10 choose 0) * (0.69)⁰ * (0.31)¹⁰] - [(10 choose 1) * (0.69)¹ * (0.31)⁹] ≈ 0.999

In summary, we have used the binomial distribution formula to calculate the probability that exactly five men, more than eight men, and at least two men will still be alive in 20 years, given that the probability that a man of this particular age will be alive in 20 years is 0.69.

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The expression that can be used to find the third number in Mark's pattern is 3[3(5) - 1]. The correct option is C.

In Mark's pattern, the rule is to subtract 1 from the previous number and then multiply the result by 3.

Starting with 5 as the first number, we can apply this rule step by step to find the subsequent numbers.

First step: Subtract 1 from 5, giving us 4.

Second step: Multiply 4 by 3, which equals 12.

So, the second number in Mark's pattern is 12.

Now, to find the third number, we apply the same rule.

First step: Subtract 1 from 12, giving us 11.

Second step: Multiply 11 by 3, which equals 33.

Therefore, the third number in Mark's pattern is 33.

Option C, 3[3(5) - 1], correctly represents this calculation, where 5 is subtracted by 1, multiplied by 3, and then multiplied by 3 again.

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If Chris bought 5 tacos and 2 burritos for $13. 25 and  Brett bought 3 tacos and 2 burritos for $10. 75, the price of one taco is $1.25, and the price of one burrito is $3.50.

Let the price of one taco be T and the price of one burrito be B. We have the following equations:

5T + 2B = $13.25
3T + 2B = $10.75

To find the prices of the taco and the burrito, we can use the system of equations. First, subtract the second equation from the first equation:

(5T + 2B) - (3T + 2B) = $13.25 - $10.75
2T = $2.50

Now, divide by 2 to find the price of one taco:

T = $1.25

Next, plug the value of T back into one of the equations (let's use the second equation):

3($1.25) + 2B = $10.75
$3.75 + 2B = $10.75

Now, subtract $3.75 from both sides:

2B = $7.00

Finally, divide by 2 to find the price of one burrito:

B = $3.50

So, the price of one taco is $1.25, and the price of one burrito is $3.50.

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Answer:

[tex]x = \$2.06[/tex]

Step-by-step explanation:

Representing the price of one juice bottle as x, we can construct the equation:

[tex]15x + \$1.93 = \$32.83[/tex]

From here, we can solve for x.

↓ subtracting $1.93 from both sides

[tex]15x = \$32.83 - \$1.93[/tex]

[tex]15x = \$30.90[/tex]

↓ dividing both sides by 15

[tex]\boxed{x = \$2.06}[/tex]

The account balance after 2 years is approximately $107.15.

The compound interest formula is given by:

A = P * [tex](1 + r/n)^(^n^*^t^)[/tex]

Where:

P = Principal

r = Annual interest rate (as a decimal)

n = Number of times interest is compounded per year

t = Time in years

A = Final account balance

In this problem, we are given:

P = $100

r = 3.5% per year = 0.035 per year

n = 365 (since interest is compounded daily)

t = 2 years

Substituting these values in the formula, we get:

A = [tex]100 * (1 + 0.035/365)^(^3^6^5^*^2^)[/tex]

A ≈ $107.15

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Using the value of λ = (18 + √130)/18, we get: x = 3λ ≈ 4.895 y = 3λ - 3 ≈ 5.316 So the point on the curve that is furthest from P is approximately (4.895, 5.316).

To use the method of Lagrange multipliers, we first need to define our objective function and our constraint. Our objective function is the distance between the point P and a point on the curve, which can be expressed as:

f(x, y) = (x - 0)^2 + (y - 3)^2 = x^2 + (y - 3)^2

Our constraint is the equation of the curve:

g(x, y) = x^2 + y^2 - 6x + 7 = 0

To use the method of Lagrange multipliers, we need to introduce a new variable λ and solve the following system of equations:

∇f = λ∇g
g(x, y) = 0

where ∇f and ∇g are the gradients of f and g, respectively.

Taking the partial derivatives of f and g with respect to x and y, we have:

∂f/∂x = 2x
∂f/∂y = 2(y - 3)
∂g/∂x = 2x - 6
∂g/∂y = 2y

Setting ∇f equal to λ∇g, we have:

2x = λ(2x - 6)
2(y - 3) = λ(2y)

Simplifying these equations, we get:

x = 3λ
y = 3λ - 3

Substituting these expressions into the equation of the curve, we get:

(3λ)^2 + (3λ - 3)^2 - 6(3λ) + 7 = 0

Simplifying this equation, we get:

18λ^2 - 36λ + 13 = 0

Solving for λ, we get:

λ = (18 ± √130)/18

Substituting these values of λ into our expressions for x and y, we get the coordinates of the points on the curve that are closest to and furthest from the point P.

To find the point that is closest to P, we need to minimize the objective function f(x, y). Using the value of λ = (18 - √130)/18, we get:

x = 3λ ≈ 1.105
y = 3λ - 3 ≈ -0.316

So the point on the curve that is closest to P is approximately (1.105, -0.316).

To find the point that is furthest from P, we need to maximize the objective function f(x, y). Using the value of λ = (18 + √130)/18, we get:

x = 3λ ≈ 4.895
y = 3λ - 3 ≈ 5.316

So the point on the curve that is furthest from P is approximately (4.895, 5.316).

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The function represents exponential decay with a percentage rate of decrease of 9.8%.

The given exponential function y = 660(0.902) represents decay because the base of the exponent is less than one.

This means that the output value of the function will decrease as the input value increases.

To determine the percentage rate of decrease, we need to find the value of the base of the exponent subtracted from one and then multiply it by 100.

The base of the exponent is 0.902, so we subtract it from one to get 0.098.

Multiplying by 100 gives us a percentage rate of decrease of 9.8%.

This means that for every unit increase in the input value, the output value of the function will decrease by approximately 9.8%.

For example, if the input value increases from 1 to 2, the output value will decrease by 9.8%, and if the input value increases from 2 to 3, the output value will again decrease by 9.8%.

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The solution of the quadratic equation is y = (x + 3)(x - 3).

The solution of the quadratic equation is calculated by applying difference of two squares as shown below;

y = x² - 9

y = x² - 3²

the difference of two square of x² - 3² = (x + 3)(x - 3)

The solution of the quadratic equation is calculated as;

y = (x + 3)(x - 3)

Thus, solution of the quadratic equation has been determined using  square root method.

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