Find f + g if f(x) = 3x-2 and g(x) = 5x + 2

The random samples of the given scores of each block represents ,

Inter quartile range of each block is Option D. Block 1 IQR = 20 and Block 2 IQR= 5.

Outliers present in each block is Option B. Block One= 25, Block Two= 100.

Each data display represents Option B. Block 1 is skewed right, Block 2 is skewed left.

Mean and standard deviation in Block 1  is Option A.  Mean = 76.5, standard deviation = 21.6.

Score of two blocks,

Block 1

25, 60, 70, 75, 80, 85, 85, 90, 95, 100.

Block 2

70, 70, 75, 75, 75, 75, 80, 80, 85, 100.

To get Interquartile range 'IQR' of each block,

First find the quartiles.

Median of Block 1 = 80

Median of Block 2 = 75

First quartile Q₁ and third quartile Q₃ of each block,

Split the data into two halves at the median and find the median of each half.

Block 1

Lower half is,

25, 60, 70, 75, 80

Q₁ = median of the lower half

    = 70

Upper half is,

85, 85, 90, 95, 100.

Q₃ = median of the upper half

     = 90

IQR = Q₃ - Q₁

      = 90 - 70

      = 20

Block 2

Lower half is,

70, 70, 75, 75, 75

Q₁ = median of the lower half

     = 75

Upper half is,

75, 80, 80, 85, 100.

Q₃ = median of the upper half

     = 80

IQR = Q₃ - Q₁

      = 80 - 75

       = 5

Option D. Block 1 IQR = 20 and Block 2 IQR= 5

Outliers in each block,

First find lower and upper bounds.

Any data point outside the bounds is considered an outlier.

The lower bound is Q₁ - 1.5(IQR),

and the upper bound is Q₃ + 1.5(IQR).

Block 1,

Q₁ = 70

Q₃ = 90

IQR = 20

Lower bound

= Q₁ - 1.5(IQR)

= 70 - 1.5(20)

= 70 - 30

= 40

Upper bound

=Q₃ + 1.5(IQR)

= 90 + 1.5(20)

= 120

The data point 25 is less than the lower bound,

so it is an outlier in Block 1.

Block 2,

Q₁ = 75

Q₃ = 80

IQR = 5

Lower bound

= 75 - 1.5(5)

= 67.5

Upper bound

= 80 + 1.5(5)

= 87.5

100 is more than the upper bounds in Block 2,

so it is an outliers of Block 2.

Option B. Block One= 25, Block Two= 100

The data displays as symmetric skewed left or skewed right,

Examine the shape of the histograms.

Block 1 has a histogram that is skewed right.

With more scores on the higher end of the range.

Block 2 has a histogram that is slightly skewed left.

With more scores on the lower end of the range.

Option B. Block 1 is skewed right, Block 2 is skewed left.

Mean and standard deviation of Block 1,

Use the formulas,

Mean = sum of scores / number of scores

Standard deviation = √ [(sum of (scores - mean)^2) / (n - 1)]

Block 1

Mean

= (25 + 60 + 70 + 75 + 80 + 85 + 85 + 90 + 95 + 100) / 10

= 76.5

Standard deviation

= √[((25-76.5)^2 + (60-76.5)^2 + ... + (100-76.5)^2) / (10 -1 )]

=√2652.25 + 272.25+ 42.25 +2.25 + 12.25 + 72.25 + 72.25 + 182.25 + 342.25 + 552.25 /9

= √4202.5/9

= 21.6

Option A.  Mean = 76.5, standard deviation = 21.6.

Therefore, for the given scores answer of the following questions are,

Inter quartile range is Option D. Block 1 IQR = 20 and Block 2 IQR= 5.

Outliers in each block is Option B. Block One= 25, Block Two= 100.

Data display is Option B. Block 1 is skewed right, Block 2 is skewed left.

In Block 1 Option A.  Mean = 76.5, standard deviation = 21.6.

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Answer:

5 units to the left

Step-by-step explanation:

Look at point A.  To get to A' you need to move 5 units to the left.

Helping in the name of Jesus.

The probability of flipping a coin 3 times consecutively and having it land on "Tails" all 3 times is 0.5 x 0.5 x 0.5, which equals 0.125 or 12.5%. This value is slightly higher than 0.10 or 10%.

The statement is true. The probability of flipping a coin and having it land on "Tails" is 0.5 or 50%. However, the probability of flipping a coin three times consecutively and having it land on "Tails" all three times is 0.5 x 0.5 x 0.5, which equals 0.125 or 12.5%. This is greater than the stated probability of being less than 0.10 or 10%. Therefore, the statement is true that the probability of flipping a coin 3 times consecutively and having it land on "Tails" all 3 times is far less than 0.10 or 10%.

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The p-value is less than the significance level of 0.05, we reject the null hypothesis.

Using the given data, we can calculate the t-test statistic as follows:

t = (117 - 138) / √[(29²/37) + (34²/35)] = -2.21

where 117 is the sample mean for the day after Thanksgiving, 138 is the sample mean for the day after Christmas, 29 is the standard deviation for the day after Thanksgiving, 34 is the standard deviation for the day after Christmas, 37 is the sample size for the day after Thanksgiving, and 35 is the sample size for the day after Christmas.

Using a t-table or a statistical software, we can find the p-value associated with this t-test statistic. At the 0.05 level of significance, the p-value is 0.016. This means that if the null hypothesis were true, we would observe a t-test statistic as extreme or more extreme than -2.21 only 1.6% of the time.

This means that we have evidence to suggest that the mean amount of money spent by shoppers on the day after Thanksgiving is less than the mean amount of money spent by shoppers on the day after Christmas.

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The general indefinite integral is F(x) = (11/3)x³ + 7x + C.

To find the general indefinite integral of ∫(5x²+6+6x²+1)dx, you can first simplify the integrand as 11x² + 7. Now, apply the Fundamental Theorem of Calculus and linearity property to find the antiderivatives of each term.

1. Simplify the integrand: 5x² + 6 + 6x² + 1 = 11x² + 7

2. Find the antiderivative of 11x²: ∫(11x²)dx = (11/3)x³ + C₁

3. Find the antiderivative of 7: ∫(7)dx = 7x + C₂

4. Add the antiderivatives and combine constants: F(x) = (11/3)x³ + 7x + (C₁ + C₂)

5. Use a single constant, C: F(x) = (11/3)x³ + 7x + C

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Answer: Since the line integral is the same for both curves, we have verified that F is the gradient of the scalar function f(x,y) = x^3y, which has partial derivatives ∂f/∂x = 3x^2y and ∂f/∂y = x^

Step-by-step explanation:

To calculate the line integral (F dr) along C1, we need to parameterize the curve. Let's use x as the parameter, so r(x) = <x, 2x^2>. Then r'(x) = <1, 4x>, and ||r'(x)|| = sqrt(1 + 16x^2). Now we can calculate:

python

Copy code

(F dr) = ∫(F(r(x)) · r'(x)) dx

      = ∫(3x(2x^2) · 1 + x^0 · 4x) / sqrt(1 + 16x^2) dx    (substituting F and r')

      = ∫(6x^3 + 4x) / sqrt(1 + 16x^2) dx

To evaluate this integral, we can use the substitution u = 1 + 16x^2, du/dx = 32x:

scss

Copy code

(F dr) = (1/32) ∫(6(u-1)/16 + 4/(32x)) du/sqrt(u)

      = (1/32) ∫(3u - 8)/u^(1/2) du

      = (1/32) [6u^(1/2) - 16ln(u^(1/2))] + C

Now we need to substitute back in for x and evaluate the integral from x = -1 to x = 1:

scss

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(F dr) = (1/32) [(6(5) - 16ln(5)) - (6(1) - 16ln(1))] = (1/32) (30 - 16ln(5)) ≈ 0.46

b. The curve C2 is the straight line that goes from (-1,2) to (1,2). Here's a graph of the curve:

markdown

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      * (1, 2)

       |

       |

       |

       |

  * (-1, 2)

To calculate the line integral (F dr) along C2, we can use the same parameterization as in part a: r(x) = <x, 2>, -1 ≤ x ≤ 1. Then r'(x) = <1, 0>, and ||r'(x)|| = 1. Now we can calculate:

python

Copy code

(F dr) = ∫(F(r(x)) · r'(x)) dx

      = ∫(3x(2) · 1 + x^0 · 0) dx    (substituting F and r')

      = ∫6x dx

      = 3x^2 + C

Now we need to evaluate this integral from x = -1 to x = 1:

scss

Copy code

(F dr) = (3(1)^2 + C) - (3(-1)^2 + C) = 6

Since the line integral is the same for both curves, we have verified that F is the gradient of the scalar function f(x,y) = x^3y, which has partial derivatives ∂f/∂x = 3x^2y and ∂f/∂y = x^

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The multiplication of 165×3 is 321

To start, we can break down 165 into 100+60+5. This is because multiplication is distributive over addition. That means we can multiply each part separately and then add the results. So,

165×3 = (100+60+5)×3

= 100×3 + 60×3 + 5×3

Now, we can use the fact that 65×3=195 to find the answer to each of these products.

100×3 = 100×(65/65)×3 = (100×65)/65×3 = 65×3

60×3 = 60×(65/65)×3 = (60×65)/65×3 = 39×3

5×3 = 5×(65/65)×3 = (5×65)/65×3 = 3×3

Putting it all together,

165×3 = 65×3 + 39×3 + 3×3

= 195 + 117 + 9

= 321

Therefore, 165×3 = 321.

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The probability that a colorblind person is from Group A is 0.952.

To find the probability that a colorblind person is from Group A, we can use Bayes' theorem. We are given the following probabilities:

1. P(C|A): Probability of being colorblind given the person is from Group A = 5% = 0.05

2. P(C|B): Probability of being colorblind given the person is from Group B = 0.25% = 0.0025

3. P(A): Probability of selecting a person from Group A

4. P(B): Probability of selecting a person from Group B

We know that P(A) + P(B) = 1, but we need more information to determine the specific probabilities of P(A) and P(B). Assuming that Group A and Group B have equal proportions in the population, then P(A) = P(B) = 0.5.

We want to find P(A|C), the probability that a colorblind person is from Group A. Using Bayes' theorem:

P(A|C) = P(C|A) * P(A) / P(C)

We need to find P(C), which can be calculated as:

P(C) = P(C|A) * P(A) + P(C|B) * P(B)

Plugging in the given values:

P(C) = (0.05 * 0.5) + (0.0025 * 0.5) = 0.02625

Now we can find P(A|C):

P(A|C) = (0.05 * 0.5) / 0.02625 = 0.95238

Rounding to three decimal places, the probability that a colorblind person is from Group A is 0.952.

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When the temperature is 29 degrees, 283 hot chocolates will be sold (according to this linear relationship).

To solve this problem, we can use the two-point form of a linear equation, which is given by:

y - y1 = (y2 - y1)/(x2 - x1) * (x - x1)

where x1 and y1 are the coordinates of one point, x2 and y2 are the coordinates of another point, x is the x-coordinate of the point we want to find, and y is the y-coordinate of the point we want to find.

In this case, we can use the points (54, 103) and (40, 173) to find the linear equation that relates temperature to the number of hot chocolates sold. We have:

y - 103 = (173 - 103)/(40 - 54) * (x - 54)

Simplifying this equation, we get:

y - 103 = -7(x - 54)

y - 103 = -7x + 378

y = -7x + 481

Now we can use this equation to find the number of hot chocolates sold when the temperature is 29 degrees. We have:

y = -7(29) + 481

y = 283

Therefore, when the temperature is 29 degrees, 283 hot chocolates will be sold (according to this linear relationship).

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The following parts can be answered by the concept of Standard deviation.

a) Child 2 is taller.

b) The z-score associated with the height of Child 1 is -0.30.

c) The z-score associated with the height of Child 2 is -0.57.

d) Child 2 is taller relative to their corresponding population

a) Based on the given heights, Child 2 is taller than Child 1 as 38.5 inches is greater than 37.3 inches.

b) To calculate the z-score for Child 1's height:

Mean height of population for Child 1's age and gender = 39.2 inches

Standard deviation of population for Child 1's age and gender = 6.3 inches

Height of Child 1 = 37.3 inches

Using the formula for calculating z-score: z = (X - μ) / σ, where X is the observed value, μ is the mean, and σ is the standard deviation.

Plugging in the values:

X = 37.3 inches

μ = 39.2 inches

σ = 6.3 inches

z = (37.3 - 39.2) / 6.3

z = -0.30 (rounded to two decimal places)

c) To calculate the z-score for Child 2's height:

Mean height of population for Child 2's age and gender = 40.5 inches

Standard deviation of population for Child 2's age and gender = 3.5 inches

Height of Child 2 = 38.5 inches

Using the formula for calculating z-score:

X = 38.5 inches

μ = 40.5 inches

σ = 3.5 inches

z = (38.5 - 40.5) / 3.5

z = -0.57 (rounded to two decimal places)

d) To determine which child is taller relative to their corresponding populations, we compare the z-scores. A z-score indicates how many standard deviations an observed value is from the mean. A higher absolute value of z-score indicates a larger deviation from the mean.

Child 1 has a z-score of -0.30, which means Child 1's height is 0.30 standard deviations below the mean height of the population for their age and gender.

Child 2 has a z-score of -0.57, which means Child 2's height is 0.57 standard deviations below the mean height of the population for their age and gender.

Since Child 1 has a smaller absolute z-score (-0.30) compared to Child 2 (-0.57), it indicates that Child 1 is closer to the mean height of their population, and hence Child 2 is taller relative to their corresponding populations.

Therefore, the correct answers are:

a) Child 2 is taller.

b) The z-score associated with the height of Child 1 is -0.30.

c) The z-score associated with the height of Child 2 is -0.57.

d) Child 2 is taller relative to their corresponding population

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(a) The 90% confidence interval for the proportion of students who plan on voting is (0.386 - 0.069, 0.386 + 0.069) = (0.317, 0.455). The smaller number is 0.317, so that is entered in the first box.
(b) The correct interpretation for the answer in part (a) is A: We can be 90% confident that the proportion of students who will vote in the upcoming election lies in the interval (0.317, 0.455).

(a) To find the 90% confidence interval, follow these steps:

1. Determine the sample proportion (p') by dividing the number of students who plan on voting by the total number of students surveyed:
  p' = 97 / (97 + 154) = 97 / 251 ≈ 0.3865

2. Find the critical value (z) for a 90% confidence interval using a z-table. For 90%, the z-value is 1.645.

3. Calculate the standard error (SE) using the formula:
  SE = √(p'(1 - p') / n) = √(0.3865(1 - 0.3865) / 251) ≈ 0.0302

4. Find the margin of error (ME) using the formula:
  ME = z × SE = 1.645 × 0.0302 ≈ 0.0497

5. Calculate the lower and upper limits of the confidence interval:
  Lower Limit = p' - ME = 0.3865 - 0.0497 ≈ 0.3368
  Upper Limit = p' + ME = 0.3865 + 0.0497 ≈ 0.4362

Confidence interval: (0.3368, 0.4362)

(b) The correct interpretation for the 90% confidence interval is:

A. We can be 90% confident that the proportion of students who will vote in the upcoming election lies in the interval (0.3368, 0.4362). This means that if we were to repeat this survey many times and construct a 90% confidence interval each time, we would expect 90% of those intervals to contain the true proportion of students who plan on voting in the election. It does not mean that there is a 90% chance that the true proportion lies in this interval, nor does it provide any information about the actual number of students who will vote.

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The area inside one loop of the lemniscate r2 = 10 sin 2θ is 5 square units.

To find the area inside one loop of the lemniscate r2 = 10 sin 2θ, we need to use the formula for the area enclosed by a polar curve:

A = (1/2) ∫[a,b] r(θ)2 dθ

Since we want the area inside one loop, we can choose the limits of integration to be from 0 to π/2, which covers one complete loop of the lemniscate. Plugging in the given polar equation, we get:

A = (1/2) ∫[0,π/2] (10 sin 2θ) dθ

Using a double angle identity, we can simplify the integrand to:

A = (1/2) ∫[0,π/2] (10 sin θ) cos θ dθ

Integrating this expression gives:

A = (1/2) [(-10/2) cos2θ] [0,π/2]

A = (1/2) [(-10/2) (1)] = -5

However, since we are looking for a positive area, we need to take the absolute value of this result, which gives:

A = 5 square units

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The expression that shows "3 less than a number" is B. n - 3.

Mathematical operations must be carried out in a specific order according to a set of rules known as the order of operations. The following is the order of events:

Operation inside brackets should come first. Expressions containing exponents should be evaluated next.

Division and Multiplication: Carry out division and multiplication from left to right. Addition and subtraction: Move from left to right when adding and subtracting. It is crucial to adhere to the order of operations since it guarantees accurate and consistent evaluation of mathematical expressions.

When we break the given phrase in to mathematical expression we have:

n - 3.

Hence, the expression that shows "3 less than a number" is B. n - 3.

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The center of the curve is c. Hyperbola; center (-5, -2). The correct option is:c.

The given equation is 9y² - 4x² - 36y - 82 = 0.

To identify the curve and find the center, we need to check the signs of the coefficients of x² and y². Here, x² has a negative coefficient and y² has a positive coefficient, which suggests that the given equation represents a hyperbola.

To find the center of the hyperbola, we need to rewrite the equation in the standard form, which is of the form:

[(y - k)² / a²] - [(x - h)² / b²] = 1

where (h, k) is the center of the hyperbola, a is the distance from the center to the vertices, and b is the distance from the center to the foci.

Rewriting the given equation in the standard form, we get:

9(y² + 4y) - 4(x² + 20x) = 328

Dividing both sides by 36, we get:

[(y + 2)² / 4²] - [(x + 5)² / (3/2)²] = 1

Comparing this equation with the standard form, we get:

Center = (-5, -2)

Therefore, the correct option is:

c. Hyperbola; center (-5, -2)

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Answer:

140J

Step-by-step explanation:

work done = Mass x acceleration x distance

mass = 7kg

acceleration = 10m/s^2

distance= 2m

work = 7 x 10 x 2

= 140j

Answer:

The answer is 140J or 140Nm

Step-by-step explanation:

Work done=mgh

W=7×10×2

W=140J or 140Nm

Answer:

  D.  y = 3sin(1/2x)

Step-by-step explanation:

You want to know the equation that corresponds to the graph.

The graph shows one cycle of a sine function. It has a peak value of 3, indicating a vertical scaling by a factor of 3. All of the answer choices have this scale factor:

  y = 3sin( )

That one cycle extends from x = -2π to x = 2π, a total of 2π -(-2π) = 4π units. The period of the parent sine function y=sin(x) is 2π, so this represents a horizontal expansion by a factor of 2. That scaling is represented in the function's equation as a divisor of x:

  y = 3sin(x/2)

  [tex]\boxed{y=\sin{\left(\dfrac{1}{2}x\right)}}[/tex]

The statement "Suppose a coin is flipped twice is false because the event of getting heads on the first toss and the event of getting heads on the second toss could be said to be mutually exclusive" is false.

Mutually exclusive events cannot occur at the same time, meaning if one event occurs, the other event cannot. In this case, getting heads on the first toss and getting heads on the second toss are not mutually exclusive because they can both occur in a single trial (i.e., flipping the coin twice).

You could have heads on the first toss and heads on the second toss as well, so these events are not mutually exclusive.

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Normalcdf and Invnorm are two statistical functions commonly used in probability and statistics.

Normalcdf is a function that calculates the cumulative probability of a standard normal distribution. It takes two arguments, the lower and upper limits, and returns the probability that a random variable falls between those limits.

This function is particularly useful for finding the area under the normal curve, which can be interpreted as the probability of a random variable taking on a certain value.

Invnorm, on the other hand, is a function that calculates the inverse normal distribution. It takes a probability value as an argument and returns the z-score (standard deviation) corresponding to that probability.

This function is useful for finding the value of a random variable that corresponds to a certain probability in a normal distribution.

Together, these functions can be used to solve a variety of statistical problems, such as calculating confidence intervals, finding probabilities of events, and determining the significance of test results.

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If we were to repeat the sampling process and construct a new confidence interval, there is a 99% chance that the new interval would also include the true population parameter.

A confidence interval is constructed using a point estimate of the population parameter, along with a margin of error.

In Question 4, we constructed a confidence interval based on a sample from the population. We found that the interval ranged from [lower bound, upper bound]. Now, we can complete the statement with 99% confidence by saying that "We are 99% confident that the interval from [lower bound, upper bound] includes the true population parameter."

It is important to note that we cannot say with 100% certainty that the true population parameter lies within the interval, but we can be highly confident based on our statistical analysis.

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Answer:

B. 4

Step-by-step explanation:

[tex]7 \sqrt{7x} = \sqrt{49 \times 7x} [/tex]

49 × 7x = 343x

14√7 = √(1372)

1372 ÷ 343 = 4 this is the vaule of x.

(a) ỹ = 5.81 + 0.497x is the best fit for the data to show their current GPAs.

To determine which regression equation best fits the data, we can calculate the correlation coefficient (r) between the two variables, which will indicate the strength and direction of the relationship.

Using a statistical software or a calculator, we can find that the correlation coefficient between entering GPA and current GPA is r = 0.737, which indicates a moderately strong positive relationship.

To choose the best regression equation, we can look at the slope coefficient (β1) of each equation, which represents the amount of change in the dependent variable (current GPA) for every one-unit increase in the independent variable (entering GPA).

a. ỹ = 5.81 + 0.497x: This equation has a slope coefficient of 0.497, indicating that for every one-unit increase in entering GPA, current GPA increases by 0.497.

b. ỹ = 3.67 + 0.0313x: This equation has a slope coefficient of 0.0313, indicating that for every one-unit increase in entering GPA, current GPA increases by only 0.0313.

c. ỹ = 2.51 + 0.329x: This equation has a slope coefficient of 0.329, indicating that for every one-unit increase in entering GPA, current GPA increases by 0.329.

d. ŷ= 4.91 + 0.0212x: This equation has a slope coefficient of 0.0212, indicating that for every one-unit increase in entering GPA, current GPA increases by 0.0212.

Based on the correlation coefficient and the slope coefficients, we can conclude that equation (a) ỹ = 5.81 + 0.497x is the best fit for the data as it has the highest slope coefficient, indicating a stronger relationship between the variables, and the y-intercept of 5.81 is closer to the average current GPA of 3.78.

Correct Question :

Which of the following equation best fit their current GPAs.

Entering GPA Current GPA

3.5 3.6.

3.8 3.7

3.6 3.9

3.6 3.6

3.5 3.9

3.9 3.8

4.0 3.7

3.9 3.9

3.5 3.8

3.7 4.0

a. ỹ = 5.81 + 0.497x

b. ỹ = 3.67 + 0.0313x

c. ỹ = 2.51 + 0.329x

d. ŷ= 4.91 + 0.0212x

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On solving the provided query we have In the circle graph, paddleboarding therefore has a centre angle of 194.4°. Paddleboarding, then, is the correct response.

It is possible to multiply, divide, add, or subtract in mathematics. The following is how an expression is put together: Number, expression, and mathematical operator The components of a mathematical expression (such as addition, subtraction, multiplication or division, etc.) include numbers, variables, and functions. It is possible to contrast expressions and phrases. An expression, often known as an algebraic expression, is any mathematical statement that contains variables, numbers, and an arithmetic operation between them. For instance, the word m in the given equation is separated from the terms 4m and 5 by the arithmetic symbol +, as does the variable m in the expression 4m + 5.

Calculating the proportion of campers who choose each activity is necessary to find the central angle of each activity in a circle graph. To achieve this, multiply the result by 100 and divide the number of campers who picked each activity by the total number of campers questioned (100).

The proportions are

15% when kayaking (15/100 x 100%).

Wakeboarding: 11% (11/100 x 100%)

Surfing the wind: 7/100 x 100% = 7%

Skipping on water: 13/100 times 100% is 13%

SUP: 54/100 times 100% is 54%.

54% x 360° = 194.4°

In the circle graph, paddleboarding therefore has a centre angle of 194.4°. Paddleboarding, then, is the correct response.

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the probability that the sample mean is less than 7 is approximately zero

Since the stress scores follow a uniform distribution from 1 to 5, the mean of the distribution is:

mu = (1 + 5) / 2 = 3

The standard deviation of the distribution is:

sigma = (5 - 1) / sqrt(12) = sqrt(2.33)

The sample size is n = 55.

The sample mean follows a normal distribution with mean mu = 3 and standard deviation sigma/sqrt(n) = sqrt(2.33)/sqrt(55) = 0.313.

To find the probability that the sample mean is less than 7, we standardize the variable:

z = (7 - 3) / 0.313 = 12.78

This is an extremely large value of z, indicating a probability that is essentially zero. Therefore, the probability that the sample mean is less than 7 is approximately zero

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Answer:

Step-by-step explanation:

Answer:

The answer is x³√5x

Step-by-step explanation:

√5x⁷=x³√5x

a) If V is a finite-dimensional vector space, then V cannot contain an infinite linearly independent subset. This statement is false.

(b) If Vi and V2 are vector spaces and dim(V1) < dim (V2), then V1 ⊂ V2. This statement is false.

(a) If xo is a solution to Ax=b, then xo is in row(A). This statement is false.

(b) If A is a 4x13 matrix, then the nullity of A could be equal to 5. This statement is false.

(a) False. A vector space can have an infinite linearly independent subset, but only if it is infinite-dimensional. For example, the vector space of polynomials has an infinite linearly independent subset {1, x, [tex]x^2[/tex], [tex]x^3[/tex], ...}.
(b) False. Just because the dimension of V1 is less than V2 does not necessarily mean V1 is a subset of V2. For example, V1 could be the x-axis in [tex]R^2[/tex] and V2 could be the entire plane.
(a) False. Just because xo is a solution to Ax=b does not mean xo is in row(A). xo could be any vector that satisfies the equation Ax=b.
(b) False. The nullity of matrix A is equal to the dimension of its null space. The null space is the set of all solutions to the homogeneous equation Ax=0. Since A has 13 columns, the maximum rank it can have is 13. Therefore, the nullity of A can be at most 13-4=9. It cannot be equal to 5.

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If  y = 5 √5x/ 5x - 3. dy/dx = 25/(2√(5x))

To find dy/dx for y = 5√(5x)/(5x-3), we need to use the chain rule.

Let u = 5x, then y = 5√(5x)/(5x-3) = 5√u/(u-3)

Using the power rule, we can find du/dx = 5

Using the chain rule, we get:

dy/dx = dy/du * du/dx

To find dy/du, we need to use the power rule and the constant multiple rule:

dy/du = (5/2) * u(-1/2)

Substituting back for u, we get:

dy/du = (5/2) * (5x)(-1/2)

Now, substituting all the values we found, we get:

dy/dx = dy/du * du/dx = (5/2) * (5x)(-1/2) * 5

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There are 479,001,600 different ways to arrange the 8 boys and 5 girls if the two particular boys wish to sit together.

To arrange the 8 boys and 5 girls with the two particular boys sitting together, follow these steps:

1. Consider the two particular boys as a single unit. So now, we have 7 boys (6 individual boys and 1 combined unit of the two particular boys), and 5 girls, making a total of 12 individuals to be arranged.
2. Arrange the 12 individuals (7 boys + 5 girls) in 12! (12 factorial) ways.
3. Since the two particular boys within their combined unit can also swap places, there are 2! (2 factorial) ways to arrange them.
4. Multiply the arrangements of the 12 individuals (12!) by the arrangements of the two particular boys (2!) to get the total number of different ways.

Total different ways = 12! × 2! = 479,001,600 ways

So, there are 479,001,600 different ways to arrange the 8 boys and 5 girls if the two particular boys wish to sit together.

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The intervals on which f(x) is concave down are (-∞,∞).

To determine the intervals on which the function f(x) = √(2x+3) is concave up or concave down, we need to find its second derivative.

First, we find the first derivative of f(x):

f'(x) = (1/2)(2x+3)^(-1/2) * 2

f'(x) = (1/√(2x+3)) * 2

f'(x) = 2/√(2x+3)

Next, we find the second derivative of f(x):

f''(x) = d/dx (2/√(2x+3))

f''(x) = -4(2x+3)^(-3/2)

f''(x) = -4/(2x+3)^(3/2)

Now, to determine where the function is concave up or down, we need to find where the second derivative is positive (concave up) or negative (concave down).

f''(x) > 0 if -4/(2x+3)^(3/2) > 0

-4 > 0

This inequality is never true for any value of x, so f(x) is never concave up.

f''(x) < 0 if -4/(2x+3)^(3/2) < 0

-4 < 0

This inequality is always true for any value of x, so f(x) is always concave down.

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The unknown angle measure based on the information is 35⁰.

An angle is a geometric figure formed by two rays (or lines) that share a common endpoint called the vertex. The rays are typically represented as line segments with the vertex at their endpoint, and the angle is measured in degrees or radians based on the amount of rotation between the two rays.

Angles are commonly denoted using the vertex as the center point and a letter at each end of the rays, for example, angle ABC, where B is the vertex and A and C are the endpoints of the two rays forming the angle.

Since the angles on a triangle are 80°, x, and 65°, the value of the unknown angle will be:

= 180 - (65 + 80)

= 35°

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The angles on a triangle are 80°, x, and 65°. What is the value of the unknown angle?