a line segment is drawn between (6,4) and (8,3). Find its gradient, midpoint and length.

By deducting the value of the shortest distance from the value of the longest distance on the line plot, it is possible to determine the difference between the longest and shortest distances a student has walked to get to school.

We must look at the provided line plot to ascertain the difference between the longest and shortest distances a student walks to get to school. Ten pupils were tracked across various distances using a line plot. The location of each student is indicated by a "X" on the map.

Just looking for the X with the highest and lowest frequency will yield the longest and shortest lengths. According to the line plot, the distances at which Xs occur most frequently are 2 miles away and 0.5 miles away, respectively. As a result, there is a 1.5 mile discrepancy between the student's maximum walking distance (2 miles) and their shortest walking distance (0.5 miles).

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Answer:

See attached graph for the two functions

y = cos(x)

y = 0.5

Solution set for cos(x) i.e. the values of x for which cos(x) = 0.5 in the interval 0 < x < 2π are
{π/3, 5π/3)

or

{1.05, 5.24}  in decimal

Step-by-step explanation:

I moved the original horizontal up to y = 0.5

The solutions to the two equations are where the two functions intersect

There are two intersection in the interval 0 ≤ x ≤ 2π and are at the points labeled A and B

The two points can be obtained by setting
cos(x) = 0.5 and solving for x

cos(x) = 0.5

=> x = cos⁻¹ (0.5)

= 60° and 300° in the range 0 ≤ x ≤ 2π where 2π = 360°

In terms of π,

Since π radians = 180°, 1° = π/180 radians

60° = π/180 x 60 = π/3 radians

300° =  π/180 x 300 = 5π/3 radians

Therefore the solutions to cos(x) = 0.5 are
x = π/3 and x = 5π/3
The solution set is written as {π/3, 5π/3}

In decimal

π/3 = 1.04719 ≈ 1.05

5π/3 = 5.23598 ≈ 5.24

Solution set in decimal: {1.05, 5.24}

The explicit functions are:

Arithmetic sequence with common difference d=3: f(n) = 3n +1

Geometric sequence with common ratio r=2: f(n) = 2^n + 2

Quadratic sequence with leading coefficient a=1, constant term c=2, and linear term b=1: f(n) = n^2 + n + 2

There are infinitely many possible sequences that satisfy the condition f(1)=4, so the function to model the value of the nth term in the sequence will depend on the specific pattern or rule that the sequence follows.

Here are some examples of possible sequences and the corresponding explicit functions:

Arithmetic sequence with common difference d=3: f(n) = 4 + 3(n-1) = 3n + 1

Geometric sequence with common ratio r=2: f(n) = 4 x 2^(n-1) = 2^n+2

Quadratic sequence with leading coefficient a=1, constant term c=2, and linear term b=1: f(n) = n^2 + n + 2

Fibonacci sequence with initial values f(1)=f(2)=1: f(n) = ((1+sqrt(5))/2)^n/sqrt(5) - ((1-sqrt(5))/2)^n/sqrt(5)

Note that these are just examples, and there are many other possible functions that could model a sequence with f(1)=4

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hope it helps you

option d

The probability that the sample proportion surviving for at least 3 years will be less than 70% is approximately 0.2734 or 27.34%.

Probability is a branch of statistics that deals with the study of random events and their likelihood of occurrence.

First, calculate the standard deviation of the sampling distribution of sample proportions using the formula.

σ = [tex]\sqrt{[(p*q)/n]}[/tex], where p is the population proportion, q = (1 - p), and n is the sample size.

In this case, p = 0.74, q = 0.26, and n = 70

Therefore, σ =  [tex]\sqrt{[(0.74*0.26)/70]}[/tex]  = 0.066

Next, we need to standardize the sample proportion using the formula,

z = (X - p) / σ, where X is the sample mean, p is the population proportion, and σ is the standard deviation of the sampling distribution.

In this case, X = 0.70, p = 0.74, and σ = 0.066

Thus, z = (0.70 - 0.74) / 0.066 = -0.606

Using a standard normal distribution table, we find that the cumulative probability for a z-score of -0.606 is 0.2734.

Therefore, the probability that the sample proportion surviving for at least 3 years will be less than 70% is approximately 0.2734 or 27.34%.

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Answer:

Yes, the triangles are similar.

Step-by-step explanation:

The triangles are similar because of the three angle measurements.

1) One of the measurements are labeled, and the other two are implied.

2) The measurement of angles HMG and JMK are equal because they are vertical angles.

3) Lastly, because two of the angles are the same, we know that the last angles, GJK and HGJ are the same because the three angles must have a sum of 180°.

a) Sample Variance = 521.646

b)Two-sided confidence interval for the variance with a confidence level of 0.98 is (5.545, 10029.794).

a) To calculate the sample variance, you will first need to calculate the sample mean. The sample mean is calculated by summing all the observations in the sample and dividing by the number of observations. For this sample, the mean is:

Mean = (32.564 + 7.57 + 21.815 − 13.971 − 15.224) / 5 = 5.168

Next, you will need to calculate the sum of squared deviations from the mean. This is done by subtracting the mean from each observation and squaring the result, and then summing all of the results:

Sum of Squared Deviations = (32.564 - 5.168)^2 + (7.57 - 5.168)^2 + (21.815 - 5.168)^2 + (-13.971 - 5.168)^2 + (-15.224 - 5.168)^2 =
= 1564.939

Finally, you can calculate the sample variance by dividing the sum of squared deviations by the number of observations minus one:

Sample Variance = 1564.939 / (5 - 1) = 521.646

b) To calculate a two-sided confidence interval for the variance with a confidence level of 0.98, you will need to find the critical value from the Chi-squared distribution with a degrees of freedom equal to the number of observations in the sample minus one. For this sample, the degrees of freedom is 4.

The critical value for this degrees of freedom at the given confidence level is 8.37.

The lower bound of the confidence interval is:

Lower bound = (521.646 / 8.37) * (1 - 0.98) = 5.545

The upper bound of the confidence interval is:

Upper bound = (521.646 / 8.37) * (1 + 0.98) = 10029.794

Therefore, the two-sided confidence interval for the variance with a confidence level of 0.98 is (5.545, 10029.794).

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The maximum value of p is 107.09 and it is obtained when x = 1.44, y = 31.02, z = 0, and w = 0.

To maximize the objective function p = 6x + 9y + 3.3z + 12w

We can use the simplex method.

First, we need to convert the inequalities into equalities by introducing slack variables:

(a) 1.2x + y + z + w + s1 = 121.5

(b) 2x + y − z − w + s2 = 30

(c) 2x + y + z + 1.2w + s3 = 31.5

We can then write the augmented matrix for the problem:

x y z w s1 s2 s3       b

1.2 1 1 1 1 0 0             121.5

2 1 -1 -1 0 1 0               30

2 1 1 1 0 0 1                 31.5

-6 -9 -3.3 -12 0 0 0  0

We choose the most negative coefficient in the bottom row, which is -12. We then select the pivot element in the column corresponding to this coefficient, which is 121.5 in the first row. We perform row operations to make this pivot element equal to 1 and all other elements in its column equal to 0:

x y z w s1 s2 s3                                            b

1 0.83 0.17 0.33 0.67 -0.50 -0.17            100.67

0 0.17 -1.33 -1.33 -0.83 0.50 -0.17   12.33

0 0.17 0.33 0.33 -0.67 -0.50 0.83 10.17

0 -6.50 -12.90 -12.00 4.00 4.50 1.00 408.00

Next, we choose the most negative coefficient in the bottom row, which is -12.9. We select the pivot element in the column corresponding to this coefficient, which is 0.33 in the third row. We perform row operations to make this pivot element equal to 1 and all other elements in its column equal to 0:

x y z w s1 s2 s3                                           b

1 0.00 0.85 0.40 1.44 -0.54 -0.08  107.09

0 0.00 -1.22 -1.67 -1.00 0.38 0.08 -12.93

0 1.00 2.45 2.33 -2.00 1.50 0.33         31.02

0 0.00 -2.19 -5.60 10.00 10.50 1.83 630.00

The objective function value at this point is p = 107.09.

The solution is x = 1.44, y = 31.02, z = 0, w = 0, and the maximum value of p is 107.09.

Therefore, the maximum value of p is 107.09, when x = 1.44, y = 31.02, z = 0, and w = 0.

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23.m∠C = 48°m∠A = 90°

Given that the triangle ABC, m∠B = 42∘, a = 10, and c = 23. Now we have to find the missing side length and angles of △ABC. Step-by-step explanation: The formula to find the missing side length of a triangle is given by the Pythagoras theorem. Hence, using the Pythagoras theorem, we get;b2 = c2 - a2 = 23² - 10²b² = 529b = √529b = 23Length of b is 23.We know that the sum of all three angles of a triangle is equal to 180°. Therefore, using this concept, we can find the remaining angles of the triangle as follows;m∠A + m∠B + m∠C = 180°m∠A + 42° + m∠C = 180°m∠A + m∠C = 138°Also, we know that the sum of the two opposite angles of a triangle is equal to the third angle. Therefore, we can use this concept to find one of the remaining angles.m∠A = 180° - m∠C - m∠Bm∠A = 180° - m∠C - 42°m∠A = 138° - m∠C/ Thus,m∠A + m∠C = 138°And,m∠A = 138° - m∠COn solving these two equations, we get,m∠C = 48°m∠A = 90°Using the Pythagoras theorem we find the missing side length of a triangle as given below;b² = c² - a² = 23² - 10²b² = 529b = √529b = 23The length of the missing side is 23.Therefore, the missing side length and angles of △ABC are:Length of b is 23.m∠C = 48°m∠A = 90°

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Answer:-14a-4

Step-by-step explanation:

Answer: Expanding the expression, we get:

-7(2a - 1) - 11 = -14a + 7 - 11

Combining like terms, we get:

-14a - 4

Therefore, the simplified expression is -14a - 4.

Enjoy!

The cost of one metre of cotton fabric is £ 7. The cost of one metre polyester fabric is  £ 2.

The ratio of the cost of one metre of polyester fabric to the cost of one metre of cotton fabric is 2: 7.

1 m polyester / 1 m cotton = 2 / 7

1 m cotton = 7/2 × 1 m polyester

for 2 m of polyester fabric, the cost is £ 4

for 1 m of polyester fabric, the cost is £ 4/2 = £ 2

for 1 m of cotton fabric = 7/2 × 1 m polyester fabric = 7/2 × £2 = £ 7

for 1 m of cotton fabric, the cost is £ 7

For 2m, polyester fabric   2 × £2  = £ 4

cotton fabric  2 × £7  = £ 14

For 6m, polyester fabric   6 × £2  = £ 12

cotton fabric  6 × £7  = £ 42

For 8m, polyester fabric   8 × £2  = £ 16

cotton fabric  8 × £7  = £ 56

For 9m, polyester fabric   9 × £2  = £ 18

cotton fabric  9 × £7  = £ 63

The complete table

                           2m        6m      8m      9m

polyester fabric   £ 4       £ 12    £ 16     £ 18

cotton fabric        £ 14     £ 42    £ 56    £ 63

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Answer:250

Step-by-step explanation:

Answer:

6

Step-by-step explanation:

[tex] \frac{2 + 3 + 5 + 7 + 13}{5} = \frac{30}{5} = 6[/tex]

Answer:

y = -9

Step-by-step explanation:

Answer:

3

Step-by-step explanation:

since 2y is adding to positive 6 we take 6 to the right side where there its like term 12. we subtract 6 from 12 which is giving us 6. we are remaining with 2y=6 we divide both sides by 2 giving us y=3

The t-statistic is less than the critical value, we fail to reject the null hypothesis

The hypothesis testing to be conducted here is a two-tailed test for the population mean, using the formula:

H0: μ = 7.6 days

Ha: μ ≠ 7.6 days

The critical value is ± 1.960, with a confidence level of 95%.

The sample mean is 7.5 days and the sample standard deviation is 0.9 days. Using the formula, we can calculate the t-statistic:

t = (7.5 - 7.6) / (0.9 / √250) = -1.105

Since the t-statistic is less than the critical value, we fail to reject the null hypothesis, suggesting that Formulation NS does not significantly speed up the healing process compared to the foregoing treatment.

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The answer is: Logan's rope is longer than Sam's rope.

A number that represents a part of a whole or a ratio between two quantities, written as a numerator over a denominator. It consists of a numerator (top) and a denominator (bottom) separated by a fraction bar. For example, 1/2 represents one-half of a whole or the ratio of one to two.

Part A:

Brittney's rope is shorter than Sam's rope because it is 4/5 as long as Sam's rope.

Logan's rope is longer than Sam's rope because it is 1 1/4 times as long as Sam's rope.

Holly's rope is equal to Sam's rope because it is 8/8 (which simplifies to 1) as long as Sam's rope.

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1. Quadratic monomials.

2.  Biquadratic five-term polynomials.

3. Quadratic trinomials.

4. x³ + 3x² - 5x - 4

5. -[tex]x^{5}[/tex] + 4[tex]x^{4}[/tex] +2x³ + 2x - 7

6. - x² + 5x + 9

7. y² - 3y - 9

8. 5(x³ + x)

9. 2x² + 2x -5

Algebraic expressions called polynomials only have non-negative integer powers for their variables. A polynomial is, for instance, 5x² - x + 1. The polynomial 3x³ + 4x + 5/x + 6[tex]x^{3/2}[/tex] is not a polynomial since one of the powers of "x" is a fraction and the other is negative.

Expressions with one or more terms that have a non-zero coefficient are called polynomials. Variables, exponents, and constants make up polynomial terms. The "leading term" refers to the first term of the polynomial in standard form.

Here in the given question,

We can see the highest degree of the variable and we can determine the name of each polynomial.

Likewise, we can just arrange the expressions as per the highest value of the power of the variable.

And simplify the expression by adding or subtracting the like terms.

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Answer: At a greengrocer, two bananas and one apple cost $1.16 .

Than the equation becomes

2x + y = 1.16

one banana and one apple cost 0.71.

Than the equation becomes

x + y = 0.71

Subtracting x + y = 0.71 from 2x + y = 1.16

2x - x + y - y = 1.16 - 0.71

x = 0.45

Put in the equation x + y = 0.71

0.45 + y = 0.71

y = 0.71 - 0.45

y = 0.26

The cost of the one apple is  0.26

Step-by-step explanation:

Her average score on the four remaining tests is indeed 90, which confirms that her lowest score was 80.

Let's assume Peace's lowest score on the five math tests is x.

According to the problem statement, her average mark on all five tests is 88. This means that the sum of her scores on all five tests is:

5 * 88 = 440

If her lowest score was dropped, then the sum of her scores on the remaining four tests would be:

4 * 90 = 360

We know that the sum of her scores on all five tests is 440, so we can write an equation:

440 - x = 360

Solving for x, we get:

x = 80

Therefore, her lowest mark was 80. We can check this by finding her average score after dropping her lowest score:

(88 + 88 + 88 + 88 + 90) / 5 = 90

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Answer:

D

Step-by-step explanation:

To find the total number of paintbrushes in the classroom, we need to multiply the number of paintbrushes on each student's desk by the number of students.

Since there are 5 students and each student has 2 paintbrushes, we can use the multiplication operation to find the total number of paintbrushes:

5 students × 2 paintbrushes per student = 10 paintbrushes

Therefore, the correct option is 4. 5 times 2.

The average speed during a given time interval can be found by calculating the change in distance over the change in time, or (d(b) - d(a))/(b-a). In this case, we can use the given function d(t) = 16t^2 to find the distance at each given time.  It is reasonable to say that this value is the speed of the object at the instant t = 9 because as the time interval approaches zero, the average speed approaches the instantaneous speed at that moment.

For the first time interval, t = a = 9 and t = b = 9.5:
d(a) = 16(9)^2 = 1296
d(b) = 16(9.5)^2 = 1444
Average speed = (1444 - 1296)/(9.5 - 9) = 148/0.5 = 296 ft/s

For the second time interval, t = a = 9 and t = b = 9.1:
d(a) = 16(9)^2 = 1296
d(b) = 16(9.1)^2 = 1324.96
Average speed = (1324.96 - 1296)/(9.1 - 9) = 28.96/0.1 = 289.6 ft/s

For the third time interval, t = a = 9 and t = b = 9.01:
d(a) = 16(9)^2 = 1296
d(b) = 16(9.01)^2 = 1300.9616
Average speed = (1300.9616 - 1296)/(9.01 - 9) = 4.9616/0.01 = 496.16 ft/s

For the fourth time interval, t = a = 9 and t = b = 9.001:
d(a) = 16(9)^2 = 1296
d(b) = 16(9.001)^2 = 1296.288016
Average speed = (1296.288016 - 1296)/(9.001 - 9) = 0.288016/0.001 = 288.016 ft/s

For the fifth time interval, t = a = 9 and t = b = 9.0001:
d(a) = 16(9)^2 = 1296
d(b) = 16(9.0001)^2 = 1296.0288016
Average speed = (1296.0288016 - 1296)/(9.0001 - 9) = 0.0288016/0.0001 = 288.016 ft/s

As the time intervals get smaller and smaller, the average speed approaches 288 ft/s.

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The possible rational roots of the polynomial are ±1/2, ±1, ±3/2, ±3, ±9/2, ±9 while the actual roots are 1, -3, 3/2

To find the possible rational roots of the polynomial 2x^3 + x^2 - 12x + 9 = 0, we can use the rational root theorem. According to the theorem, if a polynomial with integer coefficients has a rational root p/q (where p and q are integers with no common factors other than 1), then p must be a factor of the constant term (in this case, 9) and q must be a factor of the leading coefficient (in this case, 2).

The factors of 9 are ±1, ±3, and ±9, and the factors of 2 are ±1 and ±2. Therefore, the possible rational roots of the polynomial are:

±1/2, ±1, ±3/2, ±3, ±9/2, ±9

We can now use synthetic division or long division to check which of these possible roots are actual roots of the polynomial. After checking, we find that the real rational root of the polynomial are x = 1, -3, 3/2

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1. The finance charge on June 11 is $1.98.

2. The new card balance on June 12th is $341.30

To calculate the finance charge and new card balance, we first need to calculate the average daily balance for the billing cycle.

May 13 Charge toys: $129.79 (balance: $129.79)

May 15 Payment $50 (balance: $79.79)

June 1 Charge clothing $135.95 (balance: $215.74)

June 8 Charge Housewares $37.63 (balance: $253.37)

Billing cycle: May 13 to June 11 (30 days)

Average daily balance:

=  (129.79 x 18) + (79.79 x 16) + (215.74 x 10) + (253.37 x 6) / 30

= $152.49

The Finance charge is computed as:

= Average daily balance * Monthly periodic rate.

= 152.49 x (1.3/100)

= $1.98.

To calculate the new card balance on June 12th, we need to add the finance charge and any new charges to the previous balance and subtract any payments made.

Previous balance (as of June 1st) = $215.74

New charges (since June 1st) = $135.95 + $37.63 = $173.58

Payments made (since May 15th) = $50

The New card balance on June 12th will be:

= $215.74 + $173.58 + $1.98 - $50

= $341.30

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Answer: The unit circle is an essential tool in trigonometry because it helps in understanding and visualizing the relationships between angles and the values of the sine, cosine, and tangent functions.

The unit circle is a circle with a radius of one unit and centered at the origin of a coordinate plane. It is divided into 360 degrees or 2π radians. By placing this circle on the coordinate plane, we can easily determine the sine and cosine values of angles in standard position.

For any given angle θ, the sine value is the y-coordinate of the point where the terminal side of the angle intersects the unit circle, and the cosine value is the x-coordinate of that same point. The tangent function, which is the ratio of sine to cosine, can also be determined using the unit circle.

The unit circle also helps in understanding the periodicity of the sine and cosine functions. Since the circumference of the unit circle is 2π, the sine and cosine functions repeat themselves after every 2π radians or 360 degrees. This periodicity allows for the use of trigonometric identities and formulas to simplify and solve complex trigonometric equations.

In summary, the unit circle is an essential tool in trigonometry as it provides a visual representation of angles and their corresponding sine, cosine, and tangent values, and allows for the use of trigonometric identities and formulas to solve complex problems.

Answer:

50.2 is the answer as the answer came in point

Let's assume Eric's height before summer is x cm. After summer, he became 5% taller, which means his new height is 1.05x cm. We also know that his new height is 151.2 cm. So we can set up an equation:

1.05x = 151.2

To solve for x, we can divide both sides by 1.05:

x = 151.2 / 1.05

x = 144 cm

Therefore, Eric's height before summer was 144 cm.

Baby lizards can run, climb and in some cases even change color to match their surroundings, while baby snakes are generally limited to crawling and slithering.

Baby lizards have developed legs, claws and a tail to help them navigate their environment, while baby snakes have lost their legs during evolution and have developed a long, slender body to help them move around.

Additionally, some baby lizards are born with a protective membrane around their eggs that allows them to move around freely, while baby snakes are usually born inside an egg that they must break out of before hatching. Overall, baby lizards have more mobility and flexibility than baby snakes, which gives them an advantage in many environments.

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Number of seconds = Number of minutes x 60

A minute is a unit of time measurement that is equal to 60 seconds. It is commonly used to measure short periods of time, such as the duration of a conversation or the time it takes to complete a task.

Time measurement is the process of quantifying the duration of an event or the interval between two events. It involves measuring the elapsed time between a start point and an end point, or between two points in time.

To convert any number of minutes into seconds, we can use the formula:

Number of seconds = Number of minutes x 60

This formula works because there are 60 seconds in one minute. So, to convert minutes to seconds, we simply multiply the number of minutes by 60 to get the equivalent number of seconds.

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The given statement "The mode number of TVs per household for both Mrs. Jamison's class and and Mr. Zimmerman's class is 3." is true. The mode of TVs per household is 3 in both Mrs. Jamison's class and and Mr. Zimmerman's class.

Mode is a statistical measure that represents the value that appears most frequently in a dataset. It is one of the measures of central tendency, along with mean and median.

In Mrs. Jamison's class, the mode number of TVs per household is 3 because it appears twice, while all other numbers appear only once. Similarly, in Mr. Zimmerman's class, the mode number of TVs per household is 3 because it appears twice, while all other numbers appear only once.

Therefore, the statement "The mode number of TVs per household for both Mrs. Jamison's class and Mr. Zimmerman's class is 3" is true.

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