(1 point) Consider the power series 00 Σ (-4)" -(x + 6)". n=1 Vn Find the radius of convergence R. If it is infinite, type "infinity" or "inf", Answer: R= What is the interval of convergence? Answer

Answer:  B   g(x)=1/4f(x)    odd

Step-by-step explanation:

First Page:

Points from the graph

Points from f(x)                         points from g(x)

(1,2)                                                (1, 1/2)

(4, 16)                                             (4, 4)

f(x) was multiplied by 1/4 to get to g(x)

Second Page:

Even functions are symmetrical about the y-axis: . Odd functions are symmetrical about the x- and y-axis: f(x)=-f(-x).

lets test for even:  does f(x)=f(-x)   =>    f(1)=f(-1)  no   -2[tex]\neq[/tex]2

see image for plotted points

lets test for odd:  does f(x)= -f(-x)   =>    f(1) = -f(-1)  yes  -2 = -2

Answer:

[tex]\textsf{B)} \quad g(x)=\dfrac{1}{4}f(x)[/tex]

[tex]\textsf{B)} \quad \textsf{odd}[/tex]

Step-by-step explanation:

Functions f(x) and g(x) are exponential functions.

From inspection of the given points on both graphs:

When x = 4, the y-value of function f(x) is four times the y-value of function g(x). Therefore, function g(x) is ¹/₄ of function f(x):

[tex]g(x)=\dfrac{1}{4}f(x)[/tex]

[tex]\hrulefill[/tex]

And even function is symmetric about the y-axis:

According to the table, f(3) = -4 and f(-3) = 4.

Therefore, as f(x) ≠ f(-x), the function is not even.

An odd function is symmetric about the origin:

According to the table, f(-3) = 4 and f(3) = -4. So -f(3) = -(-4) = 4.

Therefore, as f(-3) = -f(3), the function is odd.

[tex]\hrulefill[/tex]

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Therefore, the equation in the form of y = mx + b that represents this linear function is: y = -3.2x + 0.1

To write an equation in the form of y = mx + b for a linear function, we need to find the slope (m) and the y-intercept (b).

We can use any two points from the given set of points to find the slope:

m = (y2 - y1) / (x2 - x1)

Let's use the first and second points:

m = (-5.5 - 2.5) / (1.75 - (-0.25))

m = -8 / 2.5

m = -3.2

Now, we can use the slope and one of the points to find the y-intercept:

y = mx + b

-5.5 = (-3.2)(1.75) + b

b = -5.5 + 5.6

b = 0.1

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The measure of the quadrilateral are

88 - 4 = 84

2 * 88 - 78 = 98

88 + 8 = 96

A quadrilateral is a polygon with four sides and four vertices (corners). There are many different types of quadrilaterals, including squares, rectangles, parallelograms, trapezoids, and kites. The angles and sides of a quadrilateral can vary depending on its specific type, but the sum of the internal angles of any quadrilateral is always 360 degrees.

x - 4 + x + 8 = 180

we have to find the value of x

x = 88

The angles would be

88 - 4 = 84

2 * 88 - 78 =

88 + 8 = 98

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The sale's person commission from the sale of $37,000 worth of furnaces is $2,220. If this is doubled, he would have $4,440.

If the salesperson's commission from the sale of $37,000 is at the rate of 6%, then we will do the following:

6/100 × $37,000 = $2,220

If sales, double, we will now record the amount, 74,000. Now 6% of 74,000 will be $4,440. So, the resultant amount, if the salesperson was to increase his sales to double the original, will be $4,440.

This follows a simple logic. When you have a number and are told to double it, you simply multiply by 2. In the same manner, we multiply the salesperson's commission by 2.

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The completed equation is:

2 + 7 = a - 2

a = 11.

We are given the following equation:

2 + b = a - 2

We need to select the correct number from the drop-down menu to complete the equation.

From the first drop-down menu, we select 7.

2 + 7 = 9

From the second drop-down menu, we select 2.

2 + b = 9 - 2

2 + b = 7

Subtracting 2 from both sides, we get:

b = 5

Therefore, from the third drop-down menu, we select 5.

So, the completed equation is:

2 + 7 = 5 - 2

9 = 3

This is not a true statement, so there must be an error in one of our selections. Upon closer inspection, we can see that the correct number to select from the first drop-down menu is 5, not 7.

2 + 5 = 7

Now, substituting 5 for b in the original equation, we get:

2 + 5 = a - 2

7 + 2 = a

a = 9

Therefore, from the third drop-down menu, we select 9.

So, the completed equation is:

2 + 5 = 9 - 2

7 = 7

This is a true statement, so we have selected the correct numbers to complete the equation.

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Looking at the spread of your data is part of the data analysis step in the statistical process.

In this step, the data is examined to identify patterns, relationships, and trends in the data.

One aspect of data analysis is understanding the distribution and spread of the data, which can be done through measures of central tendency and measures of variability.

Understanding the spread of the data can help in making decisions about what statistical analyses are appropriate and how to interpret the results.

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The value of f(4) rounded to the nearest tenth is approximately 194.9.

The value of f(4) can be found by substituting x=4 in the given function f(x) = [tex]4e^x[/tex], so we get:

f(4) = [tex]4e^4[/tex]

Using a calculator, we can evaluate this expression as:

f(4) ≈ 194.92

Rounding this to the nearest tenth gives:

f(4) ≈ 194.9

Therefore, the value of f(4) rounded to the nearest tenth is approximately 194.9.

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Answer:

(-∞, 1) ∪ (1, ∞)

Step-by-step explanation:

1 - x = 0

-x = -1

x = 1

In interval notation, the domain is (-∞, 1) U (1, ∞)

The equation of line passing through the points (8, 6) and (-3, 6) is y = 6. Since the y-coordinate is the same for both points, the line is a horizontal line at y = 6.

To find the equation of the line passing through the points (8, 6) and (-3, 6), we can use the slope-intercept form of a linear equation:

y = mx + b

where m is the slope of the line and b is the y-intercept.

First, we need to find the slope, which is given by

m = (y2 - y1) / (x2 - x1)

where (x1, y1) = (8, 6) and (x2, y2) = (-3, 6)

m = (6 - 6) / (-3 - 8)

m = 0 / -11

m = 0

Since the slope is zero, the line is a horizontal line. We can see from the given points that the line passes through y = 6. Therefore, the equation of the line is

y = 6

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Start with the base graph: y = |x|

Translate the graph one unit to the right: y = |x - 1|

---We use a minus/negative/subtraction sign when dealing with horizontal translations because it is the opposite of the way we want to go. If the translation occurs within parenthesis/absolute value bars, we always do the opposite of what we think we should.

Translate the graph one unit down: y = |x - 1| - 1

---If the translation occurs to the y-value/vertically, we use the expected operation/sign. If the translation occurs outside of parenthesis, we use the same operation/sign as the translation (+ for up, - for down).

Answer: y = |x - 1| - 1

Hope this helps!

Answer:

y=|x-1|-1

Step-by-step explanation:

The function for a v shaped graph is an absolute value function: y=|x|

Subtracting 1 from the absolute value y=|x|, we move the graph 1 unit right of the x axis

Subtracting 1 from the whole equation, y=|x-1|, we move the graph 1 unit down the y axis

So, the equation would be y=|x-1|-1

Answer:

C) 6

Step-by-step explanation:

n=6

6( 6 + 1) + 3 =45

36 + 6 + 3 = 45

36 + 9 = 45

The value for each conversion is: 12,000 g = 12 kg 120 cm = 1200 mm 2 L = 2000 mL 1,200 cm = 12 m 0.12 m = 120 mm.

Here are the completed conversions using the provided numbers:

1. 12,000 g = 12 kg (To convert grams to kilograms, divide by 1,000)

2. 120 cm = 1,200 mm (To convert centimeters to millimeters, multiply by 10)

3. 1.2 L = 1,200 mL (To convert liters to milliliters, multiply by 1,000)

4. 1,200 cm = 12 m (To convert centimeters to meters, divide by 100)

5. 0.12 m = 120 mm (To convert meters to millimeters, multiply by 1,000)

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The volume of the cone is (5/3)π(9²) cubic units ≈ 381.7 cubic units.

The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius of the circular base and h is the height of the cone. However, we are given the base area B instead of the radius, so we need to find the radius first.

We know that the area of a circle is A = πr², so if the base area of the cone is B = 5π square units, then πr² = 5π, which means r² = 5. Solving for r, we get r = √5.

Now that we have the height h = 9 units and the radius r = √5 units,

we can use the formula for the volume of a cone:

V = (1/3)πr²h.

Substituting the values, we get

V = (1/3)π(√5)²(9) = (5/3)π(9²) cubic units, which simplifies to ≈ 381.7 cubic units.

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Answer:

2

Step-by-step explanation:

To solve this problem, we need to first find the current average and median number of AP classes per student, and then use that information to determine the number of AP classes the new student is taking.

To find the current average number of AP classes per student, we can use the information in the table:

(1 AP class) x 6 students = 6 AP classes

(2 AP classes) x 9 students = 18 AP classes

(3 AP classes) x 5 students = 15 AP classes

(4 AP classes) x 4 students = 16 AP classes

Total number of AP classes = 6 + 18 + 15 + 16 = 55

Total number of students = 6 + 9 + 5 + 4 = 24

Average number of AP classes per student = Total number of AP classes / Total number of students

= 55 / 24

= 2.29 (rounded to two decimal places)

To find the current median number of AP classes per student, we need to order the number of AP classes per student from least to greatest:

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4

The median is the middle value when the data is ordered in this way. Since there are 24 students, the median is the average of the 12th and 13th values:

Median = (2 + 2) / 2

= 2

Since we know that the current average and median are not equal, the new student must be taking a number of AP classes that will bring the average up to 2. We can set up an equation to represent this:

(55 + x) / (24 + 1) = 2

where x is the number of AP classes the new student is taking. Solving for x, we get:

55 + x = 50

x = -5

This is a nonsensical answer, as the number of AP classes taken by the new student cannot be negative. Therefore, our assumption that the new student is taking a number of AP classes greater than the current average is incorrect. Instead, the new student must be taking a number of AP classes less than the current average, which will bring the average down to 2.

Let y be the number of AP classes the new student is taking. We can set up a new equation to represent this:

(55 + y) / (24 + 1) = 2 - ((2.29 - 2) / 2)

where the term on the right-hand side represents the amount by which the average needs to decrease in order to reach 2. Solving for y, we get:

55 + y = 46.5

y = 46.5 - 55

y = 8.5

So the new student is taking 8.5 AP classes. However, since the number of AP classes must be a whole number, we need to round this value to the nearest integer. Since 8.5 is closer to 9 than to 8, we round up to 9. Therefore, the answer is:

The new student is taking 9 AP classes. Answer: None of the above (not given as an option).

The number of cones that can be filled with the ice cream from the container is 10.

Let's start with the container. We are given that it is a right circular cylinder with a diameter of 12 cm and a height of 15 cm. To find the volume of this cylinder, we use the formula:

Volume of cylinder = πr²h

where r is the radius of the cylinder (which is half of the diameter), and h is the height. Substituting the given values, we get:

Volume of cylinder = π(6 cm)²(15 cm) = 540π cubic cm

So the container has a volume of 540π cubic cm.

Now, let's move on to the cones. We are given that the cones have a height of 12 cm and a diameter of 6 cm. The cones have a hemispherical shape on the top, so we can consider them as a combination of a cone and a hemisphere. The formula for the volume of a cone is:

Volume of cone = (1/3)πr²h

where r is the radius of the base of the cone, and h is the height. Substituting the given values, we get:

Volume of cone = (1/3)π(3 cm)²(12 cm) = 36π cubic cm

The formula for the volume of a hemisphere is:

Volume of hemisphere = (2/3)πr³

where r is the radius of the hemisphere. Substituting the given values (the radius is half the diameter of the cone, which is 3 cm), we get:

Volume of hemisphere = (2/3)π(3 cm)³ = 18π cubic cm

So the total volume of each cone is:

Volume of cone + hemisphere = 36π + 18π = 54π cubic cm

To find out how many cones can be filled with the ice cream from the container, we divide the volume of the container by the volume of each cone:

Number of cones = Volume of container / Volume of each cone Number of cones

=> (540π cubic cm) / (54π cubic cm) Number of cones = 10

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Complete Question:

A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream.

Loss on disposal of plant assets = $46400 - $32550

Loss on disposal of plant assets = $13850

Date Account titles and Explanation Debit Credit

Jan. 01 Accumulated depreciation-Equipment $81000

Equipment  $81000

June 30 Depreciation expense (1) $4000

Accumulated depreciation-Equipment  $4000

(To record depreciation expense on forklift)  

June 30 Cash $13000

Accumulated depreciation-Equipment (2) $28000

Equipment  $40000

Gain on disposal of plant assets (3)  $1000

(To record sale of forklift)  

Dec. 31 Depreciation expense (4) $5425

Accumulated depreciation-Equipment  $5425

(To record depreciation expense on truck)  

Dec. 31 Accumulated depreciation-Equipment (5) $32550

Loss on disposal of plant assets (6) $13850

Equipment  $46400

(To record sale of truck)  

Calculations :

(1)

Depreciation expense = (Book value - Salvage value) / Useful life

Depreciation expense = ($40000 - $0) / 5 = $8000 per year

So, for half year = $8000 * 6/12 = $4000

(2)

From Jan. 1, 2019 to June 30, 2022 i.e 3.5 years.

Accumulated depreciation = $8000 * 3.5 years = $28000

(3)

Gain on disposal of plant assets = Sale value + Accumulated depreciation - Book value

Gain on disposal of plant assets = $13000 + $28000 - $40000

Gain on disposal of plant assets = $1000

(4)

Depreciation expense = (Book value - Salvage value) / Useful life

Depreciation expense = ($46400 - $3000) / 8

Depreciation expense = $5425 per year

(5)

From Jan. 1, 2017 to Dec. 31, 2022 i.e 6 years.

Accumulated depreciation = $5425 * 6 years = $32550

(6)

Loss on disposal of plant assets = Book value - Accumulated depreciation

Loss on disposal of plant assets = $46400 - $32550

Loss on disposal of plant assets = $13850

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The ratios are: surface area 9:121, volume 27:1331, width 3:11.

(a) The ratio of the surface area of the model to the surface area of the original can be found by using the scale factor to find the ratio of the corresponding side lengths. Since surface area is proportional to the square of the side length, we can use this ratio squared to find the ratio of the surface areas.
The ratio of the corresponding side lengths is 3:11, so the ratio of the surface areas is (3/11)^2, which simplifies to 9/121.
Therefore, the ratio of the surface area of the model to the surface area of the original is 9:121.
(b) The ratio of the volume of the model to the volume of the original can be found using the same method as above, but with volume instead of surface area. Since volume is proportional to the cube of the side length, we can use this ratio cubed to find the ratio of the volumes.
The ratio of the corresponding side lengths is 3:11, so the ratio of the volumes is (3/11)^3, which simplifies to 27/1331.
Therefore, the ratio of the volume of the model to the volume of the original is 27:1331.
(c) The ratio of the width of the model to the width of the original can be found directly from the scale factor, since width is one of the corresponding side lengths.
The ratio of the corresponding side lengths is 3:11, so the ratio of the widths is 3:11.
Therefore, the ratio of the width of the model to the width of the original is 3:11.
Overall, the ratios are: surface area  9:121, volume  27:1331, width 3:11.

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The value of the tangent TU for the circle with secant through U which intersect the circle at points V and W is equal to 12

Circle theorems are a set of rules that apply to circles and their constituent parts, such as chords, tangents, secants, and arcs. These rules describe the relationships between the different parts of a circle and can be used to solve problems involving circles.

For the tangent TU and the secant through U which intersect the circle at points V and W;

TU² = UV × VW {secant tangent segments}

(5x)² = 9 × 16

(5x)² = 144

5x = √144 {take square root of both sides}

5x = 12

x = 12/5

so;

TU = 5(12/5)

TU = 12

Therefore, the value of the tangent TU for the circle with secant through U which intersect the circle at points V and W is equal to 12

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The inequality which is used to represent normal "temperature-range" for  "human-body", is (d) |t − 98.6| ≤ 1.8.

The "average-temperature" of body is = 98.6° F, and it can vary by 1.8°F.

The inequality |t − 98.6| ≤ 1.8 indicates that the absolute difference between the body temperature and the average temperature is less than or equal to 1.8° F.

This means that the body temperature t can vary within a range of 1.8° F from the average temperature of 98.6° F.

Which means, the temperature cam range from :

⇒ 98.6-1.8 ≤ t ≤ 98.6+1.8,

⇒ 96.8 ≤ t ≤ 100.4;

Therefore, the correct inequality is (d) |t − 98.6| ≤ 1.8.

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The given question is incomplete, the complete question is

The average human body temperature is 98.6° F, but it can vary by as much as 1.8° F. Write an inequality to represent the normal temperature range of the human body, where t represents body temperature.

(a) |t − 1.8| ≥ 98.6

(b) |t − 1.8| ≤ 98.6

(c) |t − 98.6| ≥ 1.8

(d) |t − 98.6| ≤ 1.8

Answer: |t − 98.6| ≤ 1.8

Step-by-step explanation: If takes then takes then takes then takes then takes.

Answer:

x's will eliminate because they have opposite coefficients

Step-by-step explanation:

Because the x's have opposite coefficients, they will eliminate when the given system of linear equations (2-var) are summed up

Answer:

x's will eliminate because they have opposite coefficients.

Step-by-step explanation:

Why the other choices are incorrect:

y's will eliminate because they have opposite coefficients - not true, they don't have opposite coefficients.

x's will eliminate because you always have to solve for x first - you can solve for y too.

y's will eliminate because why not - isn't a good explanation.

The minimum value of 9y – 45x^2 is 0, which occurs when y = 5x^2/3, so the range of V is all non-negative real numbers:

Range of V: [0, ∞)

To find the domain and range of the function V(x, y) = 9√(9y – 45x^2), we need to consider the values of x and y that make the expression under the square root non-negative, since we cannot take the square root of a negative number.

So, we have:

9y – 45x^2 >= 0

Dividing both sides by 9 and rearranging, we get:

y >= 5x^2/3

This means that the domain of V is all points (x, y) such that y is greater than or equal to 5x^2/3:

Domain of V: {(x, y) | y >= 5x^2/3}

To find the range of V, we note that the square root is always non-negative, so V(x, y) will be non-negative whenever 9y – 45x^2 is non-negative. The minimum value of 9y – 45x^2 is 0, which occurs when y = 5x^2/3, so the range of V is all non-negative real numbers:

Range of V: [0, ∞)

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The mean monthly rainfall amount for City A is approximately 338.3 inches, and for City B, it is 6 inches.

The mean absolute deviation for City A is approximately 92.8 inches, and for City B, it is 0.5 inches.

The median for City A is 5.5 inches, and for City B, it is 6 inches.

(a) To find the mean monthly rainfall amount for each city, we sum up the rainfall amounts for each city and divide by the number of months.

For City A:

Mean = (5 + 2.5 + 6 + 2008.5 + 5 + 3) / 6 = 2030 / 6 ≈ 338.3 inches per month

For City B:

Mean = (7 + 6 + 5.5 + 6.5 + 5 + 6) / 6 = 36 / 6 = 6 inches per month

So, the mean monthly rainfall amount for City A is approximately 338.3 inches, and for City B, it is 6 inches.

(b) The mean absolute deviation (MAD) is a measure of the average distance between each data point and the mean. To calculate the MAD, we first find the absolute difference between each data point and the mean, sum them up, and then divide by the number of data points.

For City A:

Absolute differences from the mean: |5 - 338.3|, |2.5 - 338.3|, |6 - 338.3|, |2008.5 - 338.3|, |5 - 338.3|, |3 - 338.3|

MAD = (333.3 + 335.8 + 332.3 + 1670.2 + 333.3 + 335.3) / 6 ≈ 557.0 / 6 ≈ 92.8

For City B:

Absolute differences from the mean: |7 - 6|, |6 - 6|, |5.5 - 6|, |6.5 - 6|, |5 - 6|, |6 - 6|

MAD = (1 + 0 + 0.5 + 0.5 + 1 + 0) / 6 ≈ 3 / 6 = 0.5

So, the mean absolute deviation for City A is approximately 92.8 inches, and for City B, it is 0.5 inches.

(c) To find the median, we arrange the rainfall amounts in ascending order and find the middle value. If there are an even number of data points, we take the average of the two middle values.

For City A: {2.5, 3, 5, 5, 6, 2008.5}

Median = (5 + 6) / 2 = 11 / 2 = 5.5 inches

For City B: {5, 5.5, 6, 6, 6.5, 7}

Median = 6 inches

So, the median for City A is 5.5 inches, and for City B, it is 6 inches.

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Answer: 55%

Step-by-step explanation:

To find the percentage of 11/20, we can use the following formula:

Percentage = (Numerator ÷ Denominator) × 100

Substituting the values from 11/20 into the formula, we get:

Percentage = (11 ÷ 20) × 100

Percentage = 0.55 × 100

Percentage = 55%

Therefore, the percentage of 11/20 is 55%.

Answer:

Solution: 11/20 as a percent is 55%

Step-by-step explanation:

First, convert the fraction into a decimal by dividing the numerator by the denominator:

11/20 = 0.55

If we multiply the decimal by 100, we will get the percentage:

00.5 * 100 = 55

We can see that 11/20 is percentage is 55.

To find the absolute value of the difference between theoretical and experimental probabilities, you need to follow these steps:

1. Calculate the theoretical probability: This is the probability of an event occurring based on the total number of possible outcomes. It can be found by dividing the number of successful outcomes by the total number of possible outcomes.

2. Calculate the experimental probability: This is the probability of an event occurring based on actual experiments or trials. It can be found by dividing the number of successful outcomes by the total number of trials conducted.

3. Find the difference: Subtract the experimental probability from the theoretical probability.

4. Take the absolute value: The absolute value is the non-negative value of a number, disregarding its sign. To find the absolute value of the difference, simply remove the negative sign if the result is negative.

By following these steps, you'll find the absolute value of the difference between theoretical and experimental probabilities, which is an important measure to assess the accuracy of experiments and predictions.

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The results of the survey suggest that the student's claim of three-quarters of 17-year-olds having their driver's licenses may be accurate, but further research would be necessary to confirm this with a larger and more representative sample.

Based on the computer output, there is no convincing evidence that the true proportion of 17-year-olds at this high school with driver's licenses is not 0.75. This is because the P-value of 0.9315 is very large, indicating that the results of the survey are not statistically significant. Additionally, the 95% confidence interval for the sample proportion of 0.755556 includes 0.75, further supporting the claim that the true proportion may be close to 0.75.

It is important to note that the correct significance test was performed, testing the null hypothesis that p = 0.75 against the alternative hypothesis that p ≠ 0.75. This is the appropriate test when the claim being tested is about a specific value of the proportion, as in this case. The alternative hypothesis being p > 0.75 or p < 0.75 would be incorrect, as it assumes a one-sided test rather than a two-sided test.

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To solve the given question, we need to use the distributive property and collect like terms. In summary, the distributive property is a useful tool in simplifying expressions.

First, we need to distribute the fraction 4/5 to the expression inside the parenthesis, which gives us 4/5 x 1/4c - 4/5 x 5. Then, we can simplify the expression by multiplying the two fractions and combining the terms. This gives us (1/5)c - 4.

Therefore, the simplified expression is (1/5)c - 4. We can use this expression to evaluate the given expression for any value of c. For example, if c = 15, then the expression becomes (1/5) x 15 - 4 = 3 - 4 = -1.

In summary, the distributive property is a useful tool in simplifying expressions.

By distributing a term to each term inside a set of parentheses, we can collect like terms and simplify the expression. In this case, we used the distributive property to simplify a fraction and a constant and then combined the like terms to obtain the final answer.

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The container could hold an additional 870 mL of broth if it is already holding the 90 mL of broth from the beaker.

To figure out if the chicken broth in the beaker will fit into the rectangular food storage container, we need to compare the volume of the beaker to the volume of the container.

The volume of the beaker can be calculated by multiplying its base area (which is the same as the area of the circle at the bottom of the beaker) by its height. The volume of the container can be calculated by multiplying its length, width, and height.

If the volume of the beaker is less than or equal to the volume of the container, then the chicken broth will fit. If it is greater, then the chicken broth will not fit.

To find out how much more broth the container could hold if it fits, we can subtract the volume of the beaker from the volume of the container.

For example, if the beaker has a diameter of 6 cm and a height of 10 cm, then its volume would be:

V_beaker = πr^2h = π(3^2)(10) = 90π cm^3 = 90 mL

If the rectangular food storage container has dimensions of 12 cm x 8 cm x 10 cm, then its volume would be:

V_container = lwh = 12(8)(10) = 960 cm^3 = 960 mL

Since 90 mL (the volume of the beaker) is less than 960 mL (the volume of the container), the chicken broth will fit in the container. The amount of broth the container can hold in addition to the beaker's contents would be:       960 mL - 90 mL = 870 mL

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To calculate the rate of change of the simple index over 1 day, we need to first calculate the index value for Monday and Tuesday.

On Monday, the value of stock A is 10,000 x $5.50 = $55,000, and the value of stock B is 8,000 x $6.25 = $50,000. The total value of the index on Monday is $55,000 + $50,000 = $105,000.

On Tuesday, the value of stock A is 10,000 x $5.80 = $58,000, and the value of stock B is 8,000 x $6.65 = $53,200. The total value of the index on Tuesday is $58,000 + $53,200 = $111,200.

To calculate the rate of change, we can use the formula:

(rate of change) = (new value - old value) / old value x 100%

Using this formula, we get:

(rate of change) = ($111,200 - $105,000) / $105,000 x 100% = 5.90%

Therefore, the rate of change of this simple index over 1 day is 5.90%.

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The numbers that are less than 3/7 are 1/8 and 3/10, while the numbers that are greater than 3/7 are 1/2 and 2/3. The decimal 0.102, which is equivalent to 51/500 as a fraction, is also less than 3/7.

To determine which of the given numbers are less than 3/7, we can convert them to fractions and compare them to 3/7.

12.5% is equivalent to 0.125 as a decimal or 1/8 as a fraction. To compare 1/8 and 3/7, we can convert them to a common denominator. The least common multiple of 8 and 7 is 56, so we can rewrite 1/8 as 7/56 and 3/7 as 24/56. Therefore, 1/8 is less than 3/7.

0.3 is equivalent to 3/10 as a fraction. To compare 3/10 and 3/7, we can also convert them to a common denominator. The least common multiple of 10 and 7 is 70, so we can rewrite 3/10 as 21/70 and 3/7 as 30/70. Therefore, 3/10 is less than 3/7.

50% is equivalent to 0.5 as a decimal or 1/2 as a fraction. To compare 1/2 and 3/7, we can convert 3/7 to a fraction with a denominator of 2. Multiplying the numerator and denominator of 3/7 by 2 gives us 6/14. Therefore, 1/2 is greater than 3/7.

2/3 is already a fraction, and we can compare it directly to 3/7. Multiplying the numerator and denominator of 3/7 by 3 gives us 9/21, and we can see that 2/3 is greater than 3/7.

0.102 is a decimal that is less than 1, but it can also be written as a fraction. To do so, we can place the decimal over 1 followed by the appropriate number of zeros. This gives us 102/1000, which can be simplified to 51/500. To compare 51/500 and 3/7, we can convert them to a common denominator. The least common multiple of 500 and 7 is 3500, so we can rewrite 51/500 as 357/3500 and 3/7 as 1500/3500. Therefore, 51/500 is less than 3/7.

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a) To verify that the points A(14, 9) and B(7, 16) are on the circle with center C(2, 4) and radius 13, we can use the distance formula:

Distance between point A and C:

d_AC = sqrt[(x_A - x_C)^2 + (y_A - y_C)^2]

    = sqrt[(14 - 2)^2 + (9 - 4)^2]

    = sqrt[144 + 25]

    = sqrt(169)

    = 13

Since the distance between point A and C is equal to the radius of the circle, point A is on the circle.

Distance between point B and C:

d_BC = sqrt[(x_B - x_C)^2 + (y_B - y_C)^2]

    = sqrt[(7 - 2)^2 + (16 - 4)^2]

    = sqrt[25 + 144]

    = sqrt(169)

    = 13

Since the distance between point B and C is equal to the radius of the circle, point B is also on the circle.

Therefore, points A and B are on the circle with center C(2, 4) and radius 13.

b) The midpoint of line segment AB can be found using the midpoint formula:

M = [(x_A + x_B)/2, (y_A + y_B)/2]

 = [(14 + 7)/2, (9 + 16)/2]

 = [10.5, 12.5]

The slope of line segment AB can be found using the slope formula:

m_AB = (y_B - y_A)/(x_B - x_A)

    = (16 - 9)/(7 - 14)

    = -7/-7

    = 1

The slope of a line perpendicular to AB will be the negative reciprocal of m_AB:

m_CM = -1/m_AB

    = -1/1

    = -1

The equation of the line passing through points C(2, 4) and M(10.5, 12.5) can be found using the point-slope form:

y - y_C = m_CM(x - x_C)

y - 4 = -1(x - 2)

y = -x + 6

The slope of line CM is -1, which is the negative reciprocal of the slope of line AB. Therefore, line CM is perpendicular to line AB.

Hence, we have shown that line segment CM is perpendicular to line segment AB.

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