The area of a circle increases at a rate of 2 cm2/s. a. How fast is the radius changing when the radius is 4 cm? b. How fast is the radius changing when the circumference is 3 cm?

The value of (f + g)(−3) given the functions f(x) = x² − 6x − 4 and g(x) = 5x + 3 is 11.

To find (f + g)(-3), we first need to add the functions f(x) and g(x) together, and then evaluate the resulting function at x = -3.

f(x) = x² - 6x - 4
g(x) = 5x + 3

Now, let's add f(x) and g(x):

(f + g)(x) = (x² - 6x - 4) + (5x + 3) = x² - x - 1

Now that we have the combined function, we can evaluate it at x = -3:

(f + g)(-3) = (-3)² - (-3) - 1 = 9 + 3 - 1 = 11

So, (f + g)(-3) = 11.

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The length of chord segment EG = 6 inches and the length of FG = √23 inches.

Given, chord EG = 8 inches and the distance from the center of J to the chord is 3 inches.

We can draw a diagram as follows:

              J

            /   \

           /     \

          /       \

         /         \

        E-----------G

              |

              |

              |

              |

              |

              F

Here, OJ is perpendicular to chord EG at point F.

As per the theorem, the length of the perpendicular from the center of the circle to a chord is half the length of the diameter intersecting the chord.

So, we can find the length of the diameter intersecting chord EG and then use it to find the radius of the circle.

Length of chord EG = 8 inches

Length of OJ = 3 inches

Using Pythagorean theorem in right triangle OFG, we get:

OG² = OF² + FG²

Let x be the length of FG

We know that OF = OJ = 3 inches

OG = radius of the circle

So, we have:

radius of circle = OG = √(OF² + FG²) = √(3² + x²)

The diameter of the circle = 2(radius) = 2√(3² + x²)

Now, using the theorem mentioned above, we can say:

Length of perpendicular from the center of the circle to chord EG = OF = 3 inches

Length of diameter intersecting chord EG = 2√(3² + x²)

So, we get:

Length of chord segment EG = 2 * length of perpendicular

                            = 2 * 3 inches

                           = 6 inches

Now, we know that the chord segment EG divides the diameter intersecting it into two equal parts.

So, we have:

Length of one part of the diameter = (2√(3² + x²))/2 = √(3² + x²)

Using Pythagorean theorem in right triangle OJF, we get:

OJ² + JF² = OF²

3² + JF² = 8²

JF² = 8² - 3² = 55

JF = √55

Using Pythagorean theorem in right triangle JFG, we get:

JG² + FG² = JF²

(√(3² + x²))² + x² = 55

9 + x² + x² = 55

2x² = 46

x² = 23

x = √23

Therefore, the length of chord segment EG = 6 inches and the length of FG = √23 inches.

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20 questions subtraction and addition of fractions only like this:

A recipe calls for 2/3 cup of flour and 1/4 cup of sugar. What is the total amount of flour and sugar needed in fraction?

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Therefore, the total area of the forest is approximately 43.10 acres.

Area is the measure of the size of a two-dimensional surface or region. It is typically measured in square units, such as square meters (m²) or square feet (ft²). The area of a shape can be calculated by multiplying the length and the width of the shape. Different shapes have different formulas for calculating their area, such as the formula for the area of a rectangle, which is length x width, or the formula for the area of a circle, which is πr², where r is the radius of the circle and π is a mathematical constant approximately equal to 3.14159.

Here,

Let's call the total area of the forest "T".

We know that 25 acres of the forest is covered by coniferous trees. Therefore, the remaining area of the forest is:

T - 25

We also know that the shrubs and bushes constitute 42% of the total area of the forest. This means that:

0.42 x T = area covered by shrubs and bushes

Putting it all together, we can set up the following equation:

T = 25 + (0.42 x T)

Simplifying, we can solve for T:

0.58 x T = 25

T = 25 / 0.58

T = 43.10 acres (rounded to two decimal places)

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Answer:

8 people

Step-by-step explanation:

Tea- 13 people

Coffee- 12 people

Neither- 3 people

20 - 3 = 17

12 + 13 = 25

25 - 17 = 8

The height of the cuboid, after calculations, in the form (a+ b√3)m, is (6√3 - 9)/47 meters.

Let the height of the cuboid be h meters. The area of the square base is given by:

(2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3 m²

The total surface area of the cuboid is the sum of the areas of the six rectangular faces. Since the base is a square, the area of each of the four vertical rectangular faces is also (2 + √3) × h = (2h + h√3) m². Therefore, we have:

Total surface area = 4(7 + 4√3) + 2(2h + h√3)(2 + √3) = 8h + 26 + (22 + 16√3)h

Since we know that one of the sides has area (2√3 - 3) m², we can set up another equation:

(2h + h√3)(2 + √3) = 2√3 - 3

Expanding the left side and simplifying, we get:

(2h + h√3)(2 + √3) = 2√3 - 3

4h + 7h√3 = 2√3 - 3

h(4 + 7√3) = 2√3 - 3

h = (2√3 - 3)/(4 + 7√3)

We can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:

h = [(2√3 - 3)/(4 + 7√3)] × [(4 - 7√3)/(4 - 7√3)]

h = (8√3 - 12 - 14√3 + 21)/(16 - 63)

h = (9 - 6√3)/(-47)

h = (6√3 - 9)/(47)

Therefore, the height of the cuboid is (6√3 - 9)/47 meters.

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The slope of the line through points (6,3) and (12,7) is 2/3.

To find the slope of a line, we use the formula:

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

In this case, we have two points: (6, 3) and (12, 7). We can label them as follows:

x1 = 6
y1 = 3
x2 = 12
y2 = 7

Now we can plug these values into the formula:

Slope = (y2 - y1) / (x2 - x1)
Slope = (7 - 3) / (12 - 6)
Slope = 4 / 6
Slope = 2/3

Therefore, the slope of the line through the points (6, 3) and (12, 7) is 2/3.

The slope of a line tells us how steep it is. A positive slope means the line goes up as you move from left to right, while a negative slope means the line goes down. In this case, since the slope is positive (2/3), we know that the line goes up as we move from left to right.

The slope also tells us how much the y-value changes for every one unit of x-value. In this case, for every one unit we move to the right, the y-value goes up by 2/3.

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Probability or p-vale that the sample mean would differ from the population mean by less than 221 miles in a sample of 56 tires is equals to zero if the manager is correct.

We have data of an operation manager at a tire manufacturing company.

Mean mileage of a tire, [tex] \mu[/tex]

= 37,014 miles

standard deviation, [tex] \sigma[/tex]

= 4617 miles.

Sample size, n = 56

We have to determine the probability that the sample mean would differ from the population mean by less than 221 miles. Using Z-score formula in normal distribution, [tex]\small z= \frac{ \bar x-\mu }{\frac{\sigma }{\sqrt{n}}},[/tex]

Plugging all known values in above formula, [tex]z = \frac{ 221 - 37,014} {\frac{4617}{ \sqrt{56}}}[/tex]

= 59.634

[tex]P( \bar x < 221) = P ( \frac{ \bar x-\mu }{\frac{\sigma }{\sqrt{n}}} < \frac{ 221 - 37,014} {\frac{4617}{ \sqrt{56}}}) \\ [/tex]

=> P ( z < 59.63) = P( \bar x < 221)

Using the Z-distribution table, probability value is equals to 0. Hence, required probability is zero.

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The number of pillows requiring [tex]\frac{2}{3}[/tex] yards that can be made from [tex]3\frac{1}{3}[/tex] yards is 5. Thus the right answer to the given question is C.

Material required for making one pillow = [tex]\frac{2}{3}[/tex] yards

Total material = [tex]3\frac{1}{3}[/tex] yards

To find the number of pillows made we have to divide the material required for one pillow by the total material available to Kevin for making pillows

Number of pillows = [tex]3\frac{1}{3}[/tex] ÷ [tex]\frac{2}{3}[/tex]

=  [tex]\frac{10}{3}[/tex] ÷ [tex]\frac{2}{3}[/tex]

To divide two fractions, we take the reciprocal of the second number and multiply it by the first number.

=  [tex]\frac{10}{3}[/tex] * [tex]\frac{3}{2}[/tex]

= 5

Thus, the number of pillows made is 5.

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Considering "line-segment" defined by points A(0, 1) and B(4,6), then  effects of reflection across the "x-axis" are :

(b) The "y-coordinate" of "end-points" become opposites in sign.

(e) The length of reflection of AB is same as length of line-segment AB.

When reflecting a line segment across the x-axis, the x-coordinates of all points remain the same, but the y-coordinates become their opposites.

So, for the line segment AB, the x-coordinate of point A remains 0, and the x-coordinate of point B remains 4. However, the y-coordinate of point A becomes -1, and the y-coordinate of point B becomes -6. This results in a new line segment A'(0, -1) and B'(4, -6).

Since the reflection is across a horizontal line, the length of the reflection is the same as the length of the original segment.

Therefore, the correct options are (b) and (e).

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Statistical measures, you can construct a box plot that shows the range, median, and quartiles of Tisha's running distances over the 10 weeks.

A statistical measure is a numerical value that provides information about a specific aspect of a dataset's distribution, such as its central tendency, spread, or variability. Box plots require several statistical measures to be constructed, including:

Minimum: The smallest value in the dataset. In this case, the minimum value is 4.

Maximum: The largest value in the dataset. In this case, the maximum value is 16.

Median: The middle value of the dataset when it is arranged in numerical order. In this case, the median is the average of the two middle values, which are 8 and 9. The median is therefore (8 + 9) / 2 = 8.5.

First Quartile (Q1): The value below which 25% of the data falls. In this case, the first quartile is the median of the first half of the data, which is 6.

Third Quartile (Q3): The value below which 75% of the data falls. In this case, the third quartile is the median of the second half of the data, which is 14.

With these statistical measures, you can construct a box plot that shows the range, median, and quartiles of Tisha's running distances over the 10 weeks.

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Part A:

The definition provided for a circle is actually correct. However, if we change the definition slightly to say that a circle is the set of all points in a plane that are the same distance from a given point, we can find an example that contradicts it.

For instance, consider a cone in three-dimensional space. If we take a cross-section of the cone that is parallel to the base, we get a circle. However, this circle is not the set of all points that are the same distance from one given point, but rather from the axis of the cone.

To make the definition more accurate, we need to specify that the circle exists in a plane.

Part B:

An example of an undefined term related to a circle is the term "point." A circle is defined as the set of all points that are the same distance from a given point, but the term "point" is not defined within this definition.

A point is typically defined as a location in space that has no size or shape. In the context of a circle, a point can be thought of as any location on the circumference of the circle. However, it is important to note that the definition of a point is not dependent on the definition of a circle, and vice versa.

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To find out the car's average fuel consumption in liters per 100 kilometers, we need to use the following formula:

Fuel consumption (liters per 100 km) = (Total fuel used / Total distance traveled) x 100

Using the information given, we know that the car traveled 2500 km and used 150 liters of fuel. Plugging these values into the formula, we get:

Fuel consumption (liters per 100 km) = (150 / 2500) x 100 = 6

Therefore, the car used an average of 6 liters of fuel per 100 kilometers on its 2500 km road trip.

It is important to note that fuel economy is an important factor in Europe, where fuel prices are generally higher than in other parts of the world.

By measuring fuel consumption in liters per 100 kilometers, European consumers can easily compare the fuel efficiency of different vehicles and make more informed purchasing decisions.

Additionally, analyzing fuel consumption data can help drivers identify ways to improve their fuel efficiency, such as reducing their speed, maintaining proper tire pressure, and avoiding excessive idling.

This not only saves money on fuel costs, but also reduces carbon emissions and contributes to a more sustainable transportation system.

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The value of the [tex]5000^{th}[/tex] term in the given arithmetic sequence K is 44995.

The sequence that is given in the question is said to be an arithmetic sequence which means the consecutive elements in the series will have common differences.

To find any term in the series first, we need to find the first term and the common difference that the series follows.

Here we know that the first and the second term of the series are 4 and 13 so from this we can find the common difference which is:

13-4=9

so the first term (a) = 4

the common difference (d) = 9

To find the [tex]n^{th}[/tex] term of the series we can use the formula:

[tex]a_n=a_1+(n-1)*d[/tex]

where [tex]a_n[/tex] is the nth term in the sequence, [tex]a_1[/tex] is the first term of the series, n is the no.of term, and d is the common difference.

So to find the 5000th term in the series

[tex]a_{5000}=4+(5000-1)*9\\a_{5000}=4+(4999*9)\\a_{5000}=4+ 44991\\a_{5000}= 44995\\[/tex]

The value of the [tex]5000^{th}[/tex] term is 44995

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The inequality that can be used to find all possible values of T, the time it would take to reach their destination if they travel at an average speed of at least "r" miles per hour, can be expressed as:

T ≤ 475 / r

This inequality states that the time taken (T) should be less than or equal to the distance traveled (475 miles) divided by the average speed (r miles per hour). By dividing the total distance by the average speed, we obtain the maximum time it would take to reach the destination. Any time less than or equal to this value would satisfy the condition of traveling at an average speed of at least "r" miles per hour.

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The value of x is  17.5 when the given diagonals of square ABCD points intersect at M and the measure 0f <MAD = 6x-15.

The diagonals are perpendicular bisectors of each other in a square and they intersect at the center of the square. By using diagonals we can divide the square into four congruent right triangles and assume that diagonals are all right angles (90 degrees).

let us consider the angle MAD as one of these angles. we can find the value of x by using the equation

6x - 15 = 90

6x = 105

x = 105/6

= 17.5.

Therefore, the value of the x is 17.5.

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The complete question is,

Given the diagonals of square ABCD points intersect at M: If the measure 0f <MAD = 6x-15, find the value of x Show the work that justifies your answer: M B Your answer

Answer:

b=18.7

Step-by-step explanation:

sin112°/37=sin28°/b

b=sin28°/(sin112°/37)

b=18.7

On babysitting for four hours, the differently earned money will be Quinton would make $5 more than Reema.

Reema's charge for babysitting for four hours = 10 × 4

Reema's charge = $40

For Quinton, keep the value of hours in expression -

Quinton's charge = 5 (4) + 25

Quinton's charge = 20 + 25

Quinton's charge = $45

As Quinton's charge is greater than Reema, she makes more money.

Amount of more money made by Quinton = 45 - 40

Subtract to find the difference

Amount of more money made by Quinton = $5

Hence, correct answer is Quinton would make $5 more than Reema.

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To get a number greater than 5,but less than 0, Peter can use the mathematical symbol of a decimal point (.) to solve this puzzle.

If Peter places decimal point between 5 and 9, he can get a number like 5.1, 5.2, 5.3... up to 8.9, which meets the conditions of being greater than 5 but less than 9.

A decimal point (.) is a mathematical symbol. When this decimal point is placed in between two numbers, suppose x and y, then x.y means x.y is greater than x and less than y.  

So to solve the puzzle, Peter can use decimal point.

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The function used to model the monthly profit for x trinkets produced are f(x) = -4(x - 50)(x - 250). The maximum profit occurs when 150 trinkets are produced.

The quadratic function that can be used to model the monthly profit for x trinkets produced is:

f(x) = -4(x - 50)(x - 250)

This is because the function is in the form of a quadratic equation, which is y = ax² + bx + c. In this case, a = -4, b = 1200, and c = 0. When we expand and simplify the equation, we get:

f(x) = -4x² + 1200x

This equation represents a parabola with a maximum value at x = 150. Therefore, the maximum profit occurs when 150 trinkets are produced.

The other answer choices are not correct because they are not quadratic functions. For example, f(x) = (x - 50)(x - 250) is a product of linear factors, and f(x) = 28(x - 50)(x + 250) and f(x) = (x + 50)(x + 250) have a coefficient of x² that is not equal to -4.

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The area of the semi-circle patio is approximately 17.91 square feet.

To find the area of the semi-circle patio, we need to use the formula for the area of a circle, which is A = πr^2, where π is 3.14 and r is the radius of the circle.

However, since we only have a semi-circle, we need to divide the result by 2 to get the area of the half-circle.

In this case, the radius of the semi-circle is half of the diameter, which is given as 6.75 feet. So, the radius is 6.75/2 = 3.375 feet.

Now, we can plug this value into the formula:

A = (π)(3.375)^2 / 2

A = (3.14)(11.390625) / 2

A = 35.81765625 / 2

A = 17.908828125 square feet

Therefore, the area of the semi-circle patio is approximately 17.91 square feet.

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The system of equations is:

Revenue from balcony seats = $10 × B

Revenue from main floor seats = $25 × M

Total revenue = $6250

B = 3M + 20

Let's define the following variables:

B = number of balcony seats sold

M = number of main floor seats sold

We know that the price of a balcony seat is $10 and the price of a main floor seat is $25.

From the given information, we can create the following equations:

The total revenue from balcony seats sold (B) is given by: Revenue from balcony seats = $10 × B

The total revenue from main floor seats sold (M) is given by: Revenue from main floor seats = $25 × M

The total revenue from the afternoon performance is $6250: Total revenue = Revenue from balcony seats + Revenue from main floor seats

The number of balcony seats sold (B) is 20 more than 3 times the number of main floor seats (M): B = 3M + 20

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The system of inequalities that has a solution set that is a line is [x + y ≥ 3; x + y ≤ 3]; option A

A system of inequalities is a set of two or more inequalities that are solved simultaneously to find the values of variables that satisfy all the inequalities in the system.

Considering the given system of inequalities:

The system of inequalities that has a solution set that is a line is:

[x + y = 3;]

This is because the solution set of this equation is a line with slope -1 passing through the point (3, 0) and (0, 3).

Therefore, the system of inequalities [x + y ≥ 3; x + y ≤ 3] has a solution set that is a line.

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The probability of choosing randomly with replacement an H or P in either selection is derived to be equal to 0.16 which makes the last option correct.

The probability of an event occurring is the fraction of the number of required outcome divided by the total number of possible outcomes.

The total possible outcome = 5

the event of selecting H = 1

probability of selecting H= 1/5

the event of selecting P = 2

probability of selecting H= 2/5

probability of choosing an H or P in either selection = 1/5 × 2/5 + 2/5 × 1/5

probability of choosing an H or P in either selection = 4/25

probability of choosing an H or P in either selection = 0.16

Therefore, the probability of choosing randomly with replacement an H or P in either selection is derived to be equal to 0.16

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The time it will take the flight is C) 5 hours.

To solve this problem, we can use the formula: distance = rate x time. In this case, we know that the distance is 1500 miles and the rate (or speed) is 300 miles per hour. We can rearrange the formula to solve for time: time = distance / rate. Plugging in the values we have, we get:

time = 1500 miles / 300 miles per hour
time = 5 hours

Therefore, the correct answer is C. It will take Penny 5 hours to fly from her starting point to Palawan at an average speed of 300 miles per hour. This calculation assumes that the plane maintains a constant speed throughout the entire flight, which may not be the case due to factors such as wind and turbulence.

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The pool has approximately 258.2 square feet of plastic material. Rounded to the nearest tenth, the answer is 258.2 square feet.+

To calculate the surface area of the pool that is lined with plastic material, we need to find the area of the curved part and the area of the top and bottom circles.

First, let's find the radius of the pool, which is half of the diameter:

radius = 11 feet / 2 = 5.5 feet

The curved part of the pool is a cylinder with a height of 2 feet and a circumference of [tex]2 * π *radius:[/tex]

curved area = height * circumference

[tex]curved area = 2 feet * 2 * 3.14 * 5.5 feet[/tex]

[tex]curved area = 68.2 square feet[/tex]

The top and bottom of the pool are two circles with a radius of 5.5 feet:

circle area = π * radius²

[tex]circle area = 3.14 * 5.5 square feet[/tex]

[tex]circle area = 95.0 square feet[/tex]

To find the total surface area of the pool that is lined with plastic material, we add the area of the curved part and the area of the top and bottom circles:

Total area = curved area + 2 * circle area

Total area = 68.2 square feet + 2 * 95.0 square feet

Total area = 258.2 square feet

The pool has approximately 258.2 square feet of plastic material. Rounded to the nearest tenth, the answer is 258.2 square feet.

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Ines has 9/10 liter of juice, if Meg has 3/10 liter of juice left in her bottle and Ines has 3 times as much juice in her bottle as Meg has .

It is need to find how much juice does Ines have.To calculate it, it is need to know how much liter of juice does Meg have. Meg have 3/10 liter of juice left in her bottle, and Ines has 3 times as much juice her bottle as Meg has.

That is Ines has 3 times means multiplying how much juice meg have by 3.

That is Juice left in bottle of Ines = juice left in bottle of Meg * 3 = 3/10 liter x 3 = 9/10 liter. Therefore, Ines has 9/10 liter of juice in her bottle.

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