Air is being pumped into a spherical balloon so that its volume increases at a rate of 100cm³/s. How fast is the radius of the balloon increasing when the diameter is 50cm?
V = 4/3 πr³

PA is tangent to circle k(O), ∠OAP is a right angle. Similarly, ∠OBP is a right angle.

To prove that m∠P=2·m∠OAB, we need to use the properties of tangents to a circle and the angle relationships between tangent lines and chords in a circle.

First, let's draw a diagram of the situation:

              P

             / \

            /   \

           /     \

          /       \

         /         \

       A-----------B

       /               \

      /                  \

     /                     \

    O                      \

    |                          \

    |                            \

    |                              \

    ----------------------------

We are given that PA and PB are tangents to circle k(O) at A and B, respectively. This means that PA and PB are perpendicular to OA and OB, respectively, at the points of tangency A and B. We can also infer that OA and OB are radii of the circle k(O).

Let ∠OAB = x. Then, ∠OBA = x (since OA = OB), and ∠APB = 180° - ∠OAB - ∠OBA = 180° - 2x.

Since PA is tangent to circle k(O), ∠OAP is a right angle. Similarly, ∠OBP is a right angle. Therefore, ∠OAP + ∠OBP = 180°.

Let ∠P = y. Then, we have:

∠OAB + ∠OBA + ∠APB + ∠P = 180°

x + x + (180° - 2x) + y = 180°

y = 2x

Therefore, we have shown that m∠P = 2·m∠OAB, as required.

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The description concerns Uruguay, option B, since it is the only country among the answer choices that is located in South America to the South of Brazil.

South America is a continent located in the western hemisphere of the Earth. It is situated south of North America, east of the Pacific Ocean, and west of the Atlantic Ocean. The continent is home to 12 independent countries and 3 dependent territories, with a total population of approximately 422 million people.

The largest country in South America is Brazil, followed by Argentina, Peru, and Colombia. The continent is characterized by diverse topography, including the Andes mountain range, the Amazon rainforest, the Atacama Desert, and the Patagonian plains. The region is also known for its rich cultural heritage, including pre-Columbian civilizations such as the Incas and the Mayas, as well as colonial influences from Spain and Portugal. Today, South America is a rapidly developing region with a diverse economy, including agriculture, mining, and manufacturing industries.

Uruguay is also a part of South America. It is located South of Brazil and, as a matter of fact, during colonization, it was a part of Brazil. We can conclude option B is the right answer.

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Answer:

$7973.26

Step-by-step explanation:

PV = $6700

i = 5.8% ÷ 365

n = 3 years · 365

Compound formula

FV = PV (1 + i)^n

FV = 6700 (1 + 5.8% ÷ 365)^(3 · 365)

FV = $7973.26 (rounded to the nearest cent)

Answer:

The value of Eva's investment after 3 years will be approximately $8,108.46. Rounded to the nearest cent, this is $8,108.45.

Step-by-step explanation:

We can use the formula for compound interest:

where:

In this case, we have:

Plugging these values into the formula, we get:

Therefore, the value of Eva's investment after 3 years will be approximately $8,108.46. Rounded to the nearest cent, this is $8,108.45.

The percent of the customers at Harris Teeter that are adults is 70%

From the question, we have the following parameters that can be used in our computation:

Customers = 160

Children = 48

using the above as a guide, we have the following:

Adults = Customers - Children

substitute the known values in the above equation, so, we have the following representation

Adults = 160 - 48

So, we have

Adults = 112

Next, we have

Percentage = 112/160 * 100%

Evaluate

Percentage = 70%

Hence, the percentage is 70%

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The exact value of secθ secθ in simplest radical form is 106/81.

The length of the hypotenuse is the distance from the origin to the point (-9, 5):

√((-9)^2 + 5^2) = √(81 + 25) = √106

cosθ = adjacent/hypotenuse = -9/√106

Therefore, secθ = 1/cosθ = -√106/9.

In order to find the value of secθ secθ, we simply multiply secθ by itself:

secθ secθ = (-√106/9) * (-√106/9) = 106/81

The exact value of secθ secθ is 106/81.

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The probability of randomly selecting a quarter from the bag is 5/16

Assuming that all the coins have the same probability of being randomly drawn, the probability of getting a quarter is equal to the quotient between the total number of quarters and the total number of coins in the bag.

There are 6 quarters, and the total number of coins is 16, then the probability of randomly selecting a quarter is:

P = 5/16

The correct option is the last one.

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The amount that the first rank takes is 1934 dinars, and the amounts that the second and third ranks take are 967 dinars and 483.5 dinars (rounded to 484 dinars), respectively.

Let's assume that there are three brothers, including Musa. Let X be the amount of money that the brother with the highest average takes, and let Y be the amount of money that the brother with the third rank takes.

According to the given conditions, we can write the following equations:

Let's simplify equation 1:

X + Y = 2900

Also, we know that:

X = (2Y + X)/2

(The amount that the third rank takes is half of what the first rank takes)

Simplifying this equation:

2X = 2Y + X

X = 2Y

Substituting this value of X in equation X + Y = 2900:

3Y = 2900

Y = 2900/3

Y ≈ 966.67

As the amount given must be a whole number between two consecutive natural numbers, we can round Y to the nearest natural number:

Y = 967

Then, X = 2Y = 2*967 = 1934

Therefore, the amount that the first rank takes is 1934 dinars, and the amounts that the second and third ranks take are 967 dinars and 483.5 dinars (rounded to 484 dinars), respectively.

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When x = 3 and dx = 0.4, the differential dy is approximately 4.056, and when x = 3 and dx = 0.8, the differential dy is approximately 8.113.

Differential,

To find the differential dy, we use the formula:

dy = f'(x) * dx

where f'(x) is the derivative of the function y = tan(5x + 5) with respect to x.

Taking the derivative, we get: f'(x) = sec^2(5x + 5) * 5 Plugging in x = 3, we get: f'(3) = sec^2(20) * 5

Now we can find the differential dy for dx = 0.4 and dx = 0.8:

When dx = 0.4: dy = f'(3) * dx dy = sec^2(20) * 5 * 0.4 dy ≈ 4.056

When dx = 0.8: dy = f'(3) * dx dy = sec^2(20) * 5 * 0.8 dy ≈ 8.113

Therefore, when x = 3 and dx = 0.4, the differential dy is approximately 4.056, and when x = 3 and dx = 0.8, the differential dy is approximately 8.113.

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Sean's percentage profit is 20% on selling 24 chocolate bars.

Sean's cost price for each chocolate bar is:

£10 / 24 bars = £0.4167 per bar

Sean sells each chocolate bar for 50p, which is £0.5

Sean's revenue from selling all 24 chocolate bars is:

24 bars x £0.5 per bar = £12

Sean's profit is the difference between his revenue and his cost:

Profit = £12 - £10 = £2

To calculate the percentage profit, we can use the following formula:

Percentage profit = (Profit / Cost price) x 100%

So, plugging in the values we get:

Percentage profit =[tex](2 / 10) x 100% = 20%[/tex]= 20

Therefore, Sean's percentage profit is 20%. He earned a profit of £2 on his initial investment of £10, which is equivalent to a 20% return on investment.

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The polynomial, a⁸ - a²·B⁶ in factored form is; a²·(a³ - B³)·(a³ + B³)

A polynomial consists of the differences or sums of terms that are the product of powers of the same variable.

The specified polynomial can be presented as follows;

a⁸ - a²·B⁶

The common factor in the terms of the polynomial is a², therefore, we get;

a⁸ - a²·B⁶ = a⁶ × a² - a²·B⁶

a⁶ × a² - a²·B⁶ = a²·(a⁶ - B⁶)

(a⁶ - B⁶) = (a³ - B³) × (a³ + B³)

The polynomial is therefore; a⁸ - a²·B⁶ = a²·(a³ - B³)·(a³ + B³)

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our final estimate for the area under the curve f(x)=1/x on [1,2] using 4 approximating rectangles and taking the average of the left and right endpoints is 0.76.

To estimate the area under the curve f(x)=1/x on [1,2], we can use 4 approximating rectangles.

a) Using left endpoints, the width of each rectangle is (2-1)/4 = 0.25. The left endpoints of the rectangles are 1, 1.25, 1.5, and 1.75. The height of each rectangle is f(x) evaluated at the left endpoint.

So, the heights are f(1) = 1/1 = 1, f(1.25) = 1/1.25 = 0.8,

f(1.5)=1/1.5 = 0.67, and f(1.75) = 1/1.75 = 0.57.

The area of each rectangle is width times height, so the areas are

0.25*1 = 0.25, 0.25*0.8 = 0.2, 0.25*0.67 = 0.1675, and 0.25*0.57 = 0.1425.

Adding these areas together, we get an estimate of the total area under the curve as

0.25 + 0.2 + 0.1675 + 0.1425 = 0.76.

b) Using right endpoints, the width of each rectangle is the same as before. The right endpoints of the rectangles are

1.25, 1.5, 1.75, and 2.

The heights are f(x) evaluated at the right endpoint. So, the heights are

f(1.25) = 0.8, f(1.5) = 0.67, f(1.75) = 0.57, and f(2) = 0.5.

The areas of the rectangles are the same as before, so adding them up, we get an estimate of the total area as

0.2 + 0.1675 + 0.1425 + 0.25 = 0.76,  

which is the same as before.
c) To get the average of the two estimates, we add them together and divide by 2:

(0.76+0.76) / 2 = 0.76.

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Mark originally had 28 shoes.

Now we can use the equation for Grace's number of shoes to find her final count:

Grace = x + 16 = 28 + 16 = 44

So Grace ends up with 44 shoes.

Let's start by setting up an equation to represent the given information:

Let the number of shoes Mark has be represented by "x"

Grace has 16 more shoes than Mark:

Grace = x + 16

Mark gives Grace 12 shoes, so Grace now has:

Grace = x + 16 + 12 = x + 28

Mark realized he has half as many shoes as Grace:

x = 0.5(x + 28)

Simplifying this equation:

x = 0.5x + 14

0.5x = 14

x = 28

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The library is about 26.1 miles from the courthouse.

To solve this problem, we will apply the Pythagorean theorem, which tells us that during a right triangle, the square of the period of the hypotenuse( the longest side) is same to the total of the places of the lengths of the different two sides.

In this instance, the park is on the right angle, and the library and courthouse are the opposite two factors. we can consider the distance among the library and the park because the length of 1 leg of the triangle, and the space among the courthouse and the park as the length of the other leg.

So, using the Pythagorean theorem, we're suitable to calculate the period of the hypotenuse( the distance among the library and the courthouse)

library- to- park distance2 courthouse- to- park distance2 = library- to- courthouse distance2

[tex](14.5)^2 + (21.7)^2 = library-to-courthouse distance^2[/tex]

[tex]210.25 + 471.29 = library-to-courthouse distance^2[/tex]

[tex]681.54 = library-to-courthouse distance^2[/tex]

Taking the square root of both aspects, we get

library- to- courthouse distance[tex]= sqrt(681.54) \approx26.1[/tex]

Accordingly, the library is about 26.1 miles from the courthouse.

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o find the coordinates of the gates in the third quadrant, we need to find the points where the circle and the line intersect in the third quadrant.

Substituting y = 2x into x² + y² - 4x = 9, we get:

x² + (2x)² - 4x = 9

5x² - 4x - 9 = 0

Using the quadratic formula, we find:

x = (-(-4) ± √((-4)² - 4(5)(-9))) / (2(5))

x = (4 ± √136) / 10

We can discard the positive root since it is in the first quadrant. The negative root corresponds to the x-coordinate of the point of intersection in the third quadrant:

x = (4 - √136) / 10 ≈ -0.433

Substituting this value into y = 2x, we get:

y = 2(-0.433) ≈ -0.866

Therefore, the ordered pair that represents the location of the gates in the third quadrant is (-0.433, -0.866).

Answer: 72 m²

Step-by-step explanation:

The simplified form of the expression 3(w+11) / 6w  is w + 11 / 2w .

Simplifying expressions mean rewriting the same algebraic expression with no like terms and in a compact manner.

In other words, we have to expand any brackets, next multiply or divide any terms and use the laws of indices if necessary, then collect like terms by adding or subtracting and finally rewrite the expression.

Therefore, let's simplify the expression:

3(w+11) / 6w

Hence, let's divide both the numerator and denominator by 3

Therefore,

3(w+11) / 6w =  w + 11 / 2w

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Sydney can row a canoe at a speed of 5 mph in still water.

Let x represent Sydney's speed in still water. When rowing upriver, her effective speed will be (x - 2) mph because she's going against the current, which flows at 2 mph. When rowing downriver, her effective speed will be (x + 2) mph, since she's going with the current.

According to the problem, the time it takes her to row 6 miles upriver is the same as the time it takes her to row 14 miles downriver. We can set up the equation using the formula time = distance / speed:

6 / (x - 2) = 14 / (x + 2)

To solve for x, first cross-multiply:

6(x + 2) = 14(x - 2)

Expand:

6x + 12 = 14x - 28

Now, rearrange and solve for x:

12 + 28 = 14x - 6x

40 = 8x

x = 5

So, Sydney can row a canoe at a speed of 5 mph in still water.

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Answer:

Since it is a right triangle, we can apply pythagores theorem.

Answer: a = 8.7 miles

Step-by-step explanation:

a^2 = c^2 - b^2

a^2 = 10^2 - 5^2

a^2 = 100 - 25

a^2 = 75

a ≈ 8.7

Therefore, the length of the missing leg is approximately 8.7 miles.

The polynomial that models the area of Molly's cardboard is 6x^2 + 13x - 5.

How can the area of Molly's cardboard be modeled with a polynomial?

First, we were given that the length of the cardboard can be modeled by 3x - 1 and the width can be modeled by 2x + 5. To find the area, we use the formula:

Area = length x width

So, we substitute the expressions for the length and width:

Area = (3x - 1) x (2x + 5)

Next, we use the distributive property of multiplication to expand the expression:

Area =[tex]6x^2 + 15x - 2x - 5[/tex]

Simplifying, we get:

Area = [tex]6x^2 + 13x - 5[/tex]

Therefore, the polynomial that models the area of Molly's piece of cardboard is [tex]6x^2 + 13x - 5.[/tex]

This polynomial gives us a way to calculate the area of the cardboard for any value of x. For example, if we know that the length of the cardboard is 5 units, we can substitute x = 2 into the polynomial to find the area:

Area =[tex]6x^2 + 13x - 5[/tex]

Area =[tex]6(2)^2 + 13(2) - 5[/tex]

Area = 24 + 26 - 5

Area = 45

So, the area of the cardboard when x = 2 (and the length is 3x - 1 = 5) is 45 square units.

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Answer: The median of the players' heights is not affected.

Step-by-step explanation: B

The median of the players' heights is increased.

we need to consider the current arrangement of heights and the positions

of the new players in relation to the existing players' heights.

If we assume that the heights of the players are sorted in ascending order,

adding two additional players can affect the median in the following ways:

If both new players have heights lower than the current median:

In this case, adding the new players would not change the median.

The median would remain the same because the new players would be

added below the existing median, and the position of the median would

not shift.

If one new player has a height lower than the current median and the other

has a height higher than the current median:

In this case, the median would be increased.

Adding a taller player would shift the median towards the higher end of the data set.

If both new players have heights higher than the current median:

In this case, the  would be increased.

Both new players would be taller than the current median, causing the

median to shift towards the higher end of the data set.

Based on these possibilities, the answer is C.

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Step-by-step explanation:

(10 x 6)/2 = 30 + 47 = 77

The solutions of equation 2sin²θ + 1 = 0 in the interval [0, 21) in radians are [tex]\theta \approx \frac{5\pi}{4}, \frac{7\pi}{4}[/tex].

Let's solve the equation and find the solutions within the given interval [0, 21) in radians.

The equation is 2sin²θ + 1 = 0.

Subtracting 1 from both sides, we get:

2sin²θ = -1

Dividing both sides by 2, we have:

sin²θ = [tex]-\frac{1}{2}[/tex]

Taking the square root of both sides, considering both the positive and negative square roots, we get:

sinθ = [tex]\± -\sqrt\frac{1}{2}[/tex]

Since the sine function is negative in the third and fourth quadrants, we only need to consider the negative square root.

sinθ = [tex]-\sqrt(\frac{1}{2})[/tex]

To find the solutions within the interval [0, 21), we need to consider the values of θ between 0 and 21 in radians.

Using a calculator or trigonometric tables, we can find the solutions for sinθ = [tex]-\sqrt(\frac{1}{2})[/tex] within the interval [0, 21):

θ ≈ 5π/4, 7π/4

Therefore, the solutions of the equation 2sin²θ + 1 = 0 in the interval [0, 21) in radians are:

[tex]\theta \approx \frac{5\pi}{4}, \frac{7\pi}{4}[/tex]

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The weight of the statue to the nearest ounce is 576 ounces.

The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where r is the radius of the sphere. Since the diameter is given as 10 inches, the radius is half of that, or 5 inches.

Using the formula, we can find the volume of the sphere:

V = (4/3)πr^3

V = (4/3)π(5^3)

V = (4/3)π(125)

V = 523.6 cubic inches (rounded to one decimal place)

Since we know the weight of the clay per cubic inch, we can find the weight of the statue by multiplying the volume by the weight per cubic inch:

Weight = Volume × Weight per cubic inch

Weight = 523.6 in^3 × 1.1 oz/in^3

Weight = 575.96 oz (rounded to two decimal places)

Therefore, the weight of the statue is approximately 576 ounces or 36 pounds (rounded to the nearest pound).

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The amount of money earned over a period of time is called 3. Income

Income refers to the money you receive from various sources such as salary, wages, investments, or any other means during a specific period of time.

Income is a fundamental concept in finance and accounting. It represents the inflow of money or economic benefits received by an individual, household, or organization.

It is typically measured and reported on a periodic basis, such as monthly, quarterly, or annually. Income can be derived from various sources, including salaries and wages, interest and dividends, rental income, profits from business activities, and capital gains.

It is an essential component in assessing an individual's or organization's financial health and is often used to calculate taxes, budgeting, and determining net worth. Understanding and effectively managing income is crucial for individuals and businesses to meet their financial goals and sustain their economic well-being.

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The probability that on a particular Monday, they each suggest a different activity is 0.11.

The probability that in a period of four consecutive Mondays, they all suggest the same activity on exactly two of the four Mondays is 0.0504.

1. Probability that on a particular Monday, they each suggest a different activity:

The probability is calculated using the formula below:

Probability = P(SMC) + P(MCS) + P(CSM)

Probability = (0.4 x 0.55 x 0.5) + (0.3 x 0.2 x 0.3) + (0.3 x 0.25 x 0.2)

Probability = 0.11

2. The probability that in a period of four consecutive Mondays, they all suggest the same activity on exactly two of the four Mondays is determined using the binomial distribution.

Let success be suggesting the same activity on exactly two of the four Mondays.

The probability of success on any Monday is:

P(success) = P(SSNN) + P(NSSN) + P(NNSS)

P(success) = 3 x (0.4 x 0.4 x 0.6 x 0.6)

P(success) = 0.3456

The probability of failure is:

P(failure) = 1 - P(success)

P(failure)= 1 - 0.3456

P(failure) = 0.6544

Choose exactly two Mondays out of four is ⁴C₂

The probability of exactly two successes = ⁴C₂ * P(success)² * P(failure)²

P(exactly 2 successes) = 6 x (0.3456)² x (0.6544)²

P(exactly 2 successes) = 0.0504 or 5.04%

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Answer:

Step-by-step explanation:

I assume you mean to solve for M in terms of a, b, x, and y.

To solve for M, we can first simplify the given polynomial:

2x^2 - (1/3)ax + by - M = 0

Multiplying through by -1 to isolate M:

M = 2x^2 - (1/3)ax + by

Therefore, the polynomial M is:

M = 2x^2 - (1/3)ax + by

Answer: Box and whisker plots are plots on number lines with a box and two lines off the edges, called whiskers. The box has a line at the upper quartile(1), one at the lower quartile(2), and one in the center of the box at the median(3). The two lines go to the ends of the data, one at the minimum(4) and one at the maximum(5).

  4     2     3       1             5

   |-----[___|____]-----------|

₀__₁__₂__₃__₄__₅__₆__₇__₈

I hope this helps.

Answer:

48 calories per (each) minute.

Step-by- Step

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