The radius of a circle is increasing uniformly at the rate of 5cm/sec. Find the rate at which the area of the circle is increasing when the radius is 6 cm.

Acute angle ⇒ 0⁰ < m < 90⁰

Straight angle ⇒  m = 180⁰

Obtuse angle ⇒ 90⁰ < m < 180⁰

Right angle ⇒ m = 90⁰

An obtuse angle is an angle that measures greater than 90 degrees but less than 180 degrees. In other words, an obtuse angle is an angle that is wider than a right angle (90 degrees), but not as wide as a straight angle (180 degrees).

When two rays or line segments intersect at a point, they form an angle. If the angle formed is less than 90 degrees, it is called an acute angle. If the angle is exactly 90 degrees, it is called a right angle.

If the angle is greater than 90 degrees but less than 180 degrees, it is called an obtuse angle. Finally, if the angle measures exactly 180 degrees, it is called a straight angle.

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The relation a into b represent  all possible combinations of elements in a and b as ordered pairs.

The relation "a into b" refers to the cartesian product of the sets a and b, which is denoted by a × b. The cartesian product of two sets is a set of all possible ordered pairs, where the first element of each ordered pair comes from the first set, and the second element comes from the second set.

In this case, a = {0, 1, 2} and b = {-1, 1, 2}. So, their cartesian product a × b is the set of all possible ordered pairs (a, b), where a is an element of a and b is an element of b. Therefore, we have:

a × b = {(0, -1), (0, 1), (0, 2), (1, -1), (1, 1), (1, 2), (2, -1), (2, 1), (2, 2)}

This means that the relation a into b consists of all possible combinations of elements in a and b as ordered pairs.

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Answer: 1. Multiplying Powers with same Base

For example: x² × x³, 2³ × 2⁵, (-3)² × (-3)⁴

In multiplication of exponents if the bases are same then we need to add the exponents.

Consider the following:

1. 2³ × 2² = (2 × 2 × 2) × (2 × 2) = 23+2

= 2⁵

2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 34+2

= 3⁶

3. (-3)³ × (-3)⁴ = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)]

                       = (-3)3+4

                       = (-3)⁷

4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m)

                 = m5+3

                 = m⁸

From the above examples, we can generalize that during multiplication when the bases are same then the exponents are added.

aᵐ × aⁿ = am+n

In other words, if ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, then

aᵐ × aⁿ = am+n

Similarly, (ab

)ᵐ × (ab

)ⁿ = (ab

)m+n

(ab)m×(ab)n=(ab)m+n

Note:

(i) Exponents can be added only when the bases are same.

(ii) Exponents cannot be added if the bases are not same like

m⁵ × n⁷, 2³ × 3⁴

Step-by-step explanation:

Answer:

2^4

Step-by-step explanation:

doing this type of division is like subtracting the powers, 9-5=4, 2^4.

the opposite applies for multiplaction, it's like addition for the powers,

2^9*2^5=2^14.

The sample space of the number of outcomes is A = 12

Given data ,

To create a tree diagram and list the sample space, we need to consider the possible outcomes at each stage of the event.

First, we have three options for the spinner: Red, Blue, and Yellow.

Now, let's consider the possible outcomes when a letter is chosen from the word MATH

The total number of outcomes A = 12 outcomes

where A = { RM , BM , YM , RA , BA , YA , RT , BT , YT , RH , BH , YH }

       Red             Blue            Yellow

       / \             / \             / \

      M   A           M   A           M   A

     /     \         /     \         /     \

    T       H       T       H       T       H

Hence , the number of outcomes is A = 12 and the tree diagram is solved

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Answer:

4.

Step-by-step explanation:

What is elimination method?

In the elimination method you either add or subtract the equations to get an equation in one variable

Lets solve for X first.

We have the equations:

5x+3y=8

4x+y=8

We take the GCM (3) and multiply everything on the bottom by -3 and multiply everything on the top by 1 (which in this case, dont touch the top) which we should now have:

5x+3y=8

-12x-3y=-36

We can now eliminate the Y and add like terms from the top to bottom, from which we should now have remaining:

-7x=-28

Solve for X

x=4

Now that we have 4, plug it in to one of the equations, which we should plug it in to 4x+y=12

4(4)+y=12

Simplify:

16+y=12

Subtract 16 to the other side:

y=-4

So now we got our x and y vaule, from which is how we get the answer (4,-4)

The measure of the angle of the near arc of the monument that her lines of sight intersect with is 124°

The angle of an arc of a circle is the angle formed by the two radii of the circle that intersects with the boundaries of the arc

The distance the park maintenance person stands from the monument = 16 m

The angle the lines of sight from the maintenance person that are tangent with the monument make where they intersect = 56°

Whereby the tangent lines from the monument to the maintenance person intersect and form an angle of 56°, we get that the tangent lines form two right triangles, please see the attached figure which is created with MS Excel;

The right triangles ΔABO and ΔACO are congruent by Leg Hypotenuse, LH, congruence rule

Therefore; ∠OAC ≅ ∠OBC

m∠OAC = m∠OBC (Definition of congruent angles)

Similarly, m∠BOA = m∠COA

However, m∠BAC = m∠OAC  + m∠OBC (Angle addition postulate)

m∠BAC = 2 × m∠OAC = 56°

m∠OAC = 56° ÷ 2 = 28°

m∠BOA = 90° - m∠OBC (Acute angles of a right triangle)

m∠BOA = 90° - 28° = 62°

Therefore, m∠BOA = m∠COA = 62°

The angle at the center = m∠BOC = m∠BOA + m∠COA

m∠BOC = 62° + 62° = 124°

Angle formed at the center of the monument, m∠BOC = 124°

The arc angle of a circle = The angle the radius of the arc forms at the center of the circle.

The measure of the arc close to the park maintenance person is 124°

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Applying the Equidistant Chords Theorem, the value of x in the circle given above is calculated as: b. 50.

The Equidistant Chords Theorem states that If two chords in a circle, or in congruent circles, are equally distant from the center, then the chords are congruent.

The image given reveals that the two chords are equidistant from the center of the circle, therefore:

x = 2(25)

x = 50.

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The type of galaxy matched to its description.

Elliptical galaxy: B) Forms a perfect sphere or an ellipse and is flattened to some degree

Galaxy: D) Is a collection of several billion stars and interstellar matter isolated in space

Lens galaxy: A) Has a large flattened core

Spiral galaxy: C) Has a central core from which curved arms spiral outward

Elliptical galaxy: An elliptical galaxy is a type of galaxy that typically forms a perfect sphere or an ellipse shape. It is characterized by its smooth and featureless appearance, lacking the distinct spiral arms seen in spiral galaxies. Elliptical galaxies often have a flattened shape due to their rotation and gravitational interactions with other galaxies.

Galaxy: A galaxy refers to a vast collection of stars, interstellar gas, dust, and dark matter, all held together by gravity. Galaxies come in various shapes and sizes, and they can contain billions or even trillions of stars. They are the building blocks of the universe and are distributed throughout the cosmos.

Lens galaxy: A lens galaxy is a type of galaxy that has a large flattened core. It gets its name from the gravitational lensing effect it produces. Gravitational lensing occurs when the gravitational field of the lens galaxy bends and distorts the light from objects behind it, creating a lens-like effect.

Spiral galaxy: A spiral galaxy is a type of galaxy that has a central core or bulge from which curved arms spiral outward. These arms are made up of stars, gas, and dust, and they give spiral galaxies their distinct appearance. Spiral galaxies often have a flattened disk shape with a central bulge and extended arms that can stretch out across the disk.

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The value of the surface area of Cylinder B is, 357 cm²

We have to given that;

Cylinder A is similar to Cylinder B with a scale factor of 3:7.

And, the surface area of Cylinder A.

Let us assume that,

The value of the surface area of Cylinder B is, y.

Hence, We can formulate;

3x : 7x = 153 : y

By comparing,

3x = 153

x = 51

Thus, The value of the surface area of Cylinder B is,

y = 7x

y = 7 x 51

y = 357 cm²

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The total distance travelled by dust in the given time of 4.0 minutes at the spin rate of 500rpm is equal to 26,400cm.

Spin rate of the cd is equal to 500rpm

distance of piece of dust from the center = 2.1 cm

Distance traveled by a point on the CD that is 2.1 cm from the center in one revolution =  circumference of the circle with a radius of 2.1 cm,

C = 2πr

   = 2π(2.1 cm)

   ≈ 13.2 cm

Distance traveled by the dust in one revolution is 13.2 cm.

Calculate the number of revolutions that the CD makes in 4.0 minutes.

  1 minute = 60 seconds

⇒ 4.0 minutes = 4.0 x 60

⇒4.0 minutes = 240 seconds

The CD rotates at a rate of 500 revolutions per minute,

In 4.0 minutes it will rotate,

500 x 4.0

= 2000 times.

The total distance traveled by the dust in 4.0 minutes is,

distance traveled per revolution x number of revolutions

= 13.2 cm/rev x 2000 rev

= 26,400 cm

Therefore, the total distance traveled by the dust in 4.0 minutes is 26,400 cm.

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The inequality that can be used to determine the number of outfits Jason can purchase while staying within his budget is 68.54 o ≤ 274.16.

Solving the inequality gives:

o ≤ 4

The total cost of items that Jason spends on :

= 217. 34 + 36. 32 + 12. 18

= $ 265. 85

The amount that Jason has left from the $ 540 is:

= 540 - 265. 85

= $ 274. 16

If each outfit costs $ 68. 54 then the inequality that would help stay in budget is:

68.54 o ≤ 274.16

Solving this, gives:

o ≤ 274. 16 / 68.54

o ≤ 4

In conclusion, Jason can purchase up to 4 biking outfits while staying within his budget.

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Each side of the square has a length of approximately 12.02.

Without additional information or a diagram, it is difficult to determine what values are missing. However, we can use the Pythagorean theorem to solve for the length of the sides of the square.

Let's assume that TR is a diagonal of the square, and let x be the length of each side. Then, by the Pythagorean theorem,

TR^2 = x^2 + x^2

17^2 = 2x^2

289 = 2x^2

x^2 = 144.5

x ≈ 12.02

We can also solve this as:

Assuming that PQRS is square and TR is a diagonal of the square, we can use the Pythagorean theorem to find the length of each side of the square. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

In this case, if we draw a diagonal TR in the square, it divides the square into two right triangles, each with sides of length x (the length of one side of the square) and TR/√2 (half the diagonal). Applying the Pythagorean theorem to one of these right triangles, we get:

(x)^2 + (TR/√2)^2 = TR^2

Simplifying and solving for x, we get:

x = √(TR^2 - (TR/√2)^2) = TR/√2 = TR * √2 / 2

Plugging in TR = 17, we get:

x = 17 * √2 / 2 ≈ 12.02

Therefore, each side of the square has a length of approximately 12.02. Without additional information or a diagram, we cannot determine any other missing values.

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The actual length of the truck is 15.6 meters and the actual width is 3.24 meters.

The actual length of the truck can be calculated by multiplying the length of the model by the scale factor:

Similarly, the actual width of the truck can be calculated as:

Therefore, the actual length of the truck is 15.6 meters and the actual width is 3.24 meters.

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As k goes to infinity, the expression (k / (k+1)) approaches 1. Therefore, the limit becomes: lim (k -> infinity) 10 * (1^k) = 10
Since the limit is greater than 1, the Ratio Test indicates that the series is divergent.

To determine if the given series is convergent or divergent, we can use the Ratio Test. The series is given by:

Σ(5^(k-1) * 2^(k+1) / k^k) from k=1 to infinity

First, let's find the ratio of consecutive terms, a_(k+1)/a_k:

a_(k+1)/a_k = [(5^k * 2^(k+2)) / (k+1)^(k+1)] * [k^k / (5^(k-1) * 2^(k+1))]

Now, let's simplify the expression:

a_(k+1)/a_k = (5 * 2) * (k^k / (k+1)^(k+1))

Now, let's take the limit as k goes to infinity:

lim (k -> infinity) a_(k+1)/a_k = lim (k -> infinity) 10 * (k^k / (k+1)^(k+1))

We can rewrite the expression as:

lim (k -> infinity) 10 * ((k / (k+1))^k)

As k goes to infinity, the expression (k / (k+1)) approaches 1. Therefore, the limit becomes:

lim (k -> infinity) 10 * (1^k) = 10

Since the limit is greater than 1, the Ratio Test indicates that the series is divergent.

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The probability of selecting an Ace, not replacing it, and then selecting a King is 4/663 or approximately 0.006 or 0.6%.

The probability of selecting an Ace, not replacing it, and then selecting a King can be calculated using the rules of conditional probability.

First, we need to determine the probability of selecting an Ace from a standard deck of 52 cards. There are four Aces in the deck, so the probability of selecting an Ace is 4/52, which can be simplified to 1/13.

Next, we need to consider the fact that the Ace is not replaced before selecting the King. This means that the deck now contains 51 cards, with only three remaining Aces. Therefore, the probability of selecting a King after selecting an Ace without replacement is 4/51.

To determine the overall probability of selecting an Ace and then a King, we multiply the probability of selecting an Ace (1/13) by the probability of selecting a King after selecting an Ace without replacement (4/51).

The calculation is as follows:

(1/13) x (4/51) = 4/663

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If the first, last, middle car in the parking-row are blue, then the total number of different arrangements are 144.

There are 7 different models of cars, out of which 3 are blue and 4 are red.

If the first, middle and last car in the row is blue, which means that the the position of blue cars is fixed and these 3 "blue-cars" can be arranged in 3! ways.

Now, only the 4 spots are left, so, the remaining 4 red-cars can be arranged in the remaining 4 spots in = 4! ways.

Therefore, the total number of ways is = 3! × 4! = 6 × 24 = 144.

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The angular velocity of the point on the rim of the wheel is 25 radians per second.

The linear velocity of a point on a wheel's rim is determined by:

v = rω

v = linear velocity

r = radius of the wheel

ω = angular velocity.

In this case, the diameter of the wheel is 8 feet, so the radius is 4 feet. The linear velocity is given as 100 feet per second, so we have:

100 = 4ω

Solving for ω, we get:

ω = 25 radians per second

Therefore, the angular velocity of the point on the rim of the wheel is 25 radians per second.

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Quadrilaterals have four sides, two diagonals, and the sum of their interior angles is 360 degrees.

Quadrilaterals are four-sided polygons that have two diagonals connecting opposite vertices. One of the most important properties of quadrilaterals is that the sum of their interior angles is always equal to 360 degrees. This means that a quadrilateral can be divided into two triangles, each of which has a total of 180 degrees. This relationship between triangles and quadrilaterals is useful when calculating the area of a quadrilateral.

The formula for calculating the area of a quadrilateral is A = B x H, where A is the area, B is the base, and H is the height. This formula is applicable to all types of quadrilaterals, regardless of their shape or size. However, different types of quadrilaterals have unique properties and formulas for calculating their area.

For example, a square is a type of quadrilateral that has four sides of equal length and four right angles. The formula for finding the area of a square is A = s², where s is the length of the side. A rectangle is a type of quadrilateral with two pairs of parallel sides and four right angles. The formula for calculating the area of a rectangle is A = L x W, where L is the length and W is the width.

A rhombus is another type of quadrilateral that has four sides of equal length, but its angles are not necessarily right angles. The formula for finding the area of a rhombus is A = (D₁ x D₂) / 2, where D₁ and D₂ are the lengths of the diagonals.

A trapezoid is a quadrilateral with one pair of parallel sides. The formula for finding the area of a trapezoid is A = ((B₁ + B₂) / 2) x H, where B₁ and B₂ are the lengths of the parallel sides, and H is the height between them.

Kites are quadrilaterals with two pairs of adjacent equal-length sides. The formula for finding the area of a kite is A = (D₁ x D₂) / 2, where D₁ and D₂ are the lengths of the diagonals.

In summary, all quadrilaterals share some common characteristics, such as having four sides, two diagonals, and the sum of their interior angles being equal to 360 degrees. However, different types of quadrilaterals have distinct properties and formulas for finding their area. By understanding these properties and formulas, one can solve problems involving different types of quadrilaterals.

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The total number of yellow cups and blue cups paint that Jordy need is 10 t and (20/3) t, under the condition that he created the shade of green paint by mixing 1/2 t of yellow paint with 1/3 t of blue paint.

Here we have to apply the principles regarding basic multiplication of fraction.

Now to get 20 cups of green paint, Jordy requires 10 cups of yellow paint and 20/3 cups of blue paint.
Then the process of evaluating the number of blue paint cups needed is
Jordy created the shade of green paint by mixing 1/2 t of yellow paint with 1/3 t of blue paint.
To get 1 unit of green paint, Jordy needs 1/2 t of yellow paint and 1/3 t of blue paint.
To get 20 units of green paint,
Jordy needs
20 ×(1/2) t = 10 t of yellow paint
20 ×(1/3) t = (20/3) t of blue paint.
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Using the fundamental counting principle, there are 20 possible outcomes for the fitness tracker, considering its battery life and color options.

To find the total number of possible outcomes for the given criteria, we can use the fundamental counting principle. This principle states that if there are m ways to perform one task and n ways to perform another task, then there are m x n ways to perform both tasks together.

In this case, we have 4 options for battery life (1 day, 3 days, 5 days, and 7 days) and 4 options for color (silver, green, blue, pink, and black). Using the fundamental counting principle, we can multiply the number of options for battery life by the number of options for color to find the total number of possible outcomes:

4 options for battery life x 5 options for color = 20 possible outcomes.

Therefore, there are 20 possible combinations of battery life and color for a fitness tracker.

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you will have to walk approximately 17.2 miles back home.

A right triangle is a triangle in which one of the angles measures 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.  the length of the hypotenuse. Right triangles have many important applications in mathematics, science, and engineering, particularly in trigonometry, which is the study of the relationships between the sides and angles of triangles.

You have formed a right triangle with legs of length 10 and 14. To find the length of the hypotenuse, which is the distance back to your starting point, we can use the Pythagorean theorem:

[tex]c^2 = a^2 + b^2[/tex]

[tex]c^2 = 10^2 + 14^2[/tex]

[tex]c^2 = 100 + 196[/tex]

[tex]c^2 = 296[/tex]

c = sqrt(296)

c ≈ 17.2

Therefore, you will have to walk approximately 17.2 miles back home.

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This value will represent the range within which the middle 50% of Emilia's daily miles are distributed

Hi! The box plot represents the miles Emilia ran after school for 21 days. To find the IQR (Interquartile Range), you can use the box plot by identifying the values of the first quartile (Q1) and the third quartile (Q3).

The IQR is calculated by subtracting Q1 from Q3 (IQR = Q3 - Q1). By examining the box plot, locate Q1 and Q3 on the plot, and perform the subtraction to find the IQR.

This value will represent the range within which the middle 50% of Emilia's daily miles are distributed.

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The solution to the expression is x² + x - 20

(x-4)(x+5)

open the bracket

x² + 5x - 4x - 20

x² + x - 20

Hence the solution to the expression leads to quadratic equation which is written is x² + x -20

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The experimental probability of the computer generating a 1 is D. 40 %.

First, add up the outcomes to see the total number of times the integers were generated ;

= 36 + 11 + 13 + 12 + 18

= 90

The number of times 1 was generated was 36.

The experimental probability is therefore;

= number of times 1 was generated / total number of outcomes

= 36 / 90

= 0. 4

= 40 %

Therefore, the experimental probability of the computer generating a 1 is 40 %.

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Answer:

m<V=100.5°

u=84.86u

v=154.86u

Step-by-step explanation:

m<V= 180-46.9-32.6=100.5°

sin46.9°/115=sin32.6°/u

sin46.9°u=115sin32.6°

u=115sin32.6°/sin46.9°

u=84.86u

sin46.9°/115=sin100.5°/v

sin46.9°v=115sin100.5°

v=115sin100/5°/sin46.9°

v=154.86

i hope this helps you.

The probability that we want to get is 0.167

To get this probability, we need to take the quotient between the number of students that preffers athletics and literature and the total number of students in the table.

We have:

Total number = 5 + 3 + 2 +8 = 18

number that preffers athletics and literature = 3

Then the probability is:

P = 3/18 = 0.167

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