rolling a die and tossing a coin simultaneously. how many possible out comes are there?

Therefore, the area between the large loop and the small loop of the curve r = 2 + 4cos(3θ) is 70π/3.

To find the area between the large loop and the small loop of the curve, we need to find the points of intersection of the curve with itself.

Setting the equation of the curve equal to itself, we have:

2 + 4cos(3θ) = 2 + 4cos(3(θ + π))

Simplifying and solving for θ, we get:

cos(3θ) = -cos(3θ + 3π)

cos(3θ) + cos(3θ + 3π) = 0

Using the sum to product formula, we get:

2cos(3θ + 3π/2)cos(3π/2) = 0

cos(3θ + 3π/2) = 0

3θ + 3π/2 = π/2, 3π/2, 5π/2, 7π/2, ...

Solving for θ, we get:

θ = -π/6, -π/18, π/6, π/2, 5π/6, 7π/6, 3π/2, 11π/6

We can see that there are two small loops between θ = -π/6 and π/6, and two large loops between θ = π/6 and π/2, and between θ = 5π/6 and 7π/6.

To find the area between the large loop and the small loop, we need to integrate the area between the curve and the x-axis from θ = -π/6 to π/6, and subtract the area between the curve and the x-axis from θ = π/6 to π/2, and from θ = 5π/6 to 7π/6.

Using the formula for the area enclosed by a polar curve, we have:

A = 1/2 ∫[a,b] (r(θ))^2 dθ

where a and b are the angles of intersection.

For the small loops, we have:

A1 = 1/2 ∫[-π/6,π/6] (2 + 4cos(3θ))^2 dθ

Using trigonometric identities, we can simplify this to:

A1 = 1/2 ∫[-π/6,π/6] 20 + 16cos(6θ) + 8cos(3θ) dθ

Evaluating the integral, we get:

A1 = 10π/3

For the large loops, we have:

A2 = 1/2 (∫[π/6,π/2] (2 + 4cos(3θ))^2 dθ + ∫[5π/6,7π/6] (2 + 4cos(3θ))^2 dθ)

Using the same trigonometric identities, we can simplify this to:

A2 = 1/2 (∫[π/6,π/2] 20 + 16cos(6θ) + 8cos(3θ) dθ + ∫[5π/6,7π/6] 20 + 16cos(6θ) + 8cos(3θ) dθ)

Evaluating the integrals, we get:

A2 = 80π/3

Therefore, the area between the large loop and the small loop of the curve r = 2 + 4cos(3θ) is:

A = A2 - A1 = (80π/3) - (10π/3) = 70π/3

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Answer:

The sum of three and xy is equal to 5

or

The difference of xy and 3 is 5

Step-by-step explanation:

\({ \tt{xy = 5 + 3}} \\ \)

Answer:

b. two solutions

Step-by-step explanation:

Answer:

Two Solutions.

Step-by-step explanation:

Answer:

H)

Step-by-step explanation:

1/7 in decimal form has no three's.

Answer:

\(0.6x - 0.1x = 7 + 5 \\ 0.5x = 12 \\ x = \frac{12}{0.5} = 24\)

x = 20.4 it take to fill the entire tank.

Calculate a tank's total capacity and the volume that is filled in gallons and litres for tanks like water and oil storage. assumes the tank's interior measurements.

You can enter metric dimensions in metres (m) or centimetres, or U.S. dimensions in feet (ft) or inches (in) (cm). Results are given in fluid US gallons, fluid Imperial (UK) gallons, fluid cubic feet (ft3), fluid metric litres (fl), and fluid cubic metres (m3).

x = total volume of the entire tank.

We can suggest the following equation:

6 minutes

6 minutes to fill 3.4

x=6*3.4=20.4

x=20.4

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Answer:

-3/10

Step-by-step explanation:

The fraction would be -30/100 because the man is descending.

-30/100 = -3/10 simplified.

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The number line represents all possible numbers of signatures Ali could collect is Number line A.

We have,

Ali currently has 520 signatures.

Now, number of signatures Ali need

= 1,000 - 520

= 480

So, the possible number depending on how many weeks he wants to spend getting signatures.

480/6 = 80

480/5 = 96

480/4 = 120

480/3 = 160

480/2 = 240

480/1 = 480

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Answer:

A.) exponential decay

B.) 5575

C.) 65%

Step-by-step explanation:

a. The percent of Benny's mom's age to his age is 400%.

b.  Benny's mom's age is 400% of his age.

What is a percent of numbers?

The percent of a number is its fraction that is expressed as percentage. Thus the percentage of a number or part of a number can be determined.

Considering the given question, the years of Benny and that of his mom were given as;

Benny's age = 12 years

His mom's age = 48 years

Thus we can deduce that;

a. The percent of his mom's age to that of Benny's = 48/ 12 * 100%

                                            = 400%

b. Benny's mom's age is 400% of his age.

Therefore, the correct option to be selected is D.

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Answer:

x = -12

Step-by-step explanation:

Let x be the number.

The product of 5 and x increased by 9 is equal to 15 less than 3 times x.

5x + 9 = 3x - 15.

We can isolate x, getting (5-3)x = -15-9

Then we can solve:

2x = -24

x = -12.

The required number is -12.

An equation is a mathematical statement that shows that two mathematical expressions are equal.

Given that, The product of five and a number, increased by 9 is equal to fifteen less than three times then number.

Let x be the number.

Establishing the equation,

5x + 9 = 3x - 15.

(5-3)x = -15-9

Then we can solve:

2x = -24

x = -12.

Hence, the number is -12

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Answer:

In 8 years, you will have $11,094.81

Step-by-step explanation:

Answer:

V=11095.3771≈11095.38

Step-by-step explanation:

you do 479 x 2 = 958 there is your answer your welcome

According to the solving  function f could describe the bacteria population d days after the scientists first measured it, assuming it grows exponentially F(d) = 10000 - a\(^\frac{d}{5}\).

A mathematical expression, rule, or law known as a function defines the relationship between two independent variables and a dependent variable (the dependent variable). In mathematics, functions are often utilised, and they are essential for creating physical connections in the sciences.

Substituting d = 5 and f(d)

Solution: fid) = 10000⋅ ad

{substitute d=5 and fid)=20000 into fid=10000. ady

10000- a⁵ =20000

a⁵ =2\(^\frac{1}{5}\)

a = 2

F(d) = 10000 - a\(^\frac{d}{5}\)

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There are 2300 ways by which we can select 3 students out of 25 students.                        

Combinations are selections made by taking some or all of a number of objects, irrespective of their arrangements.

Here total number of Students = 25

         Number of selection = 3

Now,

Total number of ways by which we can select 3 out of 25  = ²⁵C₃

                                                                                               = 25!/3!.22!

                                                         = 25 X 24 X 23 X 22! / 3X2X1 X 22!

                                                         = 13800 / 6

                                                         = 2300

Thus, there are 2300 ways by which we can select 3 students out of 25 students.                

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Answer:

W PM^2 − WP(M−S)^2=T

Step-by-step explanation:

this is what mathaway said

The required z-score is -0.5 and the probability that the firm WILL have a negative EPS is 30.85%.

Firm has a loss level EBIT of $175m and an expected EBIT of $200m with a standard deviation of $50m.

Z-score = (EBIT - Expected EBIT) / Standard deviation= (175 - 200) / 50= -0.5

Probability of having a negative EPS:

To find the probability that the firm will have a negative EPS, we need to find the area to the left of Z-score. As the Z-score is negative, we will be finding the left-tail probability using the standard normal table from the link below:

The area to the left of Z-score (0.5) is 0.3085 or 30.85%.

Therefore, the probability that the firm WILL have a negative EPS is 30.85%.

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The balloon will be collapsed due to the excess pressure.

if we press on a balloon that is already filled by air stretched over the jar, the air of the jar will enter the balloon and create excess pressure beyond its tolerance and finally the balloon will be collapsed.

as the air-filled balloon is pressed directly over a jar, there is no way to leakage of air, besides the air of jar enter the balloon causes high pressure on the wall of balloon and finally it is destroyed. The result represents the presence of gases inside any empty bottle as well as any jar.

hence, we understand that every empty jar or bottle has gaseous substances with them.

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Answer:

Question 2: 6x-21

Question 1: -9x+3

Step-by-step explanation:

1) The value of P(A∩B) = 0.

2) The value of P(A|B) = 0.

3) P(A|B) is not equal to P(A). A and B are independent events.

4) The statement "Mutually exclusive events and independent events are really the same, and that if events are mutually exclusive they must be independent." is not accurate.

5) We can conclude that mutually exclusive events have a probability of intersection of zero and are not independent.

1) Since A and B are mutually exclusive, they cannot occur at the same time, so P(A∩B) = 0.

2) The conditional probability of A given B is defined as the probability of A occurring given that B has occurred. Since A and B are mutually exclusive, if B has occurred, then A cannot occur. Therefore, P(A|B) = 0.

3) No, P(A|B) is not equal to P(A). Since B has occurred, the sample space has been reduced to only those outcomes in which B has occurred. Therefore, the probability of A given that B has occurred can only be determined from the portion of the sample space in which both A and B occur.

However, since A and B are mutually exclusive, this probability is zero. Therefore, P(A|B) ≠ P(A).

4) The statement is not accurate. Mutually exclusive events and independent events are different concepts. Two events are mutually exclusive if they cannot occur at the same time, whereas two events are independent if the occurrence of one event does not affect the probability of the other event.

If two events are mutually exclusive, they are not independent because the occurrence of one event precludes the occurrence of the other event.

5) Mutually exclusive events have a relationship where if one event occurs, the other cannot. In contrast, independent events are events where the occurrence of one event does not affect the probability of the other event occurring.

Therefore, we can conclude that mutually exclusive events and independent events are not the same concept, and we must be careful not to confuse them.

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Answer:

46°

Step-by-step explanation:

from large triangle:

let the third unknown angle be 'a'

then,

a+x+7+85=180

a=88-x

now,from small triangle,

let the third unknown angle be 'b'

then,

b+x+2x=180

b=180-3x

b=a (vertically opposite angles)

then,

180-3x=88-x

2x=92

x=46

Step-by-step explanation:

Root of the function, where y = 0

f(x) = x² + 2x - 1

0 = x² + 2x - 1

x² + 2x - 1 = 0

x² + 2x = 1

x² + 2x + 1 = 1 + 1

(x + 1)² = 2

x + 1 = ±√2

x = -1 ± √2

Roots → x = -1 ± √2

answers are (a) If F(x, y) = ⟨cos(y), x²sin(y)⟩ and C be the rectangle with vertices (0, 0), (5, 0), (5, 2), and (0, 2) ∫ₓ F ⋅ dr = 60. (b) the curl function is

curl(F) = 2 - 2eˣy + eˣsin(z)

To compute this integral, we can use the Green's Theorem, which states that for a vector field F = ⟨P, Q⟩ and a simple closed curve C with counterclockwise orientation, the line integral ∫ₓ F ⋅ dr is equal to the double integral ∬ᵦ curl(F) ⋅ dA, where dA is the differential area element and ᵦ represents the region bounded by C.

First, let's calculate the curl of F:

curl(F) = (∂Q/∂x - ∂P/∂y) = (∂/∂x(x²sin(y)) - ∂/∂y(cos(y)))

        = (2xsin(y) - (-sin(y)))

        = (2xsin(y) + sin(y))

        = sin(y)(2x + 1).

Now, let's calculate the double integral over the region ᵦ bounded by the rectangle C:

∬ᵦ curl(F) ⋅ dA = ∬ᵦ sin(y)(2x + 1) dA.

Since C is a rectangle, we can parameterize it as follows: x = u, y = v, where 0 ≤ u ≤ 5 and 0 ≤ v ≤ 2.

Then, the differential area element dA can be expressed as dA = du dv.

Substituting the parameterization and the differential area element into the double integral:

∬ᵦ curl(F) ⋅ dA = ∫₀² ∫₀⁵ sin(v)(2u + 1) du dv.

Now, we can evaluate the integral:

∫₀² ∫₀⁵ sin(v)(2u + 1) du dv = ∫₀² [(u² + u)sin(v)]₀⁵ dv

                            = ∫₀² (5² + 5)sin(v) dv

                            = 30 ∫₀² sin(v) dv

                            = 30 [-cos(v)]₀²

                            = 30 (1 - (-1))

                            = 60.

Therefore, ∫ₓ F ⋅ dr = 60.

To compute this integral, we can use the Stokes' Theorem, which relates the surface integral of the curl of a vector field over a surface S to the line integral of the vector field around the boundary curve ∂S of the surface.

First, let's calculate the curl of F:

curl(F) = ∇ × F

       = ∂Fₓ/∂y - ∂Fᵧ/∂x + ∂/∂x(Fᵧ - F_z)

       = (2 - eˣy) - (0 - 0) + (∂/∂x(eˣsin(z) - eˣy))

       = 2 - eˣy + eˣsin(z) - eˣy

       = 2 - 2eˣy + eˣsin(z)

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The complete question is:

(a) Let F(x, y) = ⟨cos(y), x²sin(y)⟩ and C be the rectangle with vertices (0, 0), (5, 0), (5, 2), and (0, 2). Compute ∫ₓ F ⋅ dr.

(b) Let F(x, y, z) = ⟨2ycos(z), eˣsin(z), eˣy⟩ and S be the hemisphere x² + y² + z² = 9, z ≥ 0, oriented upwards. Compute ∬ₛ curl(F) ⋅ dS.

The distance between points M(6, 16) and Z(-1, 14) is approximately 7.1 units.

We can use the distance formula to find the distance between two points in a coordinate plane. The distance formula is:

d = √((x2 - x1)² + (y2 - y1)²)

where (x1, y1) and (x2, y2) are the coordinates of the two points. Substituting the coordinates of M(6, 16) and Z(-1, 14), we get:

d = √((-1 - 6)² + (14 - 16)²) = √(49 + 4) = √53 ≈ 7.1

Therefore, the distance between points M(6, 16) and Z(-1, 14) is approximately 7.1 units.

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Answer:

y+2=-1/3(x-4)

Step-by-step explanation:

Plug in the numbers:

y-___=___(x-___)

Answer:

6

Step-by-step explanation:

4 1/2 divided by 3/4 is 6. so 6 pieces

The greatest number of 3/4 foot pieces that Annie can cut from the rope

is 6.

A fraction is written in the form of p/q, where q ≠ 0.

Fractions are of two types they are proper fractions in which the numerator is smaller than the denominator and improper fractions where the numerator is greater than the denominator.

Given, Annie has 4 1/2 feet of rope.

Therefore, The greatest number of 3/4 foot pieces Annie can cut can be obtained by dividing the length of the total rope by the length of each piece which is,

= (4 1/2)/(3/4).

= (9/2)×(4/3).

= 36/6.

= 6.

So, 6 pieces are possible.

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Answer:

30 units²

Step-by-step explanation:

The area (A) of a parallelogram is calculated as

A = bh ( b is the base and h the perpendicular height )

Here

b is the horizontal measure between 2 red points, that is 6

h is the vertical measure from the lower left point to the other horizontal line, that is 5, then

A = 6 × 5 = 30 units²

An approximate solution to the equation f(x) = g(x) to the nearest quarter of a unit is; A: 2.5

We are given;

f(x) = ¹/₄x³ + 2x - 1

g(x) = [5^(x - 1)] - 3

Thus;

f(x) = g(x) gives;

¹/₄x³ + 2x - 1 = [5^(x - 1)] - 3

At x = 2, we have;

LHS = 5

RHS = 2

At x = 2.5, we have;

LHS = 7.90625

RHS = 8.180

The answer to the two sides are close and so we can adopt that.

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Answer:

-5 and 9

Step-by-step explanation:

Graph the equation on a graph.

You can also simplify the equation into y=mx+b form and then graph:

y=-3(x+5)(x-9)

y=(-3x-15)(x-9)

y=-3x^2+27x-15x+135

y=-3x^2+12x+135