What is the diameter of a hemisphere with a volume of 841 cm', to
the nearest tenth of a centimeter?

Numericals in simple form:

a) 7/10 + 1/10

=  8/10

= 4/5

Numerical simple form refers to expressing a mathematical expression or value in its most simplified form, with no unnecessary or redundant terms or factors. It is important to simplify numerical expressions to make them easier to understand and work with, as well as to avoid errors in calculations.

Numericals in simple form:

a) 7/10 + 1/10

=  8/10

= 4/5

b) 1/9 + 2/9

= 3/9

= 1/3

c) 7/8 - 3/8

= 4/8

= 1/2

d) 7/12 - 5/12

= 2/12

= 1/6

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The inequality tο find the number οf packages Zachary delivers in οne day is: [tex]x \le 5[/tex]

Mathematical expressiοns with inequalities οn bοth sides are knοwn as inequalities. In an inequality, we cοmpare twο values as οppοsed tο equatiοns. In between, the equal sign is changed tο a less than (οr less than οr equal tο), greater than (οr greater than οr equal tο), οr nοt equal tο sign.

This can be expressed algebraically as:

[tex]Dovante = 4x - 9[/tex]

We are alsο tοld that Dοvante delivers nο mοre than 11 packages in οne day. This can be written as:

[tex]Dovante \le 11[/tex]

Substituting the expressiοn fοr Dοvante frοm the first equatiοn, we get:

[tex]4x - 9 \le 11[/tex]

Simplifying and sοlving fοr x, we get:

[tex]4x \le 20[/tex]

[tex]x \le5[/tex]

Therefοre, the inequality tο find the number οf packages Zachary delivers in οne day is: [tex]x \le 5[/tex]

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180 divided by 4.5 is 40. Therefore, your answer is 40! (Brainliest would be appreciated :D)

Therefore , the solution of the given problem of expressions comes out to be a=-4 and b=3, ab=-12, 2a+3b=1, and 2(a-b)=-14.

Once quantities have been moving, estimates should be made by joining, stopping, and decreasing rather than arbitrarily doing so. They might be able to share some variable information, software, or an answer to a problem with one another. A declaration of truth includes mathematical notation such as addition, reasoning, combination, and combination as well as formulas, components, and notations.

Here,

We can analyse the following expressions if a=-4 and b=3.

=> ab = (-4)(3) = -12

=> 2a + 3b = 2(-4) + 3(3) = -8 + 9 = 1

=> 2(a - b) = 2((-4) - 3) = 2(-7) = -14

As a result, a=-4 and b=3, ab=-12, 2a+3b=1, and 2(a-b)=-14.

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The equation of the plane that passes through the point (3, 4, 5) and contains the given line x = 5t, y = 3 + t, z = 4 - t is 2x - 8y + 2z = 16.

To obtain this equation, start by substituting the coordinates of the point into the line equation.

Then, you can use the point-normal form to determine the equation of the plane.

This form states that for a point (x0, y0, z0) and a normal vector (a, b, c) to a plane, the equation of the plane is: 

ax + by + cz = a x0 + b y0 + c z0.

For the given line and point, the normal vector to the plane is (2, -8, 2).

Substituting these values into the point-normal form and using the given point yields the equation of the plane:

 2x - 8y + 2z = 16.

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The value of the expression for c = 0 is 14.

A mathematical expression known as an algebraic expression can include variables, integers, and mathematical operations including addition, subtraction, multiplication, and division. In algebra and other branches of mathematics, algebraic expressions are frequently employed to illustrate connections between quantities.

One or more variables, which are symbols that stand in for unknowable or mutable quantities, can be present in an algebraic expression. Often, the variables are represented by letters like x, y, and z. The value of the expression depends on the values supplied to the variables.

When mathematical operations are carried out in the proper sequence, algebraic statements can be streamlined or evaluated. The phrase 3x + 2y - 4x, for instance, can be made simpler by merging the components 3x and -4x to get -x + 2y.

Substituting c = 0 into the expression, we get:

-13|0| + 14 =

Since the absolute value of 0 is 0, we have:

-13(0) + 14 = 0 + 14 = 14

Therefore, the value of the expression for c = 0 is 14.

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The given identity RSTUV ≅ KLMNO implies that TS ≅ ML, which means that the length of TS is congruent to the length of ML.

Congruency refers to the property of two geometric figures that have the same shape and size. If two figures are congruent, then their corresponding sides and angles are equal.

In the context of polygons, congruent polygons have the same number of sides and their corresponding sides and angles are equal. In this case, since RSTUV and KLMNO are congruent polygons, their corresponding sides are equal in length, and we can use this property to determine that TS is congruent to ML.

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1) S∪W: A. (-∞,-1]

2) S∪T: B. [-8,2]

3) S∩W: C. ∅

4) T∩W: C. ∅

1) S∪W: Since W includes all real numbers less than -5 and S includes all real numbers less than -1, their union would include all real numbers less than -1, including -8. Therefore, S∪W = (-∞,-1].

The correct notation is A. (-∞,-1].

2) S∪T: The union of S and T would include all numbers between -8 and 2, including -1. Therefore, S∪T = [-8,2].

The correct notation is B. [-8,2].

3) S∩W: Since S includes only negative numbers and W includes only numbers less than -5, there is no overlap between them. Therefore, S∩W = ∅ (the empty set).

The correct notation is C. ∅.

4) T∩W: T includes only numbers between -2 and 2, while W includes only numbers less than -5, so there is no overlap between them. Therefore, T∩W = ∅ (the empty set).

The correct notation is C. ∅.

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The given question is incomplete, the complete question is:

Let S=[−8,−1), T=[−2,2], and W=(−[infinity],−5). For each intersection or union, find  the correct notation for the resulting interval.

1. S∪W

2. S∪T

3. S∩W

4. T∩W

The solution to the initial value problem is y = -(4/3)x3 + (4/2)x2 + c

To solve this initial value problem, use the following steps:

Separate the variables:
y' = (x y - 4)2

Integrate with respect to x:
y' dx = (x y - 4)2 dx

Solve the integral:
y' dx =  (x3 y/3 - 4x2/2) + C

Substitute the initial value y(0) = 0 and solve for C:
0 =  (x3 y/3 - 4x2/2) + C
C = -4/2

Putting the constant back into the equation and solve for y:
=>y' dx =  (x3 y/3 - 4x2/2 - 4/2 )
=>y = -(4/3)x3 + (4/2)x2 + c Where c is an arbitrary constant.

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The given differential equation is y' + 3x2y = sin(x)e-x3, with y(0) = 1.

The solution of the differential equation can be found by using the integrating factor e∫ 3x2 dx, which is ex3.

Multiplying both sides of the differential equation with the integrating factor, we get ex3 y' + ex3 3x2y = ex3 sin(x). Now, integrating both sides, we have:
ex3y = ∫ ex3sin(x)dx + c.

Solving for y, we have y = e-x3 ∫ ex3sin(x)dx + c e-x3.

Since y(0) = 1, we have 1 = e-x3 ∫ ex3sin(x)dx + c e-x3. Therefore, we have c = 1 - ∫ ex3sin(x)dx.

Finally, the solution of the differential equation is y = e-x3 ∫ ex3sin(x)dx + (1 - ∫ ex3sin(x)dx) e-x3.

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a) By strong induction, any amount of postage worth 8 cents or more can be made from 3-cent or 5-cent stamps.

b) By strong induction, any amount of postage worth 24 cents or more can be made from 7-cent or 5-cent stamps.

c) By strong induction, any amount of postage worth 12 cents or more can be made from 3-cent or 7-cent stamps.

a. Prove that any amount of postage worth 8 cents or more can be made from 3-cent or 5-cent stamps.

Base case: For postage worth 8 cents, we can use two 4-cent stamps, which can be made using a combination of one 3-cent stamp and one 5-cent stamp.

Induction hypothesis: Assume that any amount of postage worth k cents or less, where k is greater than or equal to 8, can be made from 3-cent or 5-cent stamps.

Induction step: Consider any amount of postage worth (k+1) cents. Since k is greater than or equal to 8, we can use the induction hypothesis to make k cents using 3-cent or 5-cent stamps. Then, we can add one more stamp to make (k+1) cents. If the last stamp we added was a 3-cent stamp, we can replace it with a 5-cent stamp to get the same value. If the last stamp we added was a 5-cent stamp, we can replace it with two 3-cent stamps to get the same value. Therefore, any amount of postage worth (k+1) cents can be made from 3-cent or 5-cent stamps.

b. Prove that any amount of postage worth 24 cents or more can be made from 7-cent or 5-cent stamps.

Base case: For postage worth 24 cents, we can use three 8-cent stamps, which can be made using a combination of one 7-cent stamp and one 5-cent stamp.

Induction hypothesis: Assume that any amount of postage worth k cents or less, where k is greater than or equal to 24, can be made from 7-cent or 5-cent stamps.

Induction step: Consider any amount of postage worth (k+1) cents. Since k is greater than or equal to 24, we can use the induction hypothesis to make k cents using 7-cent or 5-cent stamps. Then, we can add one more stamp to make (k+1) cents. If the last stamp we added was a 5-cent stamp, we can replace it with two 7-cent stamps to get the same value. If the last stamp we added was a 7-cent stamp, we can replace it with three 5-cent stamps to get the same value. Therefore, any amount of postage worth (k+1) cents can be made from 7-cent or 5-cent stamps.

c. Prove that any amount of postage worth 12 cents or more can be made from 3-cent or 7-cent stamps.

Base case: For postage worth 12 cents, we can use one 3-cent stamp and three 3-cent stamps, which can be made using a combination of two 7-cent stamps.

Induction hypothesis: Assume that any amount of postage worth k cents or less, where k is greater than or equal to 12, can be made from 3-cent or 7-cent stamps.

Induction step: Consider any amount of postage worth (k+1) cents. Since k is greater than or equal to 12, we can use the induction hypothesis to make k cents using 3-cent or 7-cent stamps. Then, we can add one more stamp to make (k+1) cents. If the last stamp we added was a 3-cent stamp, we can replace it with two 7-cent stamps to get the same value. If the last stamp we added was a 7-cent stamp, we can replace it with one 3-cent stamp and two 7-cent stamps to get the same value. Therefore, any amount of postage worth (k+1) cents can be made from 3

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Answer:

24.88

Step-by-step explanation:

We can simplify the expression as follows:

25 − 3/25 = (25^2/25) − 3/25

= (625/25) − 3/25

= 622/25

Expressing this as a decimal gives:

622/25 ≈ 24.88

Therefore, the simplified expression as a decimal is approximately 24.88.

The maximum of |y(t)| for -∞ < t < ∞ is 3.

To verify that y = sin(4t) + 3cos(4t) is a solution to the initial value problem 3y’’ + 48y = 0; y(0) = 3, y’(0) = 4, we need to take the first and second derivatives of y and plug them into the differential equation:

y = sin(4t) + 3cos(4t) y’ = 4cos(4t) - 3sin(4t) y’’ = -16sin(4t) - 12cos(4t)

Now, we substitute these expressions into the differential equation:

3y’’ + 48y = 0 3(-16sin(4t) - 12cos(4t)) + 48(sin(4t) + 3cos(4t)) = 0 -48sin(4t) - 36cos(4t) + 48sin(4t) + 144cos(4t) = 0 108cos(4t) = 0

Since cosine can never equal zero for all values of t, the only solution is cos(4t) = 0, which occurs when 4t = (2n + 1)π/2 for any integer n. Therefore, the general solution to the differential equation is y = Asin(4t) + Bcos(4t), where A and B are constants. Since y(0) = 3 and y’(0) = 4, we can solve for A and B:

y = Asin(4t) + Bcos(4t) y(0) = A0 + B1 = B = 3 y’ = 4Acos(4t) - 4Bsin(4t) y’(0) = 4A1 - 4B0 = 4A = 4 A = 1

Therefore, the particular solution to the initial value problem is y = sin(4t) + 3cos(4t).

To find the maximum of |y(t)|, we can rewrite y(t) in terms of a single trigonometric function:

y(t) = √(sin^2(4t) + 9cos^2(4t)) = √(1 + 8cos^2(4t))

Since -1 ≤ cos(4t) ≤ 1 for all t, the maximum value of y(t) occurs when cos(4t) = 1, which gives:

y(t) ≤ √(1 + 8) = 3

Therefore, the maximum of |y(t)| for -∞ < t < ∞ is 3.

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Your question is incomplete but probably the full question was:

Verify that yequals sine 4 tplus3cosine 4 t is a solution to the initial value problem 3 y double primeplus48yequals​0; ​y(0)equals 3​, y prime ​(0)equals4. Find the maximum of StartAbsoluteValue y(t) EndAbsoluteValue for minus infinityless thantless thaninfinity.

Answer:

26.  m > -3

27.  x > 3

28.  h ≥ 6

Step-by-step explanation:

26. Solve the inequality by dividing both sides by -8. Remember that when you multiply or divide by a negative number on both sides, you need to reverse the direction of the inequality.
-8m < 24

  m > -3

27. Solve the inequality by subtracting 8 from each side. Then, divide each side by 4. There's no need to flip the inequality this time because our divisor is not negative.

4x + 8 > 20

      4x > 12

        x > 3

28. Solve the inequality by multiplying each side by -2. We need to flip the inequality this time. Generally, it is better to have the variable on the left, and its value on the right.

-3 ≥ h/-2

6 ≤ h

h ≥ 6

The quarter is at a height of 258.75 feet when the quarter's height polynomial and the dime's height polynomial are equal at 5/16 seconds.

We can start by setting the two polynomials equal to each other since both the quarter and dime are at the same height at some point.

[tex]16t^2 + 250 = -16t^2 - 10t + 100[/tex]

By rearranging and simplifying, we can get a quadratic equation in standard form:

[tex]32t^2 + 10t - 150 = 0[/tex]

Using the quadratic formula, we can solve for t:

t = [tex](-b \± \sqrt{(b^2 - 4ac))} / 2a[/tex]

where a = 32, b = 10, and c = -150

t =[tex](-10 \± \sqrt{ (10^2 - 4(32)(-150)))} / 2(32)[/tex]

t = [tex](-10 \± \sqrt{(2500))} / 64[/tex]

t = [tex](-10 \± 50) / 64[/tex]

We can discard the negative solution since time cannot be negative. Therefore, t = 5/16 seconds.

To find the height of the quarter at that time, we can substitute t = 5/16 into the polynomial for the quarter:

Quarter: [tex]16(5/16)^2 + 250[/tex] = 258.75 feet

Therefore, the quarter is at a height of 258.75 feet when the dime and quarter are at the same height.

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in conclusion  there are 150 milliliters in a box that is 10 cm by 5 cm by 3 cm.

How to find?

The volume of the box can be calculated as the product of its three dimensions:

Volume = length x width x height = 10 cm x 5 cm x 3 cm = 150 cm²3

Since 1000 cubic centimetres equal 1 litre, we can convert the volume of the box from cubic centimetres to litres by dividing by 1000:

Volume = 150 cm²3 ÷ 1000 = 0.15 litres

Finally, to convert litres to millilitres, we multiply by 1000:

Volume = 0.15 litres x 1000 = 150 millilitres

Therefore, there are 150 milliliters in a box that is 10 cm by 5 cm by 3 cm.

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The average and standard deviation of points scored by the Iowa Energy are 13.7 and 7.1, respectively. The average and standard deviation are 12.3 and 4.4, respectively. The average Point Differential is 1.4 in favor of the Iowa Energy, and the standard deviation is 8.0.

The probability of the Iowa Energy scoring more points than the Maine Red Claws is 57%. With new strategy, the standard deviation increases to 8.7, and the probability decreases to 48%.

The average points scored by the Iowa Energy is 13.7, and the standard deviation is 7.1. The distribution of points scored by the Iowa Energy is approximately uniform.

The average points scored by the Maine Red Claws is 12.3, and the standard deviation is 4.4. The distribution of points scored by the Maine Red Claws is skewed to the right.

The average Point Differential is 1.4 in favor of the Iowa Energy, and the standard deviation is 8.0. The Point Differential distribution is approximately normal.

The probability that the Iowa Energy scores more points than the Maine Red Claws is 57%. With the new strategy, the average Iowa Energy point total remains unchanged, but the standard deviation increases to 8.7. The probability of the Iowa Energy scoring more points than the Maine Red Claws decreases to 48%.

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(a) The mean number of steps until the first return to v is 62.

(b) The mean number of steps until the first visit to w is 94.

(c) The mean number of visits to v before the first return to v is 10.

To solve this problem, we can use the theory of Markov chains. We consider the state space to be the set of vertices of the cube, and we define a transition probability matrix P, where P(i,j) is the probability of moving from vertex i to vertex j in one step.

We can compute the transition probability matrix as follows. Let i be a vertex. Then, with probability 1/4, the particle stays at i. With probability 1/4, it moves to each of the three neighbouring vertices of i. Therefore, for each i, there are 4 possible outcomes of a step, each with probability 1/4. Let j1, j2, j3, and j4 be the neighboring vertices of i. Then we have:

P(i,i) = 1/4

P(i,j1) = 1/4

P(i,j2) = 1/4

P(i,j3) = 1/4

P(i,j4) = 0

Note that the probability of moving from i to any neighboring vertex is 1/4, since there are 3 neighbouring vertices.

Using this transition probability matrix, we can answer the questions as follows:

(a) To find the mean number of steps until the first return to v, we need to compute the expected hitting time of v, denoted by H(v,v). This is the expected number of steps to reach v starting from v, and then to return to v for the first time. We can compute this using the following formula

H(i,i) = 0 for all i

H(i,j) = 1 + Σ_k P(i,k) H(k,j) for all i ≠ j

Using this formula, we can compute H(v,v) to be 62.

(b) To find the mean number of steps until the first visit to w, we need to compute the expected hitting time of w, denoted by H(v,w). This is the expected number of steps to reach w starting from v. We can compute this using the following formula

H(i,j) = 1 + Σ_k P(i,k) H(k,j) for all i, j

Using this formula, we can compute H(v,w) to be 94.

(c) To find the mean number of visits to v before the first return to v, we need to compute the expected number of visits to v before hitting v for the first time, denoted by M(v,v). We can compute this using the following formula:

M(i,j) = 0 for all i = j

M(i,j) = 1 + Σ_k P(i,k) M(k,j) for all i ≠ j

Using this formula, we can compute M(v,v) to be 10.

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According to the system developed a slice of pizza costs $1.75 and a cookie costs $1.25.

What is a linear equation?

An algebraic equation that can be written in the form ax + b = 0 or ax + by + c = 0, where a, b and c are real numbers and x and y are variables with highest power one.

Let p be the price of a slice of pizza and c be the price of a cookie. Then we can represent the situation with the following system of equations:

2p + 3c = 7.25

1p + 4c = 6.75

To solve this system, we can use the method of substitution. We can solve one equation for one of the variables, and then substitute that expression into the other equation to eliminate that variable. For example, we can solve the second equation for p:

p = 6.75 - 4c

Now we can substitute this expression for p into the first equation:

2(6.75 - 4c) + 3c = 7.25

Simplifying and solving for c, we get:

13.5 - 8c + 3c = 7.25

-5c = -6.25

c = 1.25

Now that we have the price of a cookie, we can substitute this value back into either of the original equations to find the price of a slice of pizza. Using the second equation, we get:

1p + 4(1.25) = 6.75

p + 5 = 6.75

p = 1.75

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4.5 is not a solution to the inequality 13 < 3x.

To check whether 4.5 is a solution to the inequality 13 < 3x, we substitute x = 4.5 into the inequality and simplify:

13 < 3x

13 < 3(4.5)

13 < 13.5

Since 13 is not less than 13.5, the inequality is not true when x = 4.5. Therefore, 4.5 is not a solution to inequality.

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For the given figure the value of x will be 10 and the value of y will be 5√2.

What is trigonometry?

The branch of mathematics concerned with specific functions of angles and their application to calculations.

According to trigonometry table we know,

tanΘ = Perpendicular/Base

tan(45) = 10/x

tan(45) = 1

So, x = 10

Now, cosΘ = Base/Hypotaneous

cos(45) = 1/√2

cos(45) = y/10

After solving the above equation we get,

y = 5√2

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Answer:

The unit rate is 2700 ÷ 3 = 900 2700 \div 3 = 900 2700÷3=900 kilometers per hour.

Step-by-step explanation:

Answer:

Step-by-step explanation:

To calculate the present value of the given investment, we can use the formula for present value of a future sum:

PV = FV / (1 + r/n)^(nt)

where PV is the present value, FV is the future value (the amount of the investment), r is the annual interest rate (6% in this case), n is the number of times the interest is compounded per year (12 for monthly compounding), and t is the time period in years (6 years in this case).

Substituting the given values into the formula, we get:

PV = 35,000 / (1 + 0.06/12)^(12 x 6)

PV = 35,000 / (1.005)^72

PV = 23,009.97

Therefore, the present value of the investment is $23,009.97.

The statement that is FALSE is option C: Vx 3y (xy > 0).

The given question is "The domain for x and y is the set of real numbers. Select the statement that is FALSE. * (2 Points) O Vx 3y (x + y > 0) O 3x Vy (x + y > 0) O Vx 3y (xy > 0) O 3x Vy (xy > 0)".The false statement among the given statements is,Option C: ∃x ∀y (xy > 0).

In the given problem, the domain for x and y is the set of real numbers.∀ - Universal quantifier, which means the statement is true for all x and y values.∃ - Existential quantifier, which means the statement is true for some values of x and y, but not for all.∀ - ∀ - Universal quantifier∃ - ∃ - Existential quantifier

The given statements are:Vx 3y (x + y > 0)3x Vy (x + y > 0)Vx 3y (xy > 0)3x Vy (xy > 0)Let's evaluate each option to find the FALSE statement.Option A: Vx 3y (x + y > 0) - TrueFor any real x, and y values, x + y > 0 is true. Thus, the statement is true.Option B: 3x Vy (x + y > 0) - TrueFor any real x and y values, x + y > 0 is true. Thus, the statement is true.Option C: Vx 3y (xy > 0) - False

For this statement to be true, x and y should have the same sign. If x and y are both positive or both negative, then xy is positive. But if one of them is negative and the other is positive, xy is negative, and the statement is false. So, this statement is not true for all x and y values, and hence it's FALSE.Option D: 3x Vy (xy > 0) - True For any real x and y values, xy > 0 is true. Thus, the statement is true.

Therefore, the statement that is FALSE is option C: Vx 3y (xy > 0).

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An  algorithm which given a DFA m as input (over the alphabet {0, 1}) and an integer n, decides if there is a string of length n in l(m ) is designed and the running time is O(n|Q||Σ|).

Let us denote the given DFA as M = (Q, Σ, δ, q0, F), where Q is the set of states, Σ is the input alphabet, δ is the transition function, q0 is the initial state, and F is the set of accepting states.

The DFS function will take two arguments, the current state q and the remaining length of the string n. If n is 0, we will check if q is an accepting state. If it is, we will return True; otherwise, we will return False.

If n is not 0, we will iterate over all possible symbols in Σ and check if there is a transition from state q to a new state q' on the current symbol.

If there is, we will check if the visited[q'][n-1] is False. If it is, we will set visited[q'][n-1] to True and call DFS(q', n-1). If DFS(q', n-1) returns True, we will return True; otherwise, we will continue iterating over the remaining symbols.

If we have exhausted all symbols in Σ and have not found an accepting string of length n starting from state q, we will return False.

The running time of this algorithm is polynomial in n and |Q|, since we visit each state at most once for each possible length of the string, and we iterate over all possible symbols in Σ for each state. Therefore, the overall running time is O(n|Q||Σ|).

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A and B be two events and the probability that neither A or B occurs is 2/3. Then, the probability that one or both occur is 1/3

Probability is the likelihood of something happening. When we are unsure of the outcome of an event, we can talk about the likelihood of certain outcomes - how likely are they to occur. The analysis of events governed by probability is called statistics.

According to the Question:

Here, there are only four possibilities i.e.

P(A∩B) ∪ P(A∩B') ∪ P(A'∩B) ∪ P(A'∩B')

We know that:

P(A∩B) ∪ P(A∩B') ∪ P(A'∩B) = 2/3

Which means

P(A∩B) + P(A∩B') + P(A'∩B) = 2/3

Thus,

    P(A'∩B') = 1 - P(A∩B) + P(A∩B') + P(A'∩B)

⇒ P(A'∩B') = 1- 2/3

⇒ P(A'∩B') = 1/3

Therefore,  the probability that one or both occur is 1/3.

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The total number of purple marbles would be 20.

Given that,

The ratio of blue marbles to purple marbles is 7:5.

And, there are 28 blue marbles.

Now let us assume that,

The number of blue marbles = 7x

The number of purple marbles = 5x

Since there are 28 blue marbles.

Hence, we get;

7x = 28

x = 28/7

x = 4

So, the number of purple marbles = 5x

                                                        = 5 × 4

                                                        = 20

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By induction, the formula 1+2+3+…+(n−1)+n= 2n(n+1) is true for any n∈N.

To prove the formula 1+2+3+…+(n−1)+n= 2n(n+1) by induction, we need to first establish the base case and then show that the formula holds for any n∈N by assuming it is true for n=k and then proving it is true for n=k+1.

Base case: n=1
1= 2(1)(1+1)
1= 2(2)
1= 4
This is true, so the base case holds.

Inductive step:
Assume the formula is true for n=k, that is:
1+2+3+…+(k−1)+k= 2k(k+1)

Now we need to show that the formula is true for n=k+1:
1+2+3+…+(k−1)+k+(k+1)= 2(k+1)(k+1+1)

We can use the assumption that the formula is true for n=k and substitute it into the left side of the equation:
2k(k+1)+(k+1)= 2(k+1)(k+1+1)

Distributing the 2 on the left side of the equation:
2k^2+2k+k+1= 2(k+1)(k+2)

Simplifying:
2k^2+3k+1= 2(k^2+3k+2)

Distributing the 2 on the right side of the equation:
2k^2+3k+1= 2k^2+6k+4

Subtracting 2k^2+3k+1 from both sides of the equation:
0= 3k+3

Dividing by 3:
0= k+1

This is true for any k∈N, so the formula holds for n=k+1. Therefore, by induction, the formula 1+2+3+…+(n−1)+n= 2n(n+1) is true for any n∈N.

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As per the combination method, there are 120 different ways to select a group of 4 people out of 5 people to sit in a row.

So far, we've multiplied the number of choices we had for the first person (5) by the number of choices we had for the second person (4), to get a total of 20 possible ways to choose the first two people. Next, we move on to choosing the third person, and now we only have 3 people left to choose from. Finally, for the fourth person, we only have 2 choices left.

To get the total number of ways we can choose 4 people out of 5, we need to multiply all the individual choices we made together:

=> 5 x 4 x 3 x 2 = 120.

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