Prove the following statement directly from the definition of divisibility. For all integers a, b, c, and d, if aſc and bld then ab|cd. Proof: Let a, b, c, and d be any integers such that alc and b|d. Then by definition of divisibilityq c= ar and d = bs for some integers r and s. Then cd equals the product of ab and a number that can be written in terms of r and s as follows: cd = ab: , which is an integer because products A of integers are integers. (Simplify your answer completely.) Thus cd = ab · (an integer), and so ablcd by definition of divisibility.

The probability that more than 8 hours are needed to grade all of the exams is about 52%

The probability of a standard normal distribution is the area under the curve of the normal distribution function within a specified interval.

Let X represent the random variable to grade an exam, and let Y represent the total time to grade all exams

The number of students = 50

Therefore;

Y = 50·X

The properties of the normal distribution indicates that we get;

E(Y) = E(50·X) = 50·E(X) = 50 × 9.7 = 485

Var(Y) = Var(50·X) = 50²·Var(X) = 50² × 1.92² = 9025

The standard deviation, SD(Y) = √(Var(Y)) = √(9025) = 95

The probability that more than 8 hours are needed can be found using the z-score of the normal distribution as follows;

8 hours = 480 minutes

Z = (480 - 485)/95 ≈ -0.0526

The probability obtained from a standard normal table, is therefore;

P(Z > -0.0526) = 1 - 0.48006 ≈ 0.52

The probability that more than 8 hours are needed to grade all students is therefore about 0.52 or 52%

Learn more about the standard normal table here: https://brainly.com/question/28991504

#SPJ1

Step-by-step explanation:

No,

if the sprinkler covers a distance of 12 m meaning the 12 m is the diameter...then to find the area that it covers we use the formula for the circle since it's circular

A=πr2

A=3.142*36

A=113.112 cm3

Answer:

Step-by-step explanation:

The product of (4x + 7) and (3x - 8) is 12x² + 7x - 56.

WHAT IS BOX METHOD ?

The box method, also known as the grid method, is a visual method used to multiply two numbers or two binomials. It involves creating a grid or box and filling it in with the products of the digits in each row and column. The method works for both single-digit and multi-digit numbers.

To use the box method for multiplying two numbers, we draw a box with two rows and two columns. We write one number along the top row and the other number along the left column. Then, we multiply the digits in each row and column and write the products in the corresponding cell of the box. Finally, we add the numbers in each cell of the box to get the product of the two numbers.

The box method can be used to multiply two binomials, such as (4x + 7) and (3x - 8). To use the box method, we draw a box with four cells, and we write the two binomials along the top and left sides of the box, like this:

     |  4x  |  7

-------------------

 3x  |      |

-------------------

    |      |

Then, we fill in the four cells of the box by multiplying the corresponding terms. For example, the top-left cell is filled by multiplying 4x and 3x, which gives 12x². The other cells are filled in a similar way:

     |  4x  |  7

-------------------

 3x  | 12x² | 28x

-------------------

    | -21x | -56

Next, we combine the terms in each row and column of the box, and write the final answer as the sum of these terms:

12x² + 28x - 21x - 56

Simplifying this expression gives:

12x² + 7x - 56

Therefore, the product of (4x + 7) and (3x - 8) is 12x² + 7x - 56.

To know more about binomials visit :-

https://brainly.com/question/25566343

#SPJ1

The solution for the other variable, according to the stated statement, is[tex]X = 6^y[/tex] and [tex]y = 21 * 2^X[/tex]

Exponential numbers are represented by an, where an is multiplied by itself n times. An easy example is 8=2³=222. In exponential notation, an is known as the base, whereas n is known as the power, exponent, or index. Scientific notation is an example of an exponential number, with 10 usually typically serving as the base number.

Exponential functions are often employed in the biological sciences to describe the amount of a certain quantity over time, such as population size. Experiment data graphs are often created with time on the x-axis and amount on the y-axis.

a. y = log6(X)

[tex]6^y = X[/tex]

[tex]X = 6^y[/tex]

b. [tex]X = log2(y/21)[/tex]

[tex]2^X = y/21[/tex]

[tex]21 * 2^X = y[/tex]

[tex]y = 21 * 2^X[/tex]

To know more about exponential visit:

https://brainly.com/question/28596571

#SPJ1

Answer:

Were sorry! Answer is not available right now check in later.

Step-by-step explanation:

Answer:

90 miles

Step-by-step explanation:

We know

She has driven 30 miles so far, which is one-third of the way home.

What is the total length of her drive?

We Take

30 x 3 = 90 miles

So, the total length of her drive is 90 miles.​

Answer:

  (2x -4)/(x -1)

Step-by-step explanation:

You want to simplify (2x² -2x -4)/(x² -1).

Simplifying a rational expression generally means cancelling common factors from the numerator and denominator. To do that, you must factor both the numerator and the denominator.

The numerator coefficients have a common factor of 2. Removing that can simplify the problem of factoring the numerator:

  2x² -2x -4 = 2(x² -x -2)

To factor the quadratic, you look for factors of -2 that have a sum of -1. Those would be -2 and +1. The middle term is written as a sum using these values.

  = 2(x² -2x +x -2)

This can now be factored by grouping terms in pairs, and factoring each pair.

  = 2((x² -2x) +(x -2)) = 2(x(x -2) +1(x -2))

  = 2(x +1)(x -2)

The denominator is the difference of squares, so is factored according to the pattern for that:

  a² -b² = (a -b)(a +b)

  x² -1 = (x -1)(x +1)

Now you know enough to be able to simplify the expression. The common factors (x+1) cancel.

  [tex]\dfrac{2x^2-2x-4}{x^2-1}=\dfrac{2(x+1)(x-2)}{(x+1)(x-1)}\\\\=\dfrac{2(x-2)}{x-1}=\boxed{\dfrac{2x-4}{x-1}}[/tex]

__

Additional comment

Even though the factor (x+1) has disappeared from the expression, the simplified form still carries the restriction that x ≠ -1. The graph of the original expression will have a "hole" at (-1, 3), where it is undefined. Otherwise, the graph looks like a graph of (2x-4)/(x-1).

Number of  prints Jim ordered last month is 271

Let's assume Jim ordered "P" prints last month. The cost of ordering "P" prints would be the sum of the monthly charge and the cost of each print, which is given by the equation:

Cost = Monthly Charge + (Cost per Print x Number of Prints)

Substitute the values in the equation

$17.79 = $6.95 + ($0.04 x P)

Simplifying the equation, we get:

$10.84 = $0.04 x P

Dividing both sides by $0.04, we get:

P = $10.84 / $0.04

Divide the numbers

P = 271

Therefore, Jim ordered 271 prints last month.

Learn more about equation here

brainly.com/question/14603452

#SPJ4

Function P represents the perimeter, in inches, of a square with the side length x inches. So, the perimeter of a square can be represented as P = x + x + x + x, which simplifies to P = 4x. This way is more efficient to write than the former.

Complete the table provided: x 0 1 2 3 4 5 6 P(x) 0 4 8 12 16 20 24. When x = 0, the perimeter of the square is 0.When x = 1, the perimeter of the square is P = 4(1) = 4. When x = 2, the perimeter of the square is P = 4(2) = 8. When x = 3, the perimeter of the square is P = 4(3) = 12. When x = 4, the perimeter of the square is P = 4(4) = 16. When x = 5, the perimeter of the square is P = 4(5) = 20. When x = 6, the perimeter of the square is P = 4(6) = 24.

Function P represents the perimeter, in inches : https://brainly.com/question/19110581

#SPJ11

A fast food restaurant estimate that 45% of their customers buy drinks with their purchases. Last week, 6200 customers did not buy soft drinks. They have a total of 11272 customers.

when in a percentage we take 100%

if 45% take soft drinks with their purchases

then the other 55% did not take soft drinks with their purchases

therefore, we have 6200 customers who did not buy soft drinks with their purchases.

now we know that 11272 are there total customers

therefore, now we have to subtract 6200 from 11272

we get 5072.

therefore 5072 customers who buy soft drinks with their purchases.

A fast food restaurant estimate that 45% of their customers buy drinks with their purchases. Last week, 6200 customers did not buy soft drinks. They have a total of 11272 customers

To learn more about percentage, click here:

brainly.com/question/29306119

#SPJ4

Answer:

To find the greatest common factor, we need to find the largest factor that divides both terms evenly. We can factor each term as follows:

8a³b5 = 2³ * a³ * b5

16a²b7 = 2⁴ * a² * b7

The greatest common factor is the product of the lowest exponent of each prime factor that appears in both terms. Therefore, the greatest common factor is:

GCF = 2³ * a² * b5 = 8a²b5

To find the least common multiple, we need to find the smallest multiple that both terms share. We can start by writing out the prime factorization of each term:

8a³b5 = 2³ * a³ * b5

16a²b7 = 2⁴ * a² * b7

The least common multiple is the product of the highest exponent of each prime factor that appears in either term. Therefore, the least common multiple is:

LCM = 2⁴ * a³ * b7 = 16a³b7

So, the greatest common factor is 8a²b5 and the least common multiple is 16a³b7.

 The 32 ways to get a five-card combination containing two jacks and three aces.

(a) A full house consists of three of one kind and two of another kind. Therefore, there are 13 different choices for the rank of the triplet and 4 cards of the same rank. Once the triplet has been chosen, there are 12 choices for the rank of the pair and 4 cards of the same rank. Therefore, the number of ways to get a full house is as follows:$${13}{\times}{4}{\times}{12}{\times}{4}={7488}$$Therefore, there are 7488 ways to get a full house.(b) In this case, the two jacks and three aces must be chosen out of the 4 jacks and 4 aces in the deck. Therefore, the number of ways to get a five-card combination with two jacks and three aces is as follows:$$\frac{{4\choose2}{4\choose3}{44\choose0}}{5!}={32}$$Therefore, there are 32 ways to get a five-card combination containing two jacks and three aces.

Learn more about Combination

brainly.com/question/27014146

#SPJ11

The domain ofA is Answer: "d. [tex]a(x) = -x^2 + 300x[/tex], Domain = {x|0 < x < 6000 "}

A farmer has 600 yards of fencing to enclose a rectangular garden. The area A of the rectangle as a function of the width x of the rectangle can be expressed as follows:

[tex]A = x(300 - x) = 300x - x^{2}[/tex]

The domain of A is given by the interval of values of x that can be chosen for the width of the rectangle. The width of a rectangle must be positive, so we have 0 < x. If the farmer has 600 yards of fencing, we can get the maximum value of x as follows:

600 = 2x + 300x = 600/3 = 200

So we have x < 200.

Therefore, the domain of A is {x | 0 < x < 200}.

Rewriting the equation of A, we get:

A(x) = -x² + 300x, Domain = {x | 0 < x < 200}.

In yards, the answer would be {x | 0 < x < 6000}.

You can learn more about domain at

https://brainly.com/question/26098895

#SPJ11

There are 48 toy soldiers, which is 6 x 8 of them.

The anger Eminem expresses in "Like Toy Soldiers" is a result of his personal beefs with rappers Ja Rule and Benzino, who was the editor of The Source at the time. The song, "Toy Soldiers," by Martika, was sampled on the 2004 release Encore.

Let's name Leo's collection of toy soldiers "x" the amount.

As a result of the problem statement, we are aware of:

There are no more when he arranges them in groups of four, proving that x is divisible by four.

Six remain after he divides them into groups of seven, proving that (x - 6) is divisible by seven.

He organizes them into fives. If there are still 3 after multiplying by 5, (x - 3) can be divided by 5.

We may create a system of equations based on these three conditions:

x = 4a (from the first condition)

x - 6 = 7b (from the second condition)

x - 3 = 5c (from the third condition)

where a, b, and c are integers.

4a - 6 = 7b

4a - 3 = 5c

Now we need to solve for a, b, and c.

7b = 4a - 6

7b + 6 = 4a

Since 7 and 4 are relatively prime, we know that (7b + 6) must be divisible by 4. Therefore, we can write:

7b + 6 = 4k

where k is some integer. Solving for b, we get:

b = (4k - 6) / 7

Since b is an integer, k must be 2, which gives us:

b = (4(2) - 6) / 7 = -1

We can try the next possible value of k, which is 3:

b = (4(3) - 6) / 7 = 0

x - 3 = 5c

6 - 3 = 5c

c = 1

6 divided by 4 is 1 with no remainder.

(6 - 6) divided by 7 is 0 with a remainder of 0.

(6 - 3) divided by 5 is 1 with a remainder of 0.

Therefore, the answer is 48, which is 6 times 8.

To know more about integer visit:-

https://brainly.com/question/15276410

#SPJ1

Answer: 105

Step-by-step explanation:

OBTUSE ANGLE (105°) IS FORMED AT 2:30 IN THE CLØCK.

As we know 12 division OF 360° CIRCLE GIVES 30°

AT TWO PM THE ANGLE BETWEEN HOUR AND MINUTE HAND IS 60° AFTER 30 MIN ARM ROTATES 180° AND HOUR ARM 30°/2 = 15°

SO THE ANGLE BETWEEN TWO ARMS IS (180-(60-15)

105

As a result, Lara made a $6,262.71 initial deposit into her savings account.

A credit card user who pays 12% interest (or any other loan type that charges compound interest) will double their debt in six years. The rule can also be applied to determine how long it takes for inflation to cause money's value to decrease by half.

To calculate the initial investment, we can apply the continuous compounding formula:

A = Pe(rt)

Where:

A = the current balance ($7,000.00)

P = the initial deposit (unknown)

r = the annual interest rate (11% or 0.11 as a decimal)

t = the time in years (1 year)

Plugging in these values, we get:

$7,000.00 = Pe(0.11 * 1)

A shorter version of the exponential expression:

$7,000.00 = Pe0.11

$7,000.00 = P * 1.1166 (rounded to 4 decimal places)

Dividing both sides by 1.1166:

P = $6,262.71 (rounded to the nearest cent)

To know more about initial deposit visit:-

https://brainly.com/question/21845455

#SPJ1

The answer is (001). This is because B (001) has h and k as non-zero integers, which does not match the criteria for the {100} family.

The question is asking which of the planes (A), (B), (C), and (D) is not part of the {100} family for a tetragonal crystal.A tetragonal crystal is a three-dimensional structure made up of four faces that intersect at right angles, forming a unit cell. Each face of the unit cell is defined by a Miller index. A Miller index is a set of three integers written in the form {hkl}, which describes the orientation of the face relative to the crystal lattice. In a tetragonal crystal, the {100} family is the set of faces described by {hkl} such that h = k = 0 and l ≠ 0.

Therefore, A (010), C (110), and E (all of these planes are in the {100} family) are all part of the {100} family for a tetragonal crystal, while B (001) is not.  because B (001) has h and k as non-zero integers, which does not match the criteria for the {100} family. In conclusion, the correct answer to the question is B (001).

To know more about tetragonal crystal system, click here:

https://brainly.com/question/29645362

#SPJ11

Answer: 11

Step-by-step explanation:

Turn it into an improper fraction so 8 from 8 1/4 to 33/4

Then just divide by 3/4

This will give you 11

Answer:

y = - [x - 1] + 2

Step-by-step explanation:

I think this is right

Im sorry if not :)

The followings A, C, D, and F subsets of P₂ are subspace of P₂.

The subsets of P₂ that are subspace of P₂ are A, C, D, and F.

A.{p(t) | p'(t)+1p(t)+6=0} is a subspace of P₂ because it satisfies the criteria of being closed under addition and scalar multiplication.

B.{p(t) | p('t)='p(t) for all t} is not a subspace of P₂ because the derivative of p(0) does not equal p(4).

C. {p(t) | p(4)=7} is a subspace of P₂ because it satisfies the criteria of being closed under addition and scalar multiplication.

[tex]{p(t) | \int_{-1}^{7}p(t)dt=0[/tex] is constant } is a subspace of P₂ because it satisfies the criteria of being closed under addition and scalar multiplication.

E. {p(t) | '(8) = p(0)} is not a subspace of P₂ because the derivative of p(8) does not equal 0.

F. {p(t) | p(3)=0} for all t} is a subspace of P₂ because it satisfies the criteria of being closed under addition and scalar multiplication.

Learn more about scalar multiplication here:

brainly.com/question/8349166

#SPJ12

Answer:

Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer

The mean, variance, and standard deviation for those who have purchased online goods or services are 72, 46.08, and 6.8, respectively.

That 36% of adult internet users have purchased products or services online and for a random sample of 200 adult internet users, we need to find the mean, variance, and standard deviation for the number who have purchased goods or services online.

Therefore, the mean, variance, and standard deviation for the number who have purchased goods or services online are 72, 46.08, and 6.8, respectively.

To learn more about "Variance": brainly.com/question/30112112

#SPJ11

The equation of the circle graphed in this problem is given as follows:

x² + y² = 34.

The equation of a circle of center [tex](x_0, y_0)[/tex] and radius r is given by:

[tex](x - x_0)^2 + (y - y_0)^2 = r^2[/tex]

The radius of a circle represents the distance between the center of the circle and a point on the circumference of the circle, while the diameter of the circle is the distance between two points on the circumference of the circle that pass through the center. Hence, the diameter’s length is twice the radius length.

The center of the circle is at the origin, hence:

x² + y² = r².

The diameter of the circle is given by segment AC, which is the hypotenuse of a right triangle of sides 10 and 6, hence it's length is obtained applying the Pythagorean Theroem as follows:

d² = 10² + 6²

d = sqrt(10² + 6²)

d = 11.66.

The radius is half the diameter, hence:

r = 11.66/2

r = 5.83.

Then the equation of the circle is given as follows:

x² + y² = 5.83²

x² + y² = 34.

More can be learned about the equation of a circle at https://brainly.com/question/1506955

#SPJ1

Answer: 31 pizzas

Step-by-step explanation:

add them together

The giraffe is 15 feet tall based on the relation between the length of neck of giraffe and it's height.

Firstly convert the mixed fraction to fraction.

Length of neck of giraffe = (4×2)+1/2

Length of neck = 9/2 feet

Now, let us assume the height of giraffe be x. So, equation will be -

30% × x = 9/2

Rewriting the equation

30/100 × x = 9/2

Cancelling zero

3x/10 = 9/2

Again rewriting the equation

x = 90/6

Performing division on Right Hand Side of the equation

x = 15 feet

Thus, height of giraffe is 15 feet.

Learn more about equations -

https://brainly.com/question/2972832

#SPJ4

Solution:Given, the mean speed of the cars racing = 158.9 mph standard deviation = 6.7 mph

To find:What speed represents the 30th percentile for speeds of race cars at Talladega?

We need to find the z-score for the 30th percentile.From the standard normal distribution table, the z-score for the 30th percentile is -0.52.Using the formula for z-score we havez=(x-μ)/σwhere x is the speed of the carsμ is the mean speed = 158.9σ is the standard deviation = 6.7Substituting these values in the above equation we have-0.52=(x-158.9)/6.7Rearranging we get,x - 158.9 =[tex]-0.52 × 6.7x - 158.9 = -3.524x = 158.9 - 3.524x = 155.376[/tex]The speed that represents the 30th percentile for speeds of race cars at Talladega is approximately 155.38 mph.

for such more questions on standard deviation

https://brainly.com/question/475676

#SPJ11

Answer: 201.06

Step-by-step explanation:

A=πr2=π·8^2≈201.06193

he tree nearest the house is our starting point. Our point person is taking the clinometer reading 15.24 meters from the tree's base and they get a reading of 237`. Our point person is 1.83 meters in height.

How tall is the tree, rounded to the nearest meter?The height of the tree can be determined by using the tangent formula. The tangent formula is tan θ = h/d where θ = angle of elevation, h = height of the object, and d = horizontal distance.

The clinometer reading is the angle of elevation. Hence, we can use the given data to determine the height of the tree.The point person is standing at 15.24 m from the base of the tree. Therefore, the horizontal distance (d) is 15.24 m. The angle of elevation (θ) is 237 degrees (given in the question).

Convert the degrees to radians as tan function uses radians. Convert degrees to radians:[tex]237 × (π/180) = 4.135[/tex]radians.Now we can use the tangent formula to determine the height of the tree:tan θ = h/dtan 4.135 = [tex]h/15.24h = 15.24 × tan 4.135h = 15.24 × 0.07311h ≈ 1.1132[/tex] metersThe height of the tree is 1.1132 meters. But, we have to round the answer to the nearest meter. Therefore, the height of the tree, rounded to the nearest meter, is 1 meter.

for such more questions on The angle of elevation

https://brainly.com/question/88158

#SPJ11

From the given information provided, the area of the shaded region inside the circle is 22.58

The area of a triangle is defined as the total space occupied by the three sides of a triangle in a 2-dimensional plane.

The space enclosed by the sector of a circle is called the area of the sector.

the radius of the circle is 5cm.

area of arc =  radius² × θ/2

area of the arc is =  5² × 4π/9 = 25 × 4/9 = 34.88

area of the triangle inside circle = a×b × sin(y)/2

area of triangle = 5×5 × sin(80°)/2 = 25 × 0.492

area = 12.3

area of the shaded region is = 34.88 - 12.3 = 22.58

Hence, the area of the shaded region is 22.58

Question - Find the area of the shaded region in the circle if the angle of the arc is 80 degree radius is 5cm.

Learn more about circle here: brainly.com/question/24375372

#SPJ4