Which graph represents the function f(x) = cos (4x)

(a) To show that {v1​,…,vk​,w1​,…,wℓ​} is an orthogonal set, we must show that the inner product of any two distinct vectors in this set is 0.

(b) To show that {v1​,…,vk​,w1​,…,wℓ​} is a basis of Rn, we must show that any vector in Rn can be written as a linear combination of these vectors.

(c) To show that dim V + dim V⊥ = n, we must show that the number of vectors in {v1​,…,vk​,w1​,…,wℓ​} is equal to n.

Let v and w be any two distinct vectors in the set {v1​,…,vk​,w1​,…,wℓ​}. By definition, since {v1​,…,vk​} is an orthogonal basis of a subspace V of Rn and {w1​,…,wℓ​} is an orthogonal basis of V⊥, we have that v and w are either in V or in V⊥, but not both.

Thus, if v is in V, then we have  = 0, because w is in V⊥. Similarly, if w is in V⊥, then  = 0, because v is in V. Therefore,  = 0 for any two distinct vectors v and w in {v1​,…,vk​,w1​,…,wℓ​}, which implies that {v1​,…,vk​,w1​,…,wℓ​} is an orthogonal set.

(b) Let x be any vector in Rn. Since {v1​,…,vk​} is a basis of V, x can be written as a linear combination of the vectors in {v1​,…,vk​}, that is, x = a1v1 + a2v2 + ... + akvk.

Since {w1​,…,wℓ​} is a basis of V⊥, x can also be written as a linear combination of the vectors in {w1​,…,wℓ​}, that is, x = b1w1 + b2w2 + ... + bℓwℓ.

Combining these two equations, we can write x = (a1v1 + a2v2 + ... + akvk) + (b1w1 + b2w2 + ... + bℓwℓ). Thus, any vector in Rn can be written as a linear combination of the vectors in {v1​,…,vk​,w1​,…,wℓ​}, which implies that {v1​,…,vk​,w1​,…,wℓ​} is a basis of Rn.

By definition, {v1​,…,vk​} is an orthogonal basis of a subspace V of Rn, so dim V = k. Similarly, {w1​,…,wℓ​} is an orthogonal basis of V⊥, so dim V⊥ = ℓ. Thus, the number of vectors in {v1​,…,vk​,w1​,…,wℓ​} is k + ℓ = n, which implies that dim V + dim V⊥ = n.

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The solution to the system of equations, 10x-11=-3y 5y-5=-10x, is x = 2 and y = -3.

To solve the system of linear equations by elimination, we need to eliminate one of the variables to find the value of the other variable. We can do this by multiplying one equation by a constant and adding it to the other equation.

10x - 11 = -3y  (Equation 1)

5y - 5 = -10x  (Equation 2)

We subtract the equations to eliminate the x variable:

10x - 11 - 5y + 5 = -3y + 10x

-5y + 3y = 10x - 10x + 6

-2y = 6

y = -3

Now we find the value of x:

5(-3) - 5 = -10x

-10x = -15 - 5

10x = 20

x = 20/10

x = 2

So, the solution to the system of equations is x = 2 and y = -3.

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P(A ∩ B) = P(A)P(B) and P(Ac ∩ Bc) = P(Ac)P(Bc).

The complement operator “c” refers to the inverse of an event. Therefore, for two independent events A and B in a sample space S, the inverse events Ac and Bc are also independent. This is because the probability of two independent events occurring is equal to the product of their individual probabilities. Therefore, the probability of two inverse events occurring is equal to the product of their individual probabilities.

Mathematically, this is expressed as follows: P(A ∩ B) = P(A)P(B) and P(Ac ∩ Bc) = P(Ac)P(Bc).

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Answer:

Step-by-step explanation:

ok so first u do 12 times 2 and then divide by 3

Part a) After 30 minutes since the accident, the oil slick has a radius of 15 feet (0.5 * 30 = 15). Therefore, the volume of the oil slick is V(15) = 0.08π * 15 ^ 2 = 705.66 cubic feet. Part b) The function (V∘r)(t) = 0.08π * (0.5t) ^ 2 = 0.02πt ^ 2 is obtained by combining the two functions V(r) and r(t).

Part c) The function (V∘r)(t) = 0.02πt ^ 2 tells us the volume of the oil slick in terms of time t. As time increases, the volume of the oil slick also increases exponentially. Part d) To find out after how many minutes the oil slick will be 705 cubic feet, we can set (V∘r)(t) equal to 705, and solve for t. Therefore, t = (705 / 0.02π) ^ 1/2 ≈ 16.4 minutes. Therefore, after 16 minutes and 24 seconds, the oil slick will be 705 cubic feet.

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tan 49° = x/55

tan Θ = opposite/adjacent

x - opposite

55 cm - adjacent

Answer:

To find the number of word problems on Mylie's math test, we need to multiply the total number of problems by the percentage that were word problems:

Number of word problems = 0.15 x 40

Number of word problems = 6

Therefore, there were 6 word problems on Mylie's math test.

Interest on $600 after 50 days of depositing is $4.53

In finance and economics, interest is the amount paid by a borrower or deposit-taking financial institution to a lender or depositor in excess of the repayment of the principal at a specified rate. It is different from a fee that a borrower can pay to a lender or a third party.

Given,

Principal amount = $600

Interest rate = 5.5%

Compounded daily for 50 days

By the interest table,

$1 compounded daily at 5.5% rate for 50 days = $1.00755

$600 compounded daily at 5.5% rate for 50 days

= $600 × 1.00755

= $604.53

Interest = Amount with interest - Principal

Interest = $604.53 - $600

Interest = $4.53

Hence, $4.53 is the interest on $600 after 50 days of depositing.

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Answer:

c

Step-by-step explanation:

1 2/3 = 5/3 = 20/12

3 1/4 = 13/4 =39/12

Answer:

639 visitors in October

Step-by-step explanation:

I'll assume the "23" is meant to be 2/3.  If so:

We learn that attendance for November is 2/3 that of October.  Let's put that into an equation.  Let x be the October attendance and y the November attendance.

    y = (2/3)x  [Nov = (2/3) of Oct]

We are told that y = 426

  y = (2/3)x

  426 = (2/3)x

(426)*(3/2) = x

  x = 639

October's attendance was 639 visitors.

The ratio of red to blue squares is given by the division of the number of red squares by the number of blue squares.

The ratio between two amounts, a and b, is obtained applying a proportion, as the ratio is the division of the amount a by the amount b.

The amounts for this problem are given as follows:

Hence the ratio is given by the division of the number of red squares by the number of blue squares.

For example, for 10 red and 20 blue squares, the ratio is given as follows:

r = 10:20 = 1:2.

The problem is incomplete, hence the general procedure to obtain the ratio is presented.

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Answer: C

Step-by-step explanation: LxWxH

8x8= 64x4=256i

hope this helped :)

Answer:

256

Step-by-step explanation:

start with 8x8 that will equal 64 then 64x4 which equals 256.

Answer:

1.8 i think

Step-by-step explanation:

i used a calculator

Answer:

Your answer is x = 7.

Step-by-step explanation:

<HFG + <YFG = 180°

180° - <HFG = <YFG

<H + <G + <HFG = 180°

<H + <G  = 180° - <HFG

<H + <G  = <YFG

(6x + 8) + (65°)  = (3+16x)

6x + 8 + 65 = 16x + 3

Get x on one side and the other numbers on the other side.

8 + 65 - 3 = 16x - 6x

70 = 10x

x = 7

You can plug in x into the angle to solve for their angles.

Answer:

x = 7

Step-by-step explanation:

∠HFG must equal to 180° - ∠YFG. Because there are 180° in a triangle, we can subtract the known value, 65°, and be left with 115°.

From there, an equation can be formed.

115° = 6x + 8 + 180 - (3+16x)

115°= 6x + 188 - 3 - 16x

115°= 185 - 10x

-70° = -10x

x = 7

Answer:

To find cos x, we can use the trigonometric identity:

cos^2 x + sin^2 x = 1

Rearranging this identity, we get:

cos^2 x = 1 - sin^2 x

Substituting the given value of sin x (2/3), we get:

cos^2 x = 1 - (2/3)^2

= 1 - 4/9

= 5/9

Taking the square root of both sides, we get:

cos x = ±√(5/9)

Since cosine is positive in the first quadrant, where sin x is positive, we can take the positive square root:

cos x = √(5/9)

We can simplify this expression by noting that both the numerator and denominator have a common factor of 5. We can simplify by factoring out this common factor:

cos x = √(5/9)

= √(5)/√(9)

= √(5)/3

Therefore, cos x = √(5)/3.

The mean and variance of the sales of cars from company FF in 8 different months

The mean and variance of the sales of cars from company FF in 8 different months are A. Mean = 10.5, Variance = 13.43. This can be calculated by adding up all the sales and then dividing by the number of months (6 + 7 + 8 + 9 + 11 + 12 + 15 + 16 = 84) and then dividing by 8 to get the mean (84 / 8 = 10.5). To calculate the variance, we first subtract the mean from each data point (6 - 10.5 = -4.5, 7 - 10.5 = -3.5, 8 - 10.5 = -2.5, 9 - 10.5 = -1.5, 11 - 10.5 = 0.5, 12 - 10.5 = 1.5, 15 - 10.5 = 4.5, 16 - 10.5 = 5.5) and then square each difference (-4.52 = 20.25, -3.52 = 12.25, -2.52 = 6.25, -1.52 = 2.25, 0.52 = 0.25, 1.52 = 2.25, 4.52 = 20.25, 5.52 = 30.25). We then add all of these values together and divide by the number of months (20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 2.25 + 20.25 + 30.25 = 93.5) and divide by 8 to get the variance (93.5 / 8 = 11.6875). The variance is 11.6875, which is equivalent to 13.43 when rounded to two decimal places. Therefore, the mean and variance of the sales of cars from company FF in 8 different months are A. Mean = 10.5, Variance = 13.43.

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Answer: 11.6

Step-by-step explanation: plug in x for 4. 3.2(4)-1.2 = 11.6

Answer: [tex]\frac{2}{35}[/tex] hour

Step-by-step explanation:

      We will divide the time it takes it to do 7 tasks (⅖ hour) by the number of tasks it does in that time frame (7 tasks) to find the time per task.

[tex]\frac{2}{5}[/tex] hour / 7 tasks = [tex]\frac{2}{5} *\frac{1}{7}[/tex] = [tex]\frac{2}{35}[/tex] hour or about 3.43 minutes

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The polynomial P(x) must have a degree of 4.

This is because a polynomial with real coefficients must have complex zeros in conjugate pairs. This means that if 1+i is a zero of the polynomial, then its conjugate, 1-i, must also be a zero. Similarly, if 2-i is a zero, then its conjugate, 2+i, must also be a zero. Therefore, the polynomial P(x) must have zeros at 1, 1+i, 1-i, 2-i, and 2+i. Since a polynomial's degree is equal to the number of its zeros, the polynomial must have a degree of 4.

In summary, a polynomial with real coefficients and zeros at 1, 1+i, and 2-i must have a degree of 4.

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According to the given information, the probabilites are a) 0.159, b)0.107, d) 0.893 e) 0.662 c) expected value is 12.192 f) standard deviation is 3.0121.

What is probability?

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

a) Using the binomial probability formula, we have:

P(X = 13) = (48)^13 * (0.254)^13 * (0.746)^35

= 0.159

So the probability that 13 out of 48 flowers will bloom early is 0.159.

b) To find the probability that fewer than 13 flowers will bloom early, we can find the cumulative probability up to 12:

P(X < 13) = P(X = 0) + P(X = 1) + ... + P(X = 12)

= ∑(48 choose k) * (0.254)^k * (0.746)^(48-k) from k=0 to 12

= 0.107

So the probability that fewer than 13 flowers will bloom early is 0.107.

c) The expected value of a binomial distribution is given by n*p, where n is the number of trials and p is the probability of success. In this case, we have:

E(X) = 48 * 0.254

= 12.192

So on average, we expect to observe about 12.192 early blooming flowers.

d) To find the probability that at least 13 flowers will bloom early, we can use the complement rule and find the probability that 12 or fewer flowers will bloom early, and subtract that from 1:

P(X ≥ 13) = 1 - P(X < 13)

= 1 - 0.107

= 0.893

So the probability that at least 13 flowers will bloom early is 0.893.

e) To find the probability that between 9 and 14 flowers (inclusive) will bloom early, we can find the cumulative probability from 9 to 14:

P(9 ≤ X ≤ 14) = P(X = 9) + P(X = 10) + ... + P(X = 14)

= ∑(48 choose k) * (0.254)^k * (0.746)^(48-k) from k=9 to 14

= 0.662

So the probability that between 9 and 14 flowers (inclusive) will bloom early is 0.662.

f) The variance of a binomial distribution is given by np(1-p), and the standard deviation is the square root of the variance. In this case, we have:

Var(X) = 48 * 0.254 * (1-0.254)

= 9.078

SD(X) = sqrt(Var(X))

= sqrt(9.078)

= 3.0121 (rounded to 4 decimal places)

So the standard deviation of the number of flowers that bloom early in a row of 48 flowers is approximately 3.0121.

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The smallest possible value of x is 11.

To find the smallest possible value of x, we can use the formula x = 6n + 5, where n is an integer. This formula represents the fact that x has a remainder of 5 when divided by 6.

If we plug in different values for n, we can find the smallest possible value of x that is greater than 10.

When n = 1, x = 6(1) + 5 = 11
When n = 2, x = 6(2) + 5 = 17
When n = 3, x = 6(3) + 5 = 23

Therefore, the smallest possible value of x that is greater than 10 is 11, so the answer is 11.

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The solution to the all six equations is that they have infinite many real solutions

Expression (a)

We have:

√(x^2-14x+49)=x-7

Squaring both sides we get:

x^2 - 14x + 49 = x^2 - 14x + 49

Evaluate the like terms

0 = 0

This means that the equation has infinite many solutions

Expression (b)

Here, we have:

√(4x^2-20x+25)=5-2x

Squaring both sides we get:

4x^2-20x+25 = 25 - 20x + 4x^2

Evaluate the like terms

0 = 0

This means that the equation has infinite many solutions

For the remaining expressions, we have the following (using the above steps)

Expression (c)

√(y^4+2y^2+1) = y^2 + 1

y^4 + 2y^2 + 1 = y^4 + 2y^2 + 1

0 = 0

Expression (d)

√(x^2+2x+1)=x+1

x^2 + 2x + 1 = x^2 + 2x + 1

0 = 0

Expression (e)

√(y^2-20y+100)=y-10

y^2 - 20y + 100 = y^2 - 20y + 100

0 = 0

Expression (f)

√(y^6-2y^3+1)=y^3-1

y^6-2y^3+1 = y^6-2y^3+1

0 = 0

Hence, the equations have infinite solutions

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Answer: 6 and 12

Step-by-step explanation:

Answer:

54w²

Step-by-step explanation:

42w² + 15w²-3w²

42w²+ 15w²= 57w²

57w²-3w²= 54w²

If a car is going 100 miles per hour, the car goes 100 miles in one hour.

Answer:36 seconds

Step-by-step explanation:

100 miles per hour, it would take me approximately 36 seconds to travel 1 mile.

Using Pythagoras theorem, the size of the TV will be 65 inches.

What is Pythagorean Theorem?

The Pythagorean Theorem is a fundamental concept in mathematics that describes the relationship between the sides of a right triangle. It states that in any right triangle, the sum of the squares of the lengths of the two shorter sides (the legs) is equal to the square of the length of the longest side (the hypotenuse).

In equation form, the Pythagorean Theorem can be written as:

a² + b² = c²

where a and b are the lengths of the two legs and c is the hypotenuse.

Now,

We can use the Pythagorean theorem to find the diagonal measure of the television screen.

In this case, the height and width of the screen form the legs of a right triangle, so we can use the following equation:

(diagonal)² = (height)² + (width)²

Substituting the given values, we get:

(diagonal)² = (32)² + (57)²

(diagonal)² = 1024 + 3249

(diagonal)² = 4273

Taking the square root of both sides

diagonal = √(4273) ≈ 65.37

Therefore, the size of the TV screen, as measured diagonally, is approximately 65 inches (rounded to the nearest whole number).

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Therefore , the solution of the given problem of function comes out to be the function that fits the given graph is f(x) = [x] + 1.

In addition to numbers, symbols, but also their constituent expression parts, anatomy, construction, and both real and fictitious geographic places are all covered in the study of mathematics. A function explains the connections between various variables, which all have a related outcome. A function is composed of a number of unique parts that, when combined, result in particular outputs for each input.

Here,

The largest number that is less than or equal to x is returned by the greatest integer function, denoted by [x]. For instance, [3.4] = 3, [-2.8] = 3, and [5] = 5 are all equal.

The curve of the largest integer function is moved up one unit by the function f(x) = [x] + 1.

The graph of the largest integer function is moved up by zero units by the function f(x) = [x].

The largest integer function's graph is reflected and moved down by one unit by the function f(x) = -[x] - 1.

The curve of the largest integer function is moved down by one unit by the function f(x) = [x] - 1.

Therefore, the function that fits the given graph is f(x) = [x] + 1.

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1.) The total number of ways in which the books can be arranged so that mathematics and physics cannot touch = 478,961,280.

2.) The total number of ways the books can be arranged when math and physics books touch = 479,001,600

Permutation is defined as the expression that shows how objects can be arranged in a definite order.

The total number of mathematics books = 5

The total number of astronomy books = 4

The total number of physics = 3

The total number of books that the student has in possession = 5+4+3 = 12.

The arrangement of the books so that math and physics cannot touch = 12!-8!

= 479001600 - 40320

= 478,961,280

The arrangement of the books so that math and physics can touch = 12! = 479,001,600

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The range of possible side lengths for the third side is 15 yd < x < ∞.

According to the triangle inequality, the lengths of any two sides of any triangle must add up to at least the length of the third side.

We can use the triangle inequality theorem to determine the range of possible side lengths for the third side of each triangle. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

10.5 km and 11 km:

Let's call the length of the third side x. Applying the triangle inequality theorem, we get:

5 + x > 11

10 + x > 5

Simplifying, we get:

x > 6

x > -5

Therefore, the range of possible side lengths for the third side is 6 km < x < ∞.

12 mi and 12 mi:

Let's call the length of the third side x. Applying the triangle inequality theorem, we get:

12 + x > 12

12 + x > 12

Simplifying, we get:

x > 0

x > 0

Therefore, the range of possible side lengths for the third side is 0 mi < x < ∞.

25 yd and 10 yd:

Let's call the length of the third side x. Applying the triangle inequality theorem, we get:

10 + x > 25

25 + x > 10

Simplifying, we get:

x > 15

x > -15

Therefore, the range of possible side lengths for the third side is 15 yd < x < ∞.

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