How many kilometers could the red car travel in 12 hours?

The probability of rolling a 6 exactly twice in 8 rolls of a fair die is approximately 33.5%.

The situation describes a binomial distribution where we have:

The binomial probability formula is given by:

[tex]P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k}[/tex]

First, calculate (n choose k) which is given by:

[tex]\binom{n}{k}= n! / [k! \times (n-k)!][/tex]

For our problem:

[tex]\binom{n}{k} = 8! / [2! \times (8-2)!] = 28[/tex]

Next, calculate the probability:

[tex]P(X = 2) = 28 \times (1/6)^2 \times (5/6)^6[/tex]

Evaluating the above:

P(X = 2) = [tex]28 \times (1/36) \times (15625/46656)[/tex] ≈ 0.335

Therefore, the probability that the die comes up a 6 exactly twice in 8 rolls is approximately 0.335 or 33.5%.

To prove that [tex]f(x) = x^3 - 1000x^2 + x - 1[/tex] is [tex]\omega(x^3)[/tex] and [tex]o(x^3)[/tex], we must show how the function grows in comparison to x^3 asymptotically.

To prove that [tex]f(x) = x^3 - 1000x^2 + x - 1[/tex] is [tex]\omega(x^3)[/tex] and [tex]o(x^3)[/tex], we need to show two things:

1. Proving that [tex]f(x) = x^3 - 1000x^2 + x - 1[/tex] is [tex]\omega(x^3)[/tex]

For a function to be [tex]\omega(x^3)[/tex], it needs to grow faster than [tex]x^3[/tex] asymptotically. This requires that the ratio of the function to [tex]x^3[/tex] tends towards infinity as x approaches infinity.

Consider:

[tex]\lim_{x \to \infty}[\frac{f(x)}{x^3} ] = \lim_{x \to \infty}[ \frac{x^3-1000x^2+x-1}{x^3} ][/tex]

Simplify the expression:

[tex]\lim_{x \to \infty}[ 1-\frac{1000}{x}+\frac{1}{x^2}-\frac{1}{x^3} ] =1[/tex]

Since the limit does not tend to infinity but instead tends to 1, f(x) is not ω(x^3). We made a mistake in our earlier assumption; let's correct this in the next point.

2. Proving that [tex]f(x) = x^3 - 1000x^2 + x - 1[/tex] is [tex]o(x^3)[/tex]

To show that f(x) is [tex]o(x^3)[/tex], the ratio of f(x) to [tex]x^3[/tex] should tend to zero as x tends to infinity.

Consider:

[tex]\lim_{x \to \infty} [\frac{f(x)}{x^3}] = \lim_{x \to \infty}[\frac{x^3-1000x^2+x-1}{x^3}][/tex]

Simplify the expression:

[tex]\lim_{x \to \infty}[ 1-\frac{1000}{x}+\frac{1}{x^2}-\frac{1}{x^3} ] =1[/tex]

This indicates that the limit tends to 1 and not 0, hence f(x) is not [tex]o(x^3)[/tex] either. We can see in both cases that the limits did not meet the required criteria for [tex]\omega(x^3)[/tex] or [tex]o(x^3)[/tex], implying a misunderstanding in the problem setup.

Answer:

51 hope this helps!!!!!!!!!!

Step-by-step explanation:

Final answer:

The nth term for the given sequence 1, 7, 15, 25, 37 is derived to be n² + n - 1, which fits the pattern of the sequence accurately.

Explanation:

The sequence given is 1, 7, 15, 25, 37. To find the nth term of this sequence, let's first identify the pattern of differences between the terms. Observing the differences, we see they are 6, 8, 10, and 12, which suggests an arithmetic sequence in the differences. This indicates the original sequence is quadratic.

Using the general formula for the nth term of a quadratic sequence, An² + Bn + C, we can substitute the first few terms to solve for A, B, and C. However, there's a quicker method given the pattern of differences seen in the second layer (6, 8, 10, 12, ...) that increases by 2 each time, suggesting that 2n is involved.

Through analysis, the nth-term formula can be derived as n² + n - 1.

Here's why:

considering n=1 for the first term, we get 1+1-1=1;

for n=2, the formula yields 2^2+2-1=7; and so forth, matching the given sequence precisely.

The answer is f(n) = 60 + 10n.

The composite function rule (also called the chain rule) Has taken a look at the characteristic f(x)=(x2 + 1)17. we will think about this function as being. the end result of mixing features. If g(x) = x2 + 1 and h(t) = t17 then the result of.

A function rule along with value = p + zero.08p is an equation that describes a useful relationship. If p is the charge you pay for an object and 0.08 is the income tax, the feature rule above is the value of the item. in case you are given a desk, usually, you need to cautiously study the desk to peer what the function rule is.

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Explanation on finding distance from a point to the origin as a function of x and showing invariance under rotations.

Distance from point P to the origin as a function of x:

Given point P(x, y) on y = x² – 9.

The distance d from P to the origin is d = √(x² + y²).

Substitute y = x² - 9 into the distance formula to express d as a function of x: d = √(x² + (x² - 9)²).

Invariance of distance under rotations:

The distance from P to the origin remains constant regardless of the rotation, as it depends only on the coordinates of the point and not the orientation of the coordinate system.

For a real number a, a + 0 = a.  TRUE

For a real number a, a + (-a) = 1.  FALSE

For a real numbers a and b, | a - b | = | b - a |.  TRUE

For real numbers a, b, and c, a + (b ∙ c) = (a + b)(a + c).  FALSE

For rational numbers a and b when b ≠ 0, is always a rational number. TRUE

This comes from the identity property for addition that tells us that zero added to any number is the number itself. So the number in this case is [tex]a[/tex], so it is true that:

[tex]a+0=a[/tex]

This is false, because:

[tex]a+(-a)=a-a=0[/tex]

For any number [tex]a[/tex] there exists a number [tex]-a[/tex] such that [tex]a+(-a)=0[/tex]

This is a property of absolute value. The absolute value means remove the negative for the number, so it is true that:

[tex]\mid a-b \mid= \mid b-a \mid[/tex]

This is false. By using distributive property we get that:

[tex](a + b)(a + c)=a^2+ac+ab+bc \\ \\ a^2+ab+ac+bc \neq a+(b.c)[/tex]

A rational number is a number made by two integers and written in the form:

[tex]\frac{u}{v} \\ \\ v \neq 0[/tex]

Given that [tex]a \ and \ b[/tex] are rational, then the result of dividing them is also a rational number.

Answer:

A) True

B) False

C) True

D) False

E) True

Step-by-step explanation:

We are given the following statements in the question:

A) True

For  every real number, a, a + 0 = a. 0 is known as the additive identity.

B) False

For a real number a, a + (-a) = 0.

C) True

For a real numbers a and b, [tex]|a-b| = |b-a|[/tex]

D) False

For real numbers a, b, and c, a + (b ∙ c) = (a + b)(a + c).

Counter example: For a = 2, b =  1, c = 3

[tex]a + (b.c) = (a + b)(a + c)\\2 + (1.3) \neq (2+1)(2+3)\\5\neq 15[/tex]

E) True

For rational numbers a and b, b is not equal to zero, [tex]\frac{a}{b}[/tex] is always a rational number.

The unknown numbers are 11.5 and 11.5

Let the unknown numbers be x and y

If the sum of the numbers is 23, hence;

x +  y = 23

x = 23 - y ............. 1

If the product is at maximum, hence;

xy = maximum ......... 2

Substitute equation 1 into 2

(23 - y )y = maximum

23y - y² = maximum

maximum = -y² + 23y

A(y) =  -y² + 23y

Since the product A(y) is at maximum, hence dA(y)/dy = 0 as shown:

dA(y)/dy= -2y + 23 = 0

-2y + 23 = 0

2y = 23

y = 23/2

y = 11.5

Since x + y = 23

x = 23 - y

x = 23 - 11.5

x = 11.5

Hence the unknown numbers are 11.5 and 11.5

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To find out how much stock is needed to make 4 1/2 times a recipe that calls for 4 2/3 cups, multiply 4.67 by 4.5 to get 21.015 cups of stock.

To determine how much stock will be needed if 4 1/2 times the recipe is made, we first need to multiply the amount of stock in the original recipe by 4 1/2. The original recipe calls for 4 2/3 cups of stock. To calculate the total amount required, we convert 2/3 to a decimal to make multiplication easier. Thus, 2/3 is equivalent to approximately 0.67.

Next, we multiply 4.67 (4 + 0.67 for the 2/3 cup) by 4.5 (4 1/2).

4.67 x 4.5 = 21.015 cups. Therefore, 21.015 cups of stock will be needed to make 4 1/2 times the recipe.

I believe it would be

784 miles/40 gallons = 19.6 miles per gallon

The dimensions that give the minimum surface area are a radius of approximately 2.83 cm and a height of approximately 0.64 cm.

To find the dimensions of the closed cylinder that give the minimum surface area, we need to express the surface area of the cylinder in terms of one variable.

Let's use the radius, r, as the variable.

The surface area of a closed cylinder is given by the formula: A = 2πr² + 2πrh, where h is the height of the cylinder.

Since we are given that the volume of the cylinder is 40 cm³, we can use the formula for the volume of a cylinder to express h in terms of r: V = πr²h = 40 cm³.

Substituting this expression for h in terms of r into the surface area formula, we get:

A = 2πr² + 2πr(40 / (πr²)) = 2πr² + (80 / r).

To find the dimensions that give the minimum surface area, we need to find the value of r that minimizes A.

To do this, we take the derivative of A with respect to r and set it equal to zero:

A' = 4πr - (80 / r²) = 0. Solving this equation, we find that r = √(20/π) ≈ 2.83 cm.

So, the radius that gives the minimum surface area is approximately 2.83 cm.

We can then use the volume formula to find the corresponding height: h = 40 / (π(2.83)²) ≈ 0.64 cm.

Therefore, the dimensions giving the minimum surface area are a radius of approximately 2.83 cm and a height of approximately 0.64 cm.

Final answer:

The trigonometric equation is solved using the sine difference identity, simplifying to sin(2x) = √2 sin(x). The solution involves finding values of x that satisfy the equation over the interval [0,2π). Trigonometric identities such as double angle formulas are essential in the process.

Explanation:

The student has presented a trigonometric equation involving sine and cosine functions to solve: sin(4x)cos(2x) - cos(4x)sin(2x) = √2 sin(x) over the interval [0,2π). This can be addressed by recognizing the left-hand side as the expansion of the sine difference identity: sin(A - B) = sin(A)cos(B) - cos(A)sin(B), where A = 4x and B = 2x. Therefore, the equation simplifies to sin(2x) = √2 sin(x). We solve this equation over the specified interval by looking for values of x that satisfy the condition.

However, none of the reference equations or principles provided directly align with solving the original question. Hence, we must rely solely on our knowledge of trigonometric identities, such as the double angle formulas which are relevant to this problem. The double angle identities state that sin(2θ) = 2sin(θ)cos(θ) and cos(2θ) = cos^2(θ) - sin^2(θ) or equivalently cos(2θ) = 2cos^2(θ) - 1 and cos(2θ) = 1 - 2sin^2(θ).

Answer: Parallel.

Step-by-step explanation:

Given: Lines a and c intersect at point S, creating 4 angles.

To Prove: Corresponding angles are congruent.

We are given that lines a and c intersect at point S.

Translate line a down line c until point S reaches point Q.

Call the new line through Q line b.

We know that translations preserve orientation then every point on line b is equidistant from every points on line a.

Therefore,  lines a and b are parallel by the definition of parallel lines.

Hence, the complete statement is Because translations preserve orientation, lines a and b are parallel.

40 hours regular pay

8 hours at 1.5 = 8*1.5 = 12

40+12 = 52

1248/52 = 24

A) $24 per hour

B) 14*1.5 = $36 per hour for overtime

a)  

           His hourly rate= $ 24

b)

         His hourly overtime rate= $ 36

Jim worked 40 regular hours last week, plus 8 overtime hours at the time and a half rate.

This means if he is paid $ x per hour working in regular hours.

Then he will be paid $ (1.5x) per hour working overtime.

( Since it is given that: he is paid time and a half rate for overtime)

Now, his gross pay is: $ 1248

i.e.

Amount working at regular hours for 40 hours+ Amount obtained for working 8 hours overtime= $ 1248

i.e.

40x+8(1.5x)=1248

i.e.

40x+12x=1248

i.e.

52x=1248

i.e.

x=24

a)

Hence, his hourly rate is:$ 24

b)

His hourly overtime rate is: $ (24×1.5)

i.e.

His hourly overtime rate is:  $ 36

The area of the shaded figure is (x⁴ + 19x² + 100) square meters after subtracting the area of the smaller square from the area of the larger square.

It is defined as a two-dimensional geometry that has four sides and four vertices. The sides of the square are equal in length. It is a regular quadrilateral.

It is given that:

Large square side length: (x squared plus 10)

Small square side length: x

The area of the large square = (x² + 10)(x² + 10)

The area of the large square = (x² + 10)²

The area of the small square = (x)(x)

The area of the small square = x²

The area of the shaded figure = (x² + 10)² - x²

The area of the shaded figure = (x⁴ + 19x² + 100) square meters

Thus, the area of the shaded figure is (x⁴ + 19x² + 100) square meters after subtracting the area of the smaller square from the area of the larger square.

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