The senior class has 412 students. They are assigned to different homerooms. There are 28 students in the smallest homeroom and the remaining 12 homerooms have the same number of students. How many students are in each of the remaining 12 homerooms?

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).

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.

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.

The slices of watermelon each person will get is  1.5 slices.

If the watermelon is divided equally, the number of slices each person would get can be determined by dividing the total slices of watermelon by the total number of friends.

Slice each person would get = total slices / total number of friends

6 / 4 = 1.5 slices

A similar question was answered here: https://brainly.com/question/11636279

I believe it would be

784 miles/40 gallons = 19.6 miles per gallon

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.

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.

Learn more about the square here:

https://brainly.com/question/14198272

#SPJ2

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

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.

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:

(C) is one more than the power

Step-by-step explanation:

Let us consider a binomial expansion:

[tex](x+y)^2=x^2+2xy+y^2[/tex]

The power of the above expansion is [tex]2[/tex] and the number of the terms in the above binomial expansion are [tex]3[/tex], thus the total number of terms in the binomial expansion is one more than the power.

Hence, Option C is correct.

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.

Learn more about the Function Rule here: https://brainly.com/question/1528097

#SPJ2

The Java Rectangle class implements both the Shape and Serializable interfaces. Therefore, Rectangle r can be legally assigned to Rectangle, Shape, Serializable, and Object data types. It can't be assigned to a String, Measurable, and ActionListener as Rectangle doesn't implement these.

In Java, the Rectangle class implements the Shape and Serializable interfaces. Therefore, the legal assignments would be:

The rest assignments are not legal because the Rectangle class doesn't implement Measurable, ActionListener interfaces and it cannot be assigned to a String.

https://brainly.com/question/14868991

#SPJ6