Are 37 and 65 71 82 are prime or composite number

Final answer:

To find the x-coordinates where the function f(x)=6sin(x)-sin^2(x) has a horizontal tangent line, we calculate the derivative, set it equal to zero, and solve for x, using trigonometric identities and potentially numerical methods. Thus, the solutions for \( x \) in the interval [tex]\( [0, 2\pi] \) are \( x = \frac{\pi}{2} + k\pi \), where \( k \)[/tex] is an integer. These solutions represent the points where [tex]\( \cos(x) = 0 \)[/tex], causing the equation to be satisfied.

Explanation:

The question asks to find the x-coordinates of all points on the graph of the function f(x)=6sin(x)-sin2(x) at which the tangent line is horizontal. To find these coordinates, we need to look for points where the derivative of the function is equal to zero since a horizontal tangent line has a slope of zero.

We start by finding the derivative of the function:

f'(x) = d/dx[6sin(x)-sin2(x)]

f'(x) = 6cos(x)-2sin(x)cos(x)

f'(x) = 6cos(x)-sin(2x) (using the double angle identity sin(2x)=2sin(x)cos(x))

To find when the derivative is zero, we set the derivative equal to zero:

0 = 6cos(x) - sin(2x)

We then need to solve this equation for x. This involves using trigonometric identities and potentially numerical methods if the equation cannot be solved analytically. The x-coordinates will be the solutions to this equation within the domain for which we are interested, typically 0 to 2π for one full cycle of a trigonometric function.

To solve the equation [tex]\( 0 = 6\cos(x) - \sin(2x) \)[/tex], we'll first try to rewrite it in terms of a single trigonometric function using known identities.

The double angle identity for sine states that [tex]\( \sin(2x) = 2\sin(x)\cos(x) \)[/tex]. Substituting this identity into the equation, we get:

[tex]\[ 0 = 6\cos(x) - 2\sin(x)\cos(x) \]\\Now, let's factor out \( \cos(x) \):\[ 0 = \cos(x)(6 - 2\sin(x)) \][/tex]

Now, for the equation to be true, either [tex]\( \cos(x) = 0 \) or \( 6 - 2\sin(x) = 0 \)[/tex].

1. If [tex]\( \cos(x) = 0 \), then \( x = \frac{\pi}{2} + k\pi \)[/tex], where \( k \) is an integer.

2. If [tex]\( 6 - 2\sin(x) = 0 \), then \( \sin(x) = 3 \)[/tex]. However, the sine function has a range of \([-1, 1]\), so there are no solutions for \( x \) in this case.

Thus, the solutions for \( x \) in the interval [tex]\( [0, 2\pi] \) are \( x = \frac{\pi}{2} + k\pi \), where \( k \)[/tex] is an integer. These solutions represent the points where [tex]\( \cos(x) = 0 \)[/tex], causing the equation to be satisfied.

The dilation of a rectangle with a scale factor of 4 will result in each side of the rectangle being multiplied by 4, thus the perimeter will also be quadrupled.

When a rectangle undergoes a dilation with a scale factor of 4, the lengths of all sides of the rectangle are multiplied by 4. Since perimeter is the sum of all the side lengths of a rectangle, the perimeter will also be affected by the scale factor. Therefore, if the original perimeter of the rectangle is P, the new perimeter after dilation will be 4 × P.

Using the property of dilations, the new dimensions of the rectangle will be proportional to the original dimensions. For example, if the original length l and width w of the rectangle are scaled by a factor of 4, the new dimensions will be 4l and 4w respectively. Subsequently, the new perimeter will be 2(4l + 4w) which is the same as saying 4 × (2l + 2w), proving the perimeter is scaled by the same factor of 4.

Answer:

commutative property of addition

Step-by-step explanation:

In the context of the number 1,711,799, the first 7 underlined is in the hundred thousands position, which gives it a value of seven hundred thousand (700,000).

In the given number 1,711,799, we're asked to find the value of the first underlined 7. Considering the location of the number, 7 is in the hundred thousands position. The value of a digit depends on its place in the number. Therefore, despite the fact that the digit is '7', because it is in the hundred thousands place, its value is seven hundred thousand (700,000).

https://brainly.com/question/27734142

#SPJ2

Answer:

The widest range lies between [68,100]

Step-by-step explanation:

Given is - to receive a grade of B in the class you must average exam score between 80% and 89% for all 4 exams including the final.

So, the range of the addition of all 4 scores is -

[tex]80*4=320[/tex]

[tex]89*4=356[/tex]

Now given scores addition is = [tex]83+85+84=252[/tex]

Now, the widest range of scores that you can get on the final exam in order to receive a grade of B can be found as :

[tex]320-252= 68[/tex]

[tex]356-252= 104[/tex]

So, the range lies between [68,100]

The function t(n)=8n represents the number of tires needed for each truck. By substituting n=25 into the function, we find we need 200 tires for 25 trucks.

The function that represents the number of tires needed for each truck is t(n) = 8n. This means for each truck we need 8 tires, as n is the number of trucks and 8n signifies the total number of tires needed for those trucks.

If we want to find out the number of tires we would need for 25 trucks, we substitute n = 25 into the function, yielding t(25) = 8 * 25. When you perform this multiplication, the result is 200 tires.

So, for 25 trucks we would need 200 tires. Meaning "the correct answer is C. 200 tires".

https://brainly.com/question/24875240

#SPJ2

The rate of the boat that travels 66 miles in 2 hours is approximately 2904 feet per minute.

Given:

Distance traveled = 66 miles

Time taken = 2 hours

We know that:

1 mile =  5280 feet

66 miles = 66 [tex]\times[/tex] 5280

                  348,480 feet

Next, we'll convert the time from hours to minutes.

 1 hour = 60 minutes,

2 hours = 120 minutes

Now, we can calculate the rate in feet per minute:

Rate (feet per minute) =  [tex]\dfrac{ Distance \ in\ feet}{Time\ in\ minutes}[/tex]

                    Rate             =        [tex]\dfrac{348480}{120}[/tex]

Rate   ≈ 2904 feet per minute

So, the rate of the boat is approximately 2904 feet per minute.

Learn more about Distance here :

https://brainly.com/question/15172156

#SPJ4

Answer:

First of all, each slice represents an intersecting plane at that level.

For example, y = 2 is a plane that passes thorugh that level and cuts the volume given by [tex]x^{2} -y^{2}+z^{2}=2[/tex]

We replace this value in the given volume.

[tex]x^{2} -y^{2}+z^{2}=2\\x^{2} -(2)^{2}+z^{2}=2\\x^{2}+z^{2}=2+4\\x^{2}+z^{2}=6[/tex]

So, results in a circumference with radius [tex]\sqrt{6}[/tex], because a circumference is defind as [tex]x^{2} +y^{2} =r^{2}[/tex]. (On plane XY).

We repeat the process.

[tex](1)^{2} -y^{2}+z^{2}=2\\z^{2}-y^{2}=2-1\\z^{2}-y^{2}=1[/tex]

It forms a horizontal hyperbola on plane ZY.

[tex]x^{2} -0^{2}+z^{2}=2\\x^{2}+z^{2}=2[/tex]

Another circle with radius of [tex]\sqrt{2}[/tex] on plane XZ.

[tex](2)^{2} -y^{2}+z^{2}=2\\z^{2}-y^{2}=2-4\\z^{2}-y^{2}=-2\\\frac{z^{2}-y^{2}}{-2} =\frac{-2}{-2} \\\frac{y^{2} }{2} -\frac{z^{2} }{2} =1[/tex]

It forms a hyporbola on plane YZ.

1.a. -19 at x = -4 for f(x) = 5x + 1.

2.a.  8x + y = -18 the standard form of the equation.

A line has no definite length it can extend in both the directions indefinitely.

According to the given question a functions f(x) = 5x + 1 is given and we have to find the value of this function when x = -4.

Given f(x) = 5x + 1

f(-4) = 5(-4) + 1

f(-4) = -20 + 1

f(-4) = -19.

In the second question we have to write an equation of the line passing through the point (-2,-2) with slope -8.

We know equation of a line in slope intercept form is y = mx + b.

∴ -2 = (-8)(-2) + b

  -2 = 16 + b

   b = - 18.

So, y = -8x -18 OR 8x + y = -18.

learn more about lines here :

https://brainly.com/question/12791368

#SPJ2

To write the expression 4 log x − 6 log (x + 2) as a single logarithm, apply the properties of logarithms.

To write the expression 4 log x − 6 log (x + 2) as a single logarithm, we can make use of the following logarithmic properties:

Using these properties, we can simplify the given expression as follows:

4 log x − 6 log (x + 2) = log(x^4) - log((x + 2)^6)

Therefore, the expression can be written as the single logarithm log(x^4) - log((x + 2)^6).

The inequality would be equal to n  ≥ 7.

If the equation or inequality contains variable terms, then there might be some values of those variables for which that equation or inequality might be true. Such values are called solution to that equation or inequality. Set of such values is called solution set to the considered equation or inequality.

We are given that the phrase as;

"A number n is at least 7"

Therefore, the inequality would be;

n  ≥ 7

Or

n = 7

Thus, the inequality would be equal to n  ≥ 7.

Learn more about inequalities here:

https://brainly.com/question/27425770

#SPJ2

Final answer:

Bobby can buy 5 cups of coffee at $2.25 each and 3 cups of hot chocolate at $1.50 each to meet his budget of $15.75 for 8 cups in total. We find this by solving a system of equations.

Explanation:

Bobby wants to buy a total of 8 cups of coffee and hot chocolate for his co-workers with a budget of $15.75. Since each cup of coffee costs $2.25 and each cup of hot chocolate costs $1.50, we need to find out how many cups of each he can buy.

Let's denote the number of cups of coffee as C and the number of cups of hot chocolate as H.

The total cost for coffee will be $2.25 × C and for hot chocolate $1.50 × H.

The total amount spent is $15.75, and the total number of cups is 8, so we have two equations: 2.25C + 1.50H = 15.75 and C + H = 8.

We can solve this system of equations using substitution or elimination. For simplicity, we'll use substitution.

From C + H = 8, we can express H in terms of C: H = 8 - C.

Substitute H in the cost equation: 2.25C + 1.50(8 - C) = 15.75.

Simplify and solve for C: 2.25C + 12 - 1.50C = 15.75, which simplifies to 0.75C = 3.75, and therefore C = 5.

Now, plug C = 5 into H = 8 - C to find H: H = 3.

Bobby can buy 5 cups of coffee and 3 cups of hot chocolate.

The ratio 60 cents: $2 simplifies to 3:10 after converting dollars to cents and dividing both terms by the greatest common divisor, which is 20.

To simplify the ratio 60 cents: $2, we need to first express both quantities in the same units. Since 1 dollar is equal to 100 cents, $2 is equivalent to 200 cents. Thus, the ratio becomes 60 cents : 200 cents.

Next, we simplify the ratio by dividing both sides by the greatest common divisor. In this case, both 60 and 200 are divisible by 20, which is the greatest common divisor. So we divide both sides by 20:

[tex]\frac{60}{20}[/tex] = 3

[tex]\frac{200}{20}[/tex] = 10

Therefore, the simplified ratio is 3:10.

Answer:

The scientific notation 2.15 × 10-3 has a value of

0.00215

Step-by-step explanation:

Step one

This problem bothers conversion from standard form otherwise known as scientific notation to decimal form

Step two

Now we are presented with the expression

2.15 × 10-³

To convert this it is equivalent to

2.15/10³= 2.15/1000

= 0.00215

On the other hand this can also be achieved without calculations by visually counting from the decimal point to the right 2 placements before inserting the first integer

This method saves alot of time expecialy when time is of essence.

Hence the scientific notation 2.15 × 10-3 has a value of

0.00215