Which of the following is an identity?
A. csc²x + cot²x = 1
B. (cscx + cotx)2 = 1
C. sin²x-cos²x = 1
D. sin²x sec²x + 1 = tan²x csc²x

The ratio of tablespoons of raspberries to tablespoons of sugar should be considered as the 8 to 6.

What is ratio?

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other. For example, if there is 1 boy and 3 girls you could write the ratio as: 1 : 3 (for every one boy there are 3 girls) 1 / 4 are boys and 3 / 4 are girls.

Calculation of the ratio:

Since Macy's aunt makes raspberry jam using 6 tablespoons of sugar and 8 tablespoons of raspberries.

So here the ration be like

= 8 raspberries tablespoons to 6 tablespoons of sugar

= 8 to 6

hence, the second option is correct.

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Answer:

1) 60

2)  15

3) 45

4) 105

5) 75

Step-by-step explanation:

7y + 5y = 180  A straight angle is 180  Combine like terms

12y = 180  Divide both sides by 12

y = 15

The sum of the interior angles is 180

y = 15

180 - (5y + 3y) = x

180 -(5(15) + 3(15)) = x

180 - (75 + 45) = x

180 - 120 = x

60 = x

Angle c

3y

3(15)

45

Angle DBC

7y

7(15)

105

Angle CBA

5y

5(15)

75

Kinetic energy of Usain is 4700kg-m²/s².

Kinetic energy of Usain's brother is 18800kg-m²/s².

If we double the mass kinetic energy get doubled.

What is kinetic energy ?

Kinetic energy of an object is the energy that it possess due to its motion.

It is calculated as  [tex]\frac{1}{2}mv^{2}[/tex] where m is mass and v is velocity.

In Usain's case m = 94kg and v = 10m/s.

So, kinetic energy = 4700kg-m²/s².

In Usain's brother case mass is same but velocity is doubled.

So, kinetic energy = 18800kg-m²/s².

From the formula we can see ,

if mass is doubled then kinetic energy get doubled.

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The perimeter of smaller circle is 78.5 ft

What is area?

Area is that the amount that expresses the extent of a vicinity on the plane or on a semicircular surface. space|the world|the realm} of a plane region or plane area refers to the world of a form or planate plate, whereas area refers to the world of associate degree open surface or the boundary of a three-dimensional object.

Main Body:

A new arboretum was designed to contain two circular gardens. Garden A has a diameter of 25 ft and Garden B has a diameter of 45 ft.

outline the smaller garden means circumference of the circle.

Perimeter of circle is 2πr.

diameter = 25 ft

radius = 25/2 = 12.5 ft

Radius = 2π *12.5

            = 25 π

            = 25*3.14

            = 78.5 ft

Hence the perimeter of circle is 78.5 ft

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Answer:

-36

Step-by-step explanation:

-3[tex]a^{2}[/tex]b  Substitute -2 for a and 3 for b

-3[tex](-2)^{2}[/tex](3)

-3(4)(3)

-36

Answer:

= -36

Step-by-step explanation:

-3a²b

if:

a = -2

b = 3

Then:

= -3*-2²*3

= -3(-2*-2) * 3

= -3*4*3

= -36

The formula of the volume of the rectangular prism is,

V = l * w * h

The formula of the surface area of the rectangular prism is,

A = 2(lw + wh + lh)

the surface area of a box is 72 sq.ft.

For a rectangular prism with length(l), width(w) and height(h),

The formula of the volume of the rectangular prism is,

V = l * w * h

The formula of the surface area of the rectangular prism is,

A = 2(lw + wh + lh)

For a rectangular box with a length of 6 ft a width of 3 ft and a height of 2 ft,

here, l = 6 ft, w = 3 ft and h = 2 ft

using the formula for the surface area of a rectangular prism, the surface area of box would be,

A = 2(lw + wh + lh)

A = 2(6 * 3 + 3 * 2 + 6 * 2)

A = 2(18 + 6 + 12)

A = 2(36)

A = 72 sq.ft.

Therefore, the surface area of a box is 72 sq.ft.

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The system of equations y 2 3x 3 x= -2 has a solution.

A mathematical statement that has a "equal to" symbol between two expressions with equal values is called an equation. as in 3x + 5 Equals 15, for instance.

Equations come in a variety of forms, including linear, quadratic, cubic, etc.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides. We have LHS = RHS (left hand side = right hand side) in every mathematical equation.

To determine the value of an unknown variable that represents an unknown quantity, equations can be solved. The statement is not an equation if there is no "equal to" symbol in it.

According to our question-

y = -2*2/3+3

y= -4/3+3

now simplifying,

5/3

Hence, The system of equations y 2 3x 3 x= -2 has a solution.

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Complete question-

What is the solution to the system of equations y 2 3x 3 x?

The difference of the functions, f(x) = 3x² + 4x - 6 and g(x) = 6x³ - 5x² - 2, is: D. (f - g)(x) = -6x³ + 8x² + 4x - 4.

Finding the difference of two functions involves combining like terms together and then simplify.

Given the functions:

f(x) = 3x² + 4x - 6

g(x) = 6x³ - 5x² - 2

To find (f - g)(x), it implies that we find the difference between f(x) and g(x).

(f - g)(x) = f(x) - g(x)

Substitute

f(x) - g(x) = (3x² + 4x - 6) - (6x³ - 5x² - 2)

Open the parentheses:

f(x) - g(x) = 3x² + 4x - 6 - 6x³ + 5x² + 2

Combine like terms

f(x) - g(x) = - 6x³ + 8x² + 4x - 4

Therefore, the answer is: D. (f - g)(x) = -6x³ + 8x² + 4x - 4

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The statement that correctly compares -9 and 2 is A. -9 < 2 because 9 is to the left of 2 on a number line.

In arithmetic, a number line is a depiction of a graduated straight line which functions as a graphic representation of real numbers. It is presumed that every number line point and every real number correspond to one another.

Horizontal straight lines termed number lines have integers along them spaced equally apart. All of the numbers in a series can be shown on a number line. This line extends indefinitely at both ends.

In this case, -9 is on the left of a number line since it's negative. In conclusion, the correct option is A

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Answer:

  x = (π/2 +2nπ) ± π/3

Step-by-step explanation:

You want the solutions to the radical equation involving sin(x).

  [tex]\sqrt{\sin^2{x}-4\sin{x}+4}+\sqrt{\sin^2{x}+4\sin{x}+4}=\sqrt{\sin^2{x}+7\sin{x}+12.25}\\\\\sqrt{(\sin{x}-2)^2}+\sqrt{(\sin{x}+2)^2}=\sqrt{(\sin{x}+3.5)^2}\\\\|\sin{x}-2|+|\sin{x}+2|=|\sin{x}+3.5|\\\\(2-\sin{x})+(\sin{x}+2)=\sin{x}+3.5\\\\0.5=\sin{x}[/tex]

The values of x that make this true are ...

  x = {π/6, 5π/6} +2πn . . . . . for any integer n

__

Additional comments

  √(x²) = |x|

  sin(x) -2 is always less than zero, so its absolute value is 2-sin(x).

The attachment shows the equation written so the x-intercepts give the desired solutions.

The real number solution set for the trigonometric equation is x = 2πn + π/6 or 2πn + 5π/6 where n = is an integer

A trigonometric equation is an equation that contains the trigonometric ratios.

Given the trigonometric equation

√(sin²x - 4sinx + 4) + √(sin²x + 4sinx + 4) = √(sin²x + 7sinx + 12.25)

We know that sin²x - 4sinx + 4 = (sinx - 2)².

Also, sin²x + 4sinx + 4 = (sinx + 2)².

Also, sin²x + 7sinx + 12.25 = (sinx + 3.5)².

So, substituting these into the equation, we have that

√(sin²x - 4sinx + 4) + √(sin²x + 4sinx + 4) = √(sin²x + 7sinx + 12.25)

√(sinx - 2)² + √(sinx + 2)² = √(sinx + 3.5)²

|sinx - 2| + |sinx + 2| = |sinx + 3.5|

(2 - sinx) + (sinx + 2) = (sinx + 3.5)

2 - sinx + sinx + 2 = sinx + 3.5

Colecting like terms, we have that

-sinx + sinx + 2 + 2 = sinx + 3.5

0 + 4 = sinx + 3.5

4 = sinx + 3.5

Subtracting 3.5 from both sides, we have that

sinx = 4 - 3.5

sinx = 0.5

Taking inverse sin, we have that

x = sin⁻¹(0.5)

x = π/6 or 5π/6

So, the real number solution set is x = 2πn + π/6 or 2πn + 5π/6 where n = is an integer

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The area of the rhombus is 42 square units, and the correct answer is C.

A rhombus is one type of parallelogram and has 4 sides and 4 vertices.

Every side is equal in length and opposite angles are equal.

We have given that side of the rhombus is 7 units and height is 6 units.

To find the area of rhombus:

Remember that a rhombus is any quadrilateral that has four equal sides.

As a rhombus is also a parallelogram.

Formula to calculate area of rhombus using base and height;

Area of the rhombus = base x height

We have base = 7 units

And height = 6 units.

Substituting the values to the formula,

Area of the rhombus = 7 x 6

Area of the rhombus = 42 unit squared.

Therefore, the area is 42 square units.

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we get the expected number of farms in the sample whose land value per acre is between $1800 and $2600 is 59.

The Empirical Rule (or 3 sigma rule) states that for a normal distribution (bell shaped distribution) 68% of the data falls within one standard deviation (μ ± σ), 95% percent within two standard deviations (μ ± 2σ), and 99.7% within three standard deviations from the mean (μ ± 3σ).

Given that the mean (μ) = $2200, standard deviation (σ) = $400

a) The percentage of data within one standard deviation = μ ± σ = (2200 ± 400) = (1800, 2600)

Hence 68% of the land are between ​$1800 and ​$2600.

Number of farms = 68% × number of sample = 0.68 × 80 = 54.4 ≈ 54 farms

Hence we get the expected number of farms in the sample whose land value per acre is between $1800 and $2600 is 59.

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He will spend 5 minutes bicycling at 11mph and 40 minutes bicycling at 15mph.

From the information, the person is planning to ride his bicycle for 45 minutes every day. He burns 6 calories per minute bicycling at 11 mph and 9.25 calories per minute bicycling at 15 mph

This will be illustrated as:

x + y = 45 ..... i

6x + 9.25y = 400 ..... ii

From equation i, x = 45 - y ..... iii

x = number of minutes cycling at 11mph

y = number of minutes cycling at 15mph

Put equation iii into ii

6x + 9.25y = 400

6(45 - y) + 9.25y = 400

270 - 6y + 9.25y = 400

3.25y = 130

Divide

y = 40

Since x + y = 45.

x = 45 - 50

x = 5

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x=-[tex]\sqrt{3}[/tex] ,[tex]\sqrt{3}[/tex],-2,2

What is a quadratic equation ?

Quadratics can be defined as a polynomial equation of a second degree, which implies that it comprises a minimum of one term that is squared. It is also called quadratic equations. The general form of the quadratic equation is:

ax² + bx + c = 0

Given:

y=[tex]x^{4} -7x^{2} +12[/tex]

Let [tex]x^{2}[/tex] = k

So , y = [tex]k^{2}[/tex] -7k+12

On simplifying/factorising the given equation  ,we get

y =  [tex]k^{2}[/tex] - 4k - 3k +12

On taking common ,we get

y = k(k-4) -3(k-4)

y =(k-3)(k-4)

Thus , on factorising and equating it to zero we get

(k-3)=0 or (k-4)=0

k=3 or k=4

Now , replacing k by [tex]x^{2}[/tex] ,we get

[tex]x^{2}[/tex] = 3 or  [tex]x^{2}[/tex] = 4

Further on solving ,we get

( [tex]x^{2}[/tex]-3)=0 or ( [tex]x^{2}[/tex]-4)=0

(x-[tex]\sqrt{3}[/tex])(x+ [tex]\sqrt{3}[/tex]) =0  or (x+2)(x-2)=0

x=-[tex]\sqrt{3}[/tex] ,[tex]\sqrt{3}[/tex] or x= -2,2

Thus , x=-[tex]\sqrt{3}[/tex] ,[tex]\sqrt{3}[/tex],-2,2

Hence ,the obtained values of y are the zeros of x .

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The hypothesis test that can be used to determine whether the mean annual premium in Pennsylvania is lower than the national mean annual premium is between -2.064 and 2.064.

Hypothesis testing

The standard formula to calculate the hypothesis testing is given by the z test is

Z = (X – U) / (SD / √n)

Where:

X – Sample Mean

U – Population Mean

SD – Standard Deviation

n – Sample size

Given,

Here we have the the insure website reports that the mean annual premium for automobile insurance in the united states was $1,503 in march 2014. being from Pennsylvania at that time, you believed automobile insurance was cheaper there and decided to develop statistical support for your opinion. a sample of 25 automobile insurance policies from the state of Pennsylvania showed a mean annual premium of $1,430 with a standard deviation of is $165.

Here we need to develop the a hypothesis test that can be used to determine whether the mean annual premium in Pennsylvania is lower than the national mean annual premium.

While we looking into the given question we have identified the values of

Sample mean = 1430

Sample size = 25

Population mean = 1503

Standard deviation = 164

Then the hypothesis testing is calculated as,

=> Z = (1430 – 1503) / (164 / √25)

Therefore, if we reject the null hypothesis and accept the alternative hypothesis. The z score of -2.23 is within the rejection area.

And the 2 critical values (cutoff points) are -2.064 and 2.064. Since the z score, -2.23, is outside this interval, we reject the null hypothesis. We accept the alternative hypothesis.

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Answer:

B which I assume to be a typo where you had meant 3/4

Answer:

B. [tex]\frac{3}{4}[/tex]

Step-by-step explanation:

We need to find the HCF of 18 and 24 to simplify it into its lowest terms.

The HCF of 18 and 24 is 6, therefore we need to divide 6 on both sides to simplify it into its lowest terms.

Hence, [tex]\frac{18}{24}=\frac{3}{4}[/tex]

I would appreciate it if you mark my answer as brainliest.

Answer:

44

Step-by-step explanation:

The probability of passing if the minimum passing grade is 50 %. P(pass) is 0.000035 .

Given :

A quiz consists of 20 multiple - choice questions, each with 6 possible answers. For someone who makes random guesses for all of the answers .

Probability :

Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event.

Binomial problem with n = 20 and p = 1/6 , q = 5/6

50 % of 20  = 10

P ( x = 10 ) = C ( 20 , 10 ) ( 1/6 )^10 ( 5/6 )^10

= 184756 * ( 1/6 )^10 ( 5/6 )^10

= 0.000035

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​(Option C. Students to their LRN) In this case, each student has a unique LRN, meaning that it can be used in a one-to-one function where each student is mapped to their LRN.

A one-to-one function is a type of function where each element in the domain has one and only one image in the range. In this case, each student can be mapped to their LRN (Learning Reference Number) in a one-to-one function. This means that for every student, there is a unique LRN, and for every LRN, there is a unique student. This type of function is useful for keeping track of individual student information, as each student has a unique LRN associated with them.

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The volume of the rectangular pyramid is 900 cubic yards.

The standard formula of the volume of the rectangular pyramid is written as

V = 1/3s²h

where s refers the base edge and h refers the height.

Here we have given that the square base has side lengths 15 yards and the height is 12 yards.

Now, we have to find the volume of the rectangular pyramid

As we know the the value of base edge is 15 yard and the height is 12 yards.

Now, we have to apply the values on the formula then we get,

=> V = 1/3 x 15² x 12

It can be simplified as,

=> V = 4 x 225

When we do the appropriate arithmetic operation, then we get,

=> V = 900 cubic yard.

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The value of the variable x in the circle is 18 units

From the question, we have the following parameters that can be used in our computation:

Diameter = (2x - 4) units

Radius = 16 units

These parameters can be expressed as

d = 2x - 4

r = 16

Start by calculating the radius of the circle

So, we have the following representation

Radius = Diameter/2

Substitute the known values in the above equation, so, we have the following representation

16 = (2x - 4)/2

Evaluate the quotients in the above equation

16= x - 2

Add 2 to both sides of the equation

x = 16 + 2

Evaluate the like terms

x = 18

Hence, the variable x is 18 units

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The condition which is must be true for the given triangle ABC is m∠A = m∠C. Option 2 is correct. If AB = AC, then the triangle is said to be an isosceles triangle.

A triangle is said to be an isosceles triangle if it has two equal sides and two equal angles in measurement.

The angles in an isosceles triangle are acute. That means they are less than 90° each. So, their sum results in a total of 180°.

Given a triangle, ABC has AB = BC.

So, that means the given triangle is isosceles.

As we know, in an isosceles triangle, two angles are formed between one of the similar sides with the odd side being equal.

I.e., m∠A = m∠C

So, option 2 is correct.

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To avoid bias - increase the sample size and take reviews randomly

The dining hall manager decided to do a survey to estimate the proportion of students living in the dormitories who think that the quality of the food should be improved. one evening, the manager asked the first 100 students entering the dining hall

There is a bias if taken data contain only the first 100 students, primary reasons students come in a friend circle to the dining hall, possibly that whole friend circle did/ didn't like food. Chances are the first hundred students like the food and the remaining students don't like, vice - versa.

To avoid bias - increase the sample size and take reviews randomly

Therefore, the manager should increase the sample size and take reviews randomly to avoid sampling bias.

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The quadrilateral is a parallelogram if and only if it has two pairs of parallel sides

What is parallelogram?

In two-dimensional geometry, it is a plane shape having four sides, in which two pairs of sides are parallel to each other and equal in length. The sum of all angles in a parallelogram is 360°.

It is given that:

A parallelogram is a quadrilateral with two pairs of parallel sides.

As we know,

The quadrilateral can be defined as the four-sided polygon in geometry having four edges and four corners.

The quadrilateral, has:

Two pairs of congruent sides.

It has one pair of opposite congruent angles.

The diagonals of a kite are perpendicular.

One diagonal is bisected.

The top and bottom angles are bisected but does not have congruent diagonals.

A quadrilateral is a parallelogram if and only if it has two pairs of parallel sides.

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The probability that a man who developed diabetes is very active is 0.0625.

What is probability?

Probability is that the branch of arithmetic regarding numerical descriptions of however probably an incident is to occur, or however probably it's that a proposition is true. The likelihood of an incident could be a range between 0 and 1.

Main body:

You want P(a/d) where a means active, d means diabetic. Then Bayer's rule gives

P(a/d)=P(d/a)p(a)/P(d)

We know P(d)=202/5990.

P(a)=1/4 and P(d/a) we need to figure out.

But

P(d)=1/4P(d/a)+3/4p(d/s)=1/4P(d/a)+3/4∗5P(d/a)

P(D&A)/P(A) = 1/5*P(D&A')/P(A')

I am given P(A) = 1/4, so this can be reduced to

P(D&A)/(1/4) = 1/5*P(D&A')/(3/4) 15*P(D&A) = P(D&A')

The probability I am interested in is P(A|D) = P(D&A)/P(D) = P(D&A)/(P(D&A) + P(D&A')) = P(D&A)/(P(D&A) + 15*P(D&A)) = 1/16

Hence ,The probability that a man who developed diabetes is very active is 0.0625.

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The quadratic parent function is f(x) = x^2. (Option A)

In mathematics, family of functions refers to a group of functions that have the same degree or form. The examples of families of functions include linear functions, quadratic functions, square root functions, and absolute value functions. A parent function refers to the simplest function of a family of functions that preserves the definition of the entire family where the graph does not experience any transformations. The parent function of a quadratic equation is f(x) = x^2, whose graph has a vertex at the origin and experiences no vertical stretch or shrink. The other three options would be translated or transformed, hence are not the parent function of the quadratic equation.

Note: The question is incomplete. The complete question probably is: Which of these is the quadratic parent function? A. f(x) = x^2 B. f(x) = x^2 + 3 C. f(x) = x^2 – 5 D. f(x) = –4x^2

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The equation y(x,t)=0.8[(4x+5t)2+5] , represents a moving pulse where x and y are in meter and t is in second.

i) Direction of wave propagation is negative x-axis.

ii) The wave speed is 1.25m/s.

iii)The maximum displacement from the mean position (i.e., the amplitude of the wave) is 0.16 m.

iv)The wave pulse is symmetric.

We compare the given wave equation from the standard wave equation. On comparing, direction and wave speed is found out. The Remaining two parts are found by maxima concept and changing x by −x at t=0 to get a wave is symmetric.

Moving pulse is given by

⇒y(x,t)=0.8[(4x+5t)2+5]

Comparing (4x+5t) with (ax+bt) then we get,

⇒a=4 and b=5 ⋯⋯⋯(1)

(i) When coefficients of x and t are positive then the wave pulse is moving negative x-direction. So, given an equation with coefficients a = 4 and b= 5 are positive. Therefore, the direction of the given equation will be negative x axis.

(ii) After getting the coefficient a and b, we can calculate the speed by determining the value of b/a. And we have value of a =4 and b =5 from the equation (1) .Then speed of the wave will be

=> b/a=5/4

=> speed of the wave =1.25m/s

(iii) To get the maximum displacement from the mean position we put (4x+5t)=0. As the denominator is minimum, the fraction will be maximum. Therefore displacement will be

=> y=0.85, putting the (4x+5t) is equal to zero.

After simplify we get,

=> y=0.16 m

Hence maximum displacement is 0.16m from the mean position,

(iv) We replace x by −x and also put t= 0 in the given equation. If the equation does not change, replacing the x by −x, it will be symmetric otherwise not.

Putting t=0 in the given equation y(x,t) we get,

=> y=0.8(4x)2+5

Now replace x by −x on the above equation we get, y=0.8(4x)2+5

there will not be any change. Therefore, the given equation is symmetric.

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Complete question:

If y(x,t)=0.8[(4x+5t)2+5] , represents a moving pulse where x and y are in meter and t is in second. Find

(i) Direction of wave propagation.

(ii) The wave speed.

(iii) The maximum displacement from the mean position (i.e., the amplitude of the wave)

(iv) Whether the wave pulse is symmetric or not.

The Trigonometric Expression can be written as  :

cos⁴ x = 18(3 + 4cos2x + cos4x)

Power:

In mathematics, a power defines a base number raised to an exponent. The base is the factor multiplied by itself, and the exponent is the number of times the same base is multiplied.

Reducing / Lowering Power:

Power identities are trigonometric identities, allowing us to rewrite trigonometric expressions raised to powers in terms of simpler trigonometric expressions.

According to the question:

cos2x = 2cos²x -1

⇒ cos²x = 1/2 (1 + cos2x)

⇒ cos²2x = 1/2 (1 + cos4x)

⇒ cos⁴ x = (cos²x)²  

          = 1/4(1 + 2cos2x+cos²2x)

⇒ cos⁴ x = 1/4( 1 + 2cos2x + 1/2(1+cos4x))

⇒ cos4x = 1/8(3 + 4cos2x+cos4x)

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