Please help!!

Complete the identity

cos^2 θ/2=?

Answer:

The slope of the tangent line to the curve at the given point is -11/7.

Step-by-step explanation:

Differentiation is an algebraic process that finds the gradient (slope) of a curve.  At a point, the gradient of a curve is the same as the gradient of the tangent line to the curve at that point.

Given function:

[tex]4x^2+5xy+y^4=370[/tex]

To differentiate an equation that contains a mixture of x and y terms, use implicit differentiation.

Begin by placing d/dx in front of each term of the equation:

[tex]\dfrac{\text{d}}{\text{d}x}4x^2+\dfrac{\text{d}}{\text{d}x}5xy+\dfrac{\text{d}}{\text{d}x}y^4=\dfrac{\text{d}}{\text{d}x}370[/tex]

Differentiate the terms in x only (and constant terms):

[tex]\implies 8x+\dfrac{\text{d}}{\text{d}x}5xy+\dfrac{\text{d}}{\text{d}x}y^4=0[/tex]

Use the chain rule to differentiate terms in y only. In practice, this means differentiate with respect to y, and place dy/dx at the end:

[tex]\implies 8x+\dfrac{\text{d}}{\text{d}x}5xy+4y^3\dfrac{\text{d}y}{\text{d}x}=0[/tex]

Use the product rule to differentiate terms in both x and y.

[tex]\boxed{\dfrac{\text{d}}{\text{d}x}u(x)v(y)=u(x)\dfrac{\text{d}}{\text{d}x}v(y)+v(y)\dfrac{\text{d}}{\text{d}x}u(x)}[/tex]

[tex]\implies 8x+\left(5x\dfrac{\text{d}}{\text{d}x}y+y\dfrac{\text{d}}{\text{d}x}5x\right)+4y^3\dfrac{\text{d}y}{\text{d}x}=0[/tex]

[tex]\implies 8x+5x\dfrac{\text{d}y}{\text{d}x}+5y+4y^3\dfrac{\text{d}y}{\text{d}x}=0[/tex]

Rearrange the resulting equation in x, y and dy/dx to make dy/dx the subject:

[tex]\implies 5x\dfrac{\text{d}y}{\text{d}x}+4y^3\dfrac{\text{d}y}{\text{d}x}=-8x-5y[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}(5x+4y^3)=-8x-5y[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=\dfrac{-8x-5y}{5x+4y^3}[/tex]

To find the slope of the tangent line at the point (-9, -1), substitute x = -9 and y = -1 into the differentiated equation:

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=\dfrac{-8(-9)-5(-1)}{5(-9)+4(-1)^3}[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=\dfrac{72+5}{-45-4}[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=-\dfrac{77}{49}[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=-\dfrac{11}{7}[/tex]

Therefore, slope of the tangent line to the curve at the given point is -11/7.

The area of the complex figure is 178 square meters.

Option (A) is correct.

What is an area?

In geometry, area is the measure of the amount of surface enclosed by a two-dimensional shape or a planar figure. It is typically measured in square units such as square centimeters, square inches, or square meters.

To find the area of a rectangle, we can multiply its length by its width.

Given width = 8 m and length = 16 m, the area of the rectangle is:

Area = length x width = 16 m x 8 m
Area = 128 cm^2 ------(I)

To find the area of a rectangle, we can multiply its length by its width.

Given width = 5 m and length = 6 m, the area of the rectangle is:

Area = length x width = 6 m x 5 m
Area = 30 m^2 -----(II)

The area of a triangle is given by the formula A = 1/2 x b x h, where A is the area, b is the base of the triangle, and h is its height.

Given that the height of the triangle is 8 m and its base is 5 m, the area of the triangle is:

A = 1/2 x b x h = 1/2 x 5 m x 8 m
Area = 20 square m  ---------(III)

Adding (I), (II) and (III), we get

Area = 128 + 30 + 20

Area = 178 sq.m

Hence, the area of the complex figure is 178 square meters.

Option (A) is correct.

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Answer:

The area of the complex figure is 178 square meters. Thus, Option (A) is correct.

The area of the triangle is approximately 1.987 square units.

The properties of the triangle are: The sum of all the angles of a triangle (of all types) is equal to 180°. The sum of the length of the two sides of a triangle is greater than the length of the third side. In the same way, the difference between the two sides of a triangle is less than the length of the third side.

Using sin rule of a triangle,

[tex]\frac{Sin A}{a} = \frac{Sin C}{c} = \frac{Sin B}{b}[/tex]

In our case,

[tex]\frac{Sin 18}{1.6} = \frac{Sin Q}{4.2} = \frac{Sin P}{QR}[/tex]

From equation 1

[tex]\frac{Sin 18}{1.6} = \frac{Sin Q}{4.2} \\\frac{(Sin 18)*4.2}{1.6} = {Sin Q}\\Sin \ Q =21/8* sin\ 18\\[/tex]

Thus,

∠Q = 125.79 degrees

since sum of all angles is 180 degrees

so,

∠P = 180-125.79-18

∠P = 36.21

So From the equation 1

[tex]\frac{Sin 18}{1.6} = \frac{Sin P}{QR}\\\frac{Sin 18}{1.6} = \frac{Sin 36.2}{QR}\\QR = \frac{Sin 36.2}{\frac{Sin 18}{1.6}}[/tex]

QR = 3.06 cm

To find the area of the triangle, we can use Heron's formula:

Area = √(s(s-a)(s-b)(s-c))

where a, b, and c are the side lengths and s is the semi perimeter:

s = (a + b + c)/2

Plugging in the values we get:

s = (4.2 + 1.6 + 3.052)/2

s ≈ 4.426

Now we can use Heron's formula:

Area = √(4.426(4.426-4.2)(4.426-1.6)(4.426-3.052))

Using a calculator, we find:

Area ≈ 1.987 square units

Therefore, the area of the triangle is approximately 1.987 square units.

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Complete question:

The diagram shows triangle PQR, PR =  4.2 cm, PQ = 1.6 cm Given that angle PQR is obtuse triangle, work out the area of triangle PQR Give your answer correct to 3 significant figures. Angle PRQ = 18° Diagram NOT accurately drawn​

The probability of getting tens depends on the context. If you are referring to a regular deck of cards, there are four tens out of 52 cards, so the probability is 4/52 or 1/13. If you are referring to rolling a regular six-sided die, there is only one ten, so the probability is 1/6.

The probability of getting tens depends on the context. If you are referring to a regular deck of cards, there are four tens (10 of hearts, diamonds, clubs, and spades) out of a total of 52 cards. So the probability of drawing a ten is 4/52 or 1/13. If you are referring to rolling a regular six-sided die, there is only one ten (the number 10 itself) out of a total of six possible outcomes. So the probability of rolling a ten is 1/6.

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Step-by-step explanation:

The sum of the given numbers is:

15 + 17 + x + 28 + 39 = 99 + x

To find the missing value, we can use the formula for the mean (average):

mean = (sum of numbers) / (number of numbers)

Substituting the given values:

23 = (15 + 17 + x + 28 + 39) / 5

Multiplying both sides by 5:

115 = 15 + 17 + x + 28 + 39

Simplifying:

x + 99 = 115

x = 16

Therefore, the missing value is 16.

Survey is a method used to collect the information from the group by asking  relevant questions.

It is simply how  likely the event can be occurred or in other words it is a likelihood of an event occurring

Let the events be:

A----Cooked food P(A)=10

B----washed dishes P(B)=13

C----- operated the cash register.

P(C)=25

P(A∩B)=5

(cooked + washed dishes)

P(A∩C)=5

(cooked & operated the cash register)

P(B∩C)=8

(washed dishes and operated the cash register)

P(A∩B∩C)=3 did All three jobs

P(A∩B∩C)'=3 did none of these jobs

P(AUBUC) = P(A) + P(B) + P(C) - P(A∩B)-P(A∩C)-P(B∩C) + P(A∩B∩C).

A.) How many employees were surveyed?

=P(A) + P(B) + P(C) - P(A∩B)-P(B∩C)-P(C∩A) + P(A∩B∩C)

=(10+13+25)-(5-5-8)+3

=48-(-8)+3

= 48 + 8 + 3

= 59

= 59+ 3 (members who did none of the jobs)

=62

62 employees were surveyed

x=A∩B∩C

=3

ab= A∩B-x

=5-3

=2

bc=B∩C-x

=8-3

=5

ca=A∩C-x

5-3

=2

b) only cooked food:

=A-(ab+x+ca)

= 10-(2+3+2)

=10-7

=3

3 employees only cooked food

c)only operated the cash register:

=C-(bc+x+ca)

=25-(5+3+2)

=25-10

=15

15 employees only operated the cash register

d) employees cooked food or operated the cash register:

P(A or C)=P(A)+P(C)

=10+25

=35

e) washed dishes or operated the cash register

P(B or C)= P(B)+P(C)

=13+25

=38

f)only washed dishes:

=B - P(bc+x+ab)

=13-(5+3+2)

=13-10

=3

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this question is incomplete

The complete question is:

In a survey of employees at a fast food​ restaurant, it was determined that 10 cooked​ food, 13 washed​ dishes, 25 operated the cash​ register, 5 cooked food and washed​ dishes, 5 cooked food and operated the cash​ register, 8 washed dishes and operated the cash​ register, 3 did all three​ jobs, and 3 did none of these jobs. Complete parts A) through F) below,

A.) How many employees were surveyed?

B.) How many of the employees only cooked food?

C.) How many of the employees only operated the cash register?

D.) How many of the employees washed dishes or operated the cash register?

E.) How many of the employees washed dishes or operated the cash register ?

F.) How many of the employees only washed dishes?

Answer:42

Step-by-step explanation:took the quiz

Answer:

The answer that you are looking for is 70º

Step-by-step explanation:

Usually, in three-sided figures, there is a total of 3 angles all adding up to 180º.

So in turn, since most quadrilaterals are made up of 2 triangles all the sides have to equal to 360º.

We can use the equation 360 = 110×2+q2

To find the value of q.

The First step to do is to simplify all terms with 110×2 being equal to 220 and q2 staying the same.

We then use the opposite symbol to remove 220 leaving q2 by itself while 360 is subtracted by 220 and turned into 140.

The only thing left to do is to do the opposite operation of multiplying, which is dividing and dividing both sides by 2 leaving q = 70.

You can find the equation below:
360 =  110×2+q2
360 =  220 + q2
-220 = -220 + q2
 140 = q2
  ÷2 = ÷2
  70 = q

I hope this was helpful!

Answer:

14.4 - proofs attached to answer

Step-by-step explanation:

proofs attached to answer

Answer:

the correct option is: L′(-1, 11).

Step-by-step explanation:

To translate the preimage 9 units up, we need to add 9 to the y-coordinate of each vertex of the polygon. Thus, the image of polygon JKLM after the translation will have vertices J′(−2, 4), K′(−4, 9), L′(−1, 11), M′(0, 8). Therefore, the image coordinates of L′ are (-1, 11).

After translation the image coordinates of L' is (-1, 11). Therefore, option B is the correct answer.

Given that, Polygon JKLM is drawn with vertices J(-2, -5), K(-4, 0), L(-1, 2), M (0, -1).

When the shape is moved up by k units, then replace y with y + k.

The translation is as follows:

Here, J(-2, -5) → (-2, -5+9) → J'(-2, 4)

K(-4, 0) → (-4, 0+9) → K'(-4, 9)

L(-1, 2) → (-1, 2+9) → (-1, 11)

M(0, -1) → (0, -1+9) → (0, 8)

Therefore, option B is the correct answer.

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Answer: 57.50$

Step-by-step explanation:

if you have more questions like this there are good commission calculators id recommend look them up

15% of 50 is 7.50

add them together and you get

57.50$

hope this helps

The solution of the inequality is x > -4. In the interval notation is (-4,∞).

What is inequality?

In mathematics, inequalities specify the relationship between two non-equal values. Equal does not imply inequality. Typically, we use the "not equal symbol (≠)" to indicate that two values are not equal. But various inequalities are used to compare the values, whether it is less than or greater than.

An inequality exists when two real numbers or algebraic expressions are connected by the symbols “>”, “<”, “≥”, “≤”.

The given inequality is:

6x + 8 > - 16

Subtract 8 from both sides:

6x + 8 - 8 > - 16 - 8

6x  > - 24

Divide both sides by 6:

x > -24/6

x > -3

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The length of the diagonals will be WZ = XZ = 11 units.

A diagonal in geometry is a line segment that connects two polygonal or polyhedral vertices when they are not on the same edge. Any sloping line is known as a diagonal informally.

The length of the two diagonals of the rectangle is equal. The value of x will be calculated as,

6x - 7 = 3x + 2

3x = 9

x = 3

WZ = 6x - 7

WZ = 6 x 3 - 7

WZ = 18 - 7

WZ = 11

XZ = 3x + 2

XZ = 3 x 3 + 2

XZ = 11

Therefore, the length of the diagonals will be WZ = XZ = 11 units.

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Answer:

20 units

Step-by-step explanation:

six puls six plus four plus four is twenty

The distance between points D(1,3) and H(9,8) found using the distance formula is 9.43 units.

To find the distance between two points, the coordinates of two points in space are used in the formula. In coordinate geometry, a two-dimensional or three-dimensional space can be used to calculate the distance between two points using the distance formula. The Pythagoras theorem is sometimes used to calculate the distance between two places. The length of the line segment bridging two points on a plane is known as the distance between the points.

d=√((x2 - x1)² + (y2 - y1)²) is a common formula to calculate the distance between two points. This equation can be used to calculate the separation between any two locations on an x-y plane or coordinate plane.

Assuming the question is the one given below.

The points given are:

D = (1,3) = (x1,y1)

H = (9,8) = (x2,y2)

The distance D between these two points can be found using the distance formula.

D = [tex]\sqrt{{(x_2-x_1)}^2 + {(y_2-y_1)}^2 } = \sqrt{{(9-1)}^2 + {(8-3)}^2 }[/tex] = √89 = 9.43 units

Therefore the distance between points D(1,3) and H(9,8) found using the distance formula is 9.43 units.

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On using unit conversion the value for 31.5 inches is obtained as 80.01 centimeters.

What is unit conversion?

Unit conversion is a process with multiple steps that involves multiplication or division by a numerical factor or, particularly a conversion factor. The process may also require selection of the correct number of significant digits, and rounding.

To convert inches to centimeters, we can multiply the number of inches by 2.54.

So to convert a 31.5 inch board to centimeters, use the arithmetic operation of multiplication.

We can multiply 31.5 by 2.54 -

31.5 inches × 2.54 centimeters/inch = 80.01 centimeters

Therefore, the board is 80.01 centimeters long.

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Answer:

Last choice

Step-by-step explanation:

the 'w-axis intercept' occurs when d = 0  ( puppy is just born day 0)

 and is equal to 4.25 ounces

Answer:

Step-by-step explanation:

i think so

The sample space will be 243, 743, 1243, 1743, 2243, 2743, 3243, 3743, 4243, and 4743.

A sample space is the collection of all potential results of a random experiment or process in probability theory. It's represented by the symbol (the uppercase Greek letter omega).

For instance, if we flip a fair coin, the sample space is H, T, where H stands for the outcome of a heads result and T stands for a tails result. The sample space on a normal six-sided dice is 1, 2, 3, 4, 5, and 6.

No matter how likely they are, all outcomes are included in the sample area. In order to accurately estimate the probability of an event, the sample space must be defined.

Now,

To choose a systematic sample of size n = 10 from a town with 5000 people, we will use a systematic sampling technique. Here are the steps to follow:

Determine the sampling interval: The sampling interval is the number of people in the population divided by the desired sample size. In this case, the sampling interval is 5000/10 = 500.

Select a random starting point: To ensure the sample is representative of the population, we need to select a random starting point between 1 and 500.

We can use a random number generator to choose a starting point. Suppose we choose the number 243 as our starting point.

Select every nth person: Starting from the chosen starting point, we will select every 500th person until we reach a sample size of 10.

So, we will select people with the following index numbers: 243, 743, 1243, 1743, 2243, 2743, 3243, 3743, 4243, and 4743.

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We get abοut 4 minutes if we rοund tο the clοsest minute. The sοlutiοn is thus A. 4 minutes οr sο.

A mathematical nοtiοn knοwn as prοpοrtiοnality describes the relatiοnship between twο quantities that cοnsistently change tοgether. When twο quantities are prοpοrtiοnal, their relatiοnship stays cοnsistent οver time. Tο put it anοther way, if twο variables, A and B, are prοpοrtiοnal, then the ratiο οf A tο B must be cοnstant and can be written as fοllοws:

A/B = k

where k is a ratiο cοnstant. This means that tο keep the ratiο cοnsistent, if οne οf the quantities is multiplied by a certain factοr, the οther quantity must alsο be multiplied by the same factοr.

given

dT/dt = -k (T - Ts)

where T is the οbject's temperature, Ts is the envirοnment's temperature, and k is a prοpοrtiοnality factοr. dT/dt stands fοr the rate οf change οf temperature.

In this instance, the rοοm is currently 70°F and the pοt οf water is οriginally 200°F. We thus have:

dT/dt = -k (200 - 70) = -k(130) (130)

The water's temperature has decreased tο 164°F after 5 minutes. Therefοre, we are aware that t = 5 at T = 164:

[tex]164 = 200e^{(-k*5)[/tex]

calculating k:

[tex]e^{(-5k)} = 164/200 = 0.82[/tex]

-5k = ln(0.82) (0.82)

k = -ln(0.82)/5

We can nοw calculate the time it takes fοr the water tο cοοl tο 150°F using this number οf k:

[tex]150 = 200e^{(-k*t)[/tex]

[tex]e^{(-k*t)}= 150/200 = 0.75-k*t = ln(0.75) (0.75)[/tex]

t = -ln(0.75)/k

By changing the amοunt οf k, we οbtain:

t = -ln(0.75)/(-ln(0.82)/5) ≈ 3.8

We get abοut 4 minutes if we rοund tο the clοsest minute. The sοlutiοn is thus A. 4 minutes οr sο.

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Using Simple interest, savings account for her needed to deposit $1,842.82 each year.

A quick and simple way to figure out interest on money is to use the simple interest technique, which adds interest at the same rate for each time cycle and always to the initial principal amount. Any bank where we deposit our funds will pay us interest on our investment. One of the different types of interest charged by banks is simple interest. Now, before exploring the idea of basic curiosity in further detail,

t can also be computed using the annuity factor of 5% for 17 years to divide the total sum of $50,000.

The 529 plan, according to available data, is a tax-advantaged or tax-qualified savings plan designed to encourage individuals to save for future education costs of their relations or for themselves.

Data and Calculations:

N (number of intervals) = 17 years

Interest every year is equal to 5%.

Present Value = $0 PV

FV (Future Value) = $50,000 ($31,327.88 + $18,672.12)

Results: PMT = $1,842.82

Sum of all periodic payments = $31,327.88 ($1,842.82 x 17)

Total Interest = $18,672.12

Hence, in order to save $50,000 in 17 years at a 5% return for his granddaughter, the man who wanted to open a 529 college savings account for her needed to deposit $1,842.82 each year.

Hence the interest the bank will collect will be $117.53.

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Answer:

the last one

Step-by-step explanation:

it passes through the most points

Answer:

7

Step-by-step explanation:

The answer to this question is as follows Hence, the appropriate expressions response would be: Sort the company's revenue gain from least to biggest percent: D, C, A, B

In mathematics, you can multiply, divide, add, or subtract. This is how an expression is put together: Expression, numerical value, and math operator Numbers, parameters, and operations make up a mathematical expression. It is feasible to use contrasting words and phrases. Any mathematical statement containing variables, numbers, and a mathematical action between them is referred to as an expression, often known as an algebraic expression. For instance, the expression 4m + 5 is composed of the expressions 4m and 5, as well as the variable m from the above equation, which are all separated by the mathematical symbol +.

Based on the information in the graph, the firms are ranked from having the least % growth in revenue to having the largest:

With a 5% rise in revenue, Company D

A 10% rise in revenue for Company C

A 15% rise in revenue for Company A

A 20% rise in revenue for Company B

Hence, the appropriate response would be:

Sort the company's revenue gain from least to biggest percent: D, C, A, B

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The area of the canvas is 8 1/8 square units.

The area of a canvas with dimensions of 2 1/2 x 3 1/4 can be calculated by multiplying the length by the breadth.

2 1/2 and 3 1/4 must first be converted into improper fractions:

2 1/2 = (2 x 2 + 1) / 2 = 5/2

3 1/4 = (3 x 4 + 1) / 4 = 13/4

The area may now be calculated by multiplying the two fractions:

(5/2) x (13/4) = (5 x 13) / (2 x 4) = 65/8

The canvas's surface area is 65/8 square units as a result. This can be expressed as a mixed number as well:

65/8 = 8 1/8

Hence, the canvas has an area of 8 1/8 square units.

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Using the method of undetermined coefficients, the particular solution is:

[tex]\mathrm{y_p(t) = 6e^{2t} \times cos(9t) + 3^{2t} \times sin(9t)}[/tex]

To use the method of undetermined coefficients, we assume that the particular solution has the same form as the forcing term, but with undetermined coefficients.

Let's start by finding a particular solution for the equation:

y′′ − 4y′ + 81y = 48[tex]e^{2t}[/tex]cos(9t) + 80[tex]e^{2t}[/tex]sin(9t) + 3[tex]e^{(-t)[/tex]

Since the forcing term contains both exponential and trigonometric functions, we'll assume that the particular solution is of the form:

[tex]y_p[/tex](t) = A*[tex]e^{2t}[/tex]cos(9t) + B[tex]e^{2t}[/tex]*sin(9t) + C[tex]e^{(-t)[/tex]

where A, B, and C are undetermined coefficients that we need to determine.

Let's find the first and second derivatives of [tex]y_p[/tex](t):

[tex]y'_p[/tex](t) = (2A + 9B)*[tex]e^{2t}[/tex]*sin(9t) + (2B - 9A)*[tex]e^{2t}[/tex]*cos(9t) - C[tex]e^{(-t)[/tex]

[tex]y''_p[/tex](t) = (4A - 81B)*[tex]e^{2t}[/tex]*cos(9t) + (4B + 81A)*[tex]e^{2t}[/tex]*sin(9t) + C[tex]e^{(-t)[/tex])

Substituting these expressions into the differential equation, we get:

(4A - 81B)*[tex]e^{2t}[/tex]*cos(9t) + (4B + 81A)*[tex]e^{2t}[/tex]*sin(9t) + C[tex]e^{(-t)[/tex]

4[(2A + 9B)*[tex]e^{2t}[/tex]*sin(9t) + (2B - 9A)*[tex]e^{2t}[/tex]*cos(9t) - C[tex]e^{(-t)[/tex]

81[A*[tex]e^{2t}[/tex]cos(9t) + B[tex]e^{2t}[/tex]*sin(9t) + C[tex]e^{(-t)[/tex]]

= 48[tex]e^{2t}[/tex]cos(9t) + 80[tex]e^{2t}[/tex]sin(9t) + 3[tex]e^{(-t)[/tex]

Simplifying and grouping terms, we get:

[(4A + 18B)cos(9t) + (18A - 4B)sin(9t)][tex]e^{2t}[/tex]+ (81C + 3)[tex]e^{(-t)[/tex] = 48[tex]e^{2t}[/tex]cos(9t) + 80[tex]e^{2t}[/tex]sin(9t) + 3[tex]e^{(-t)[/tex]

Since the two sides of the equation must be equal for all values of t, we can equate the coefficients of the exponential and trigonometric functions on both sides:

4A + 18B = 48 => A = 6

18A - 4B = 80 => B = 3

81C + 3 = 3 => C = 0

Therefore, the particular solution is:

[tex]y_p[/tex](t) = 6[tex]e^{2t}[/tex]*cos(9t) + 3[tex]e^{2t}[/tex]*sin(9t)

Thus, the general solution of the differential equation is:

[tex]\mathrm{y(t) = y_h(t) + y_p(t)}[/tex]

where [tex]y_h[/tex](t) is the complementary solution. Since the characteristic equation of the differential equation is r² - 4r + 81 = 0, it has two complex conjugate roots: r = 2 ± 9i. Therefore, the complementary solution is:

[tex]y_h[/tex](t) = [tex]c_1[/tex]*[tex]e^{2t}[/tex]cos(9t) + [tex]c_2[/tex][tex]e^{2t}[/tex]*sin(9t).

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According to the given composite function the correct answer is , (g o h)[tex](n/2) = 8n^2 + 42n + 35.[/tex]

What is a composite function?

A composite function is a function that is formed by combining two or more functions. It is obtained by taking the output of one function and using it as the input of another function.

Suppose we have two functions, f(x) and g(x). The composite function of f and g, denoted by f o g, is defined as:

(f o g)(x) = f(g(x))

In words, this means that we apply the function g to the input x, and then we apply the function f to the output of g. The result is a new function that combines the effects of both f and g.

For example, if we have [tex]f(x) = x^2[/tex] and g(x) = 2x + 1, then the composite function f o g is:

[tex](f o g)(x) = f(g(x)) = f(2x + 1) = (2x + 1)^2 = 4x^2 + 4x + 1[/tex]

The composite function f o g takes an input x, applies g to it to get 2x + 1, and then applies f to that to get [tex]4x^2 + 4x + 1[/tex] as the final output.

To compute (g o h)(n/2), we need to first evaluate h(n/2) and then substitute the result into g(n) in place of n.

So, first we evaluate h(n/2):

h(n/2) = 4(n/2) + 5 = 2n + 5

Now we substitute this expression into g(n):

[tex]g(h(n/2)) = 2(h(n/2))^2 + h(n/2)[/tex]

[tex]= 2(2n + 5)^2 + (2n + 5)[/tex]

[tex]= 2(4n^2 + 20n + 25) + 2n + 5[/tex]

[tex]= 8n^2 + 42n + 35[/tex]

Therefore, [tex](g o h)(n/2) = 8n^2 + 42n + 35.[/tex]

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The volume of the box is a function of x, given by:[tex]V(x) = 4x^3 - 32x^2 + 64x[/tex]

What is volume ?

Volume is a physical quantity that measures the amount of three-dimensional space that a substance or object occupies. It is usually measured in cubic units, such as cubic meters, cubic centimeters, cubic feet, or cubic inches.

To construct the box, we start with a square piece of cardboard that is 8 inches by 8 inches. We will cut out squares from each corner with side length x inches. This will leave us with a rectangular piece of cardboard with dimensions 8-2x by 8-2x, and we can fold up the sides to form the open box.

The height of the box will be x inches, and the length and width of the base of the box will be 8-2x inches. Therefore, the volume of the box will be:

Volume = Length x Width x Height

Volume = (8-2x) x (8-2x) x x

[tex]Volume = x(64-32x+4x^2)[/tex]

Therefore, the volume of the box is a function of x, given by: [tex]V(x) = 4x^3 - 32x^2 + 64x[/tex]

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For the given rectangle, the ordered pair that represents the location of point Z is (11,6).

What is a rectangle?

The internal angles of a rectangle, which has four sides, are all precisely 90 degrees. The two sides come together at a straight angle at each corner or vertex. The rectangle differs from a square because its two opposite sides are of equal length. Also, it is known as an equiangular quadrilateral. Rectangles can also be referred to as parallelograms because their opposite sides are equal and parallel. A rectangle's diagonals cut each other in half. It has four right angles and is a parallelogram. The complete distance that the rectangle's outside boundary covers is referred to as its perimeter. The area a two-dimensional form occupies in a plane is known as its area. It has a square unit of measurement. As a result, the rectangle's area is the space enclosed by its exterior lines. It is the same as the length times the width calculation.

The rectangle should have opposite sides of equal length.

Here,

XY = WZ

XW = YZ

From the graph, we can write the coordinates as:

X = (7,11)

Y = (11,11)

W = (7,6)

Point Z should lie directly below Y so that it is in a straight vertical line.

Then it should have the same x coordinate as Y.

It should also lie in the same horizontal line that passes through W.

So it will have the same y coordinate as W.

Therefore for the given rectangle, the ordered pair that represents the location of point Z is (11,6).

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The value of probability that they have pulse rates with mean less than 78 beats per min would be; 0.5948

The term probability refers to the likelihood of an event occurring. Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event.

Given that If 16 female are randomly selected the Mean is 75 beats per min And, Standard deviation is, 12.5

Hence, We get;

z = (78 - 75) / 12.5

z = 3/12.5

z = 0.24

Hence, The probability is, 0.5948

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