Please please help me

Use the form  

a

cos

(

b

x

−

c

)

+

d

acos(bx-c)+d

to find the variables used to find the amplitude, period, phase shift, and vertical shift.

a

=

4

a=4

b

=

3

b=3

c

=

π

4

c=π4

d

=

0

d=0

Find the amplitude  

|

a

|

|a|

.

Amplitude:  

4

4

Find the period using the formula  

2

π

|

b

|

2π|b|

.

Tap for more steps...

Period:  

2

π

3

2π3

Find the phase shift using the formula  

c

b

cb

.

Tap for more steps...

Phase Shift:  

π

12

π12

Find the vertical shift  

d

d

.

Vertical Shift:  

0

0

List the properties of the trigonometric function.

Amplitude:  

4

4

Period:  

2

π

3

2π3

Phase Shift:  

π

12

π12

(

π

12

π12

to the right)

Vertical Shift:  

0

0

i think ;-;

Answer:

[tex]\frac{\pi }{4}[/tex]

Step-by-step explanation:

The standard form of the cosine function is

y = a cos(bx + c)

where a is the amplitude, period = [tex]\frac{2\pi }{b}[/tex] and

phase shift = - [tex]\frac{c}{b}[/tex]

here b = 3 and c = - [tex]\frac{3\pi }{4}[/tex], hence

phase shift = - [tex]\frac{-\frac{3\pi }{4} }{3}[/tex] = [tex]\frac{\pi }{4}[/tex]

A would be the very first step in solving the equation for x

You have to distribute 2 to x and 7.

2(x + 7) + 3x = 12

2x + 14 +3x = 12

Answer:

A) 2x + 14 + 3x = 12

Step-by-step explanation:

The first step is to distribute the 2

2(x + 7) + 3x = 12

2x+14 +3x =12

Then we combine like terms

5x+14 =12

Subtract 14 from each side

5x+14-14=12-14

5x = -2

Then divide each side by 5

5x/4 = -2/5

x = -2/5

Answer:

Step-by-step explanation:

We know a maximum point on the height vs. time curve is at t=16 seconds. Then the height function can be written by filling in the known values in ...

  h(t) = (center height) + (wheel radius)·cos((frequency)·2π·(t -(time at max height)))

Since t is in seconds, we want the frequency in revolutions per second. That will be ...

  (3.2 rev/min)·(1 min)/(60 sec) = 3.2/60 rev/sec = 4/75 rev/sec

Then our height function is ...

  h(t) = 59 + 45·cos(8π/75·(t -16))

9 minutes is 9·60 sec = 540 sec, so we want to find the value of h(540).

  h(540) = 59 + 45·cos(8π/75·(540 -16))

  = 59 +45·cos(4192π/75)

  ≈ 59 + 45·0.944376 . . . . . calculator in radians mode

 ≈ 101.496937 . . . . feet

_____

The cosine function is a maximum when its argument is zero. We used the process of function translation to translate the maximum point to t=16 from t=0. That is, we replaced t in the usual cosine function with (t-16).

We can also evaluate the cosine function by subtracting multiples of 2π from the argument. When we do that, we find that Shirley's height at 9 minutes is the same as it is after 15 seconds. Some calculators evaluate smaller cosine arguments more accurately than they do larger argument values.

Dilation since one image is bigger than the other

A dilation transformation must be included to result in quadrilateral T.

To result in quadrilateral T, a dilation transformation must be included. A dilation is a transformation that changes the size of the figure without changing its shape. In this case, the side lengths of quadrilateral T are twice as long as those of quadrilateral S, indicating a scaling factor of 2. Therefore, a dilation is needed to scale up the size of quadrilateral S by a factor of 2 to obtain quadrilateral T.

Answer:

The height of the cone is [tex]48\ in[/tex]

Step-by-step explanation:

step 1

Find the radius of the base of cone

we know that

The volume of the cone is equal to

[tex]V=\frac{1}{3}\pi r^{2} h[/tex]

we have

[tex]V=12,288\pi\ in^{3}[/tex]

[tex]tan(30\°)=\frac{r}{h}[/tex]  ---> remember that the vertex angle of the vertical cross section is 60 degrees

so

[tex]r=(h)tan(30\°)[/tex]

[tex]r=(h)\frac{\sqrt{3}}{3}[/tex]

substitute the values and solve for h

[tex]12,288\pi=\frac{1}{3}\pi ((h)\frac{\sqrt{3}}{3})^{2} h[/tex]

[tex]36,864=\frac{h^{3}}{3}[/tex]

[tex]h^{3}=110,592[/tex]

[tex]h=48\ in[/tex]

Answer:

29

Step-by-step explanation:

50 - 20 = 30

7 - 8 = -1

30 - 1 = 29

Answer:

1. [tex]\boxed{y=x^2+16\to138sq.\:units}[/tex]

2. [tex]\boxed{y=-x^2+7x\to42sq. \:units}[/tex]

3. [tex]\boxed{y=4x+26\to 204sq.\:units}[/tex]

4.[tex]\boxed{y=-0.5x+13\to72sq.\:units}[/tex]

Step-by-step explanation:

1. The first curve is [tex]y=x^2+16[/tex]

The area under this curve on the interval [-1, 5] is given by:

[tex]\int\limits^5_{-1} {x^2+16} \, dx[/tex]

We integrate to obtain:

[tex]\frac{1}{3}x^3+16x|_{-1}^5[/tex]

We evaluate to obtain:

[tex]\frac{1}{3}(5)^3+16(5)-(\frac{1}{3}(-1)^3+16(-1))=138sq.\:units[/tex]

[tex]\boxed{y=x^2+16\to138sq.\:units}[/tex]

2. The second curve is [tex]y=-x^2+7x[/tex].

The area under this curve on the interval [-1, 5] is given by:

[tex]\int\limits^5_{-1} {-x^2+7x} \, dx[/tex]

We integrate this function to obtain:

[tex]-\frac{1}{3}x^3+\frac{7}{2}x^2|_{-1}^5[/tex]

This evaluates to

[tex]-\frac{1}{3}(5)^3+\frac{7}{2}(5)^2-(-\frac{1}{3}(-1)^3+\frac{7}{2}(-1)^2)=42[/tex] square units.

[tex]\boxed{y=-x^2+7x\to42sq. \:units}[/tex]

3.  The third curve is [tex]y=4x+26[/tex]

The area under this curve on the interval [-1, 5] is given by:

[tex]\int\limits^5_{-1} {4x+26} \, dx[/tex]

We integrate this function to obtain:

[tex]2x^2+26x|_{-1}^5[/tex]

We evaluate the limits of integration to obtain:

[tex]2(5)^2+26(5)-(2(5)^2+26(5))=204sq.\:units[/tex]

[tex]\boxed{y=4x+26\to 204sq.\:units}[/tex]

4. The fourth curve is [tex]y=-0.5x+13[/tex]

The area under this curve on the interval [-1, 5] is given by:

[tex]\int\limits^5_{-1} {-0,5x+13} \, dx[/tex]

We integrate this function to obtain:

[tex]-0.25x^2+13x|_{-1}^5[/tex]

We evaluate the limits of integration to obtain:

[tex]-0.25(5)^2+13(5)-(-0.25(-15)^2+13(-1))=72sq.\:units[/tex]

[tex]\boxed{y=-0.5x+13\to72sq.\:units}[/tex]

Answer:

C

Step-by-step explanation:

A function is even if:

f(x) = f(-x)

In other words, the function is symmetrical about the y-axis.

Function f is symmetrical, but it's not centered on the y-axis.

Function g is both symmetrical and centered on the y-axis.

So only g is an even function.  Answer C.

Answer:

question 8: 30975

question 9:

I don't know if it wants me to find the interest from the bond or the quote price of the bond.

If it is the bond it would be $1400 interest.

If it is the quote price of the bond, it would be $1,239 interest.

Answer:

Yes, no, no, no

Step-by-step explanation:

The first method is random.  There's no bias in selecting fans spaced out evenly as they enter.

The second method is not random.  It is biased towards those who hear the announcement and wish to participate.

The third method is not random.  It is biased towards those in the biggest rush to leave.

The fourth method is not random.  It is biased towards those with the best seats.

Answer:

Option D) 64

Step-by-step explanation:  

see the attached figure with letters to better understand the problem

In the right triangle BCD find BC

Applying the Pythagoras Theorem

[tex]BC^{2}=48^{2}+36^{2}\\BC^{2}=3,600\\BC=60\ units[/tex]

In the right triangle ABC

[tex]cos(C)=BC/AC[/tex]

substitute the values  

[tex]cos(C)=60/(x+36)[/tex] -------> equation A

In the right triangle BDC

[tex]cos(C)=CD/BC[/tex]

substitute the values

[tex]cos(C)=36/60[/tex] -------> equation B

equate equation A and equation B

[tex]60/(x+36)=36/60\\ \\(x+36)*36=60*60\\ \\36x+1,296=3,600\\ \\36x=3,600-1,296\\ \\36x=2,304\\ \\x=64\ units[/tex]

To find the missing length, use the Pythagorean theorem by setting up an equation with the given lengths in a right triangle. Solve for the missing length using the equation and determine the correct answer.

To find the missing length, we need to use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Let's assume the missing length is represented by 'x'. Using the theorem, we can set up the equation as follows:

We can solve for 'x' by substituting the given lengths into the equation and solving for 'x'.

x2 = a2 + b2

x2 = 162 + 302

x2 = 256 + 900

x2 = 1156

x = √1156

x = 34

Therefore, the missing length indicated is 34. Option B) 36 is not the correct answer.

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Answer:

The required system of equations is:

2x + 3.25y = 231.75

x = y + 3

where x is the number of pounds of bananas and y is the number of pounds of peaches

Explanation:

1- Defining the variables:

Assume that the number of pounds of bananas sold is x

Assume that the number of pounds of peaches sold is y

2- Setting the system of equations:

We are given that:

i. Each pound of bananas sells for $2

  Money gained from selling x pounds of bananas is 2x

ii. Each pound of peach sells for $3.23

   Money gained from selling y pounds of peaches is 3.25y

iii. Nicole made $231.75 in total. This means that:

    2x + 3.25y = 231.75 .................> equation I

iv. Nicole sold 3 more pounds of bananas than pounds of peaches. This

    means that:

    pounds of bananas = pounds of peaches + 3

    x = y + 3 ..................> equation II

3- Solving the equations (if required):

In equation II, we have: x = y + 3

Substitute with equation II in equation I and solve for y as follows:

2x + 3.25y = 231.75

2(y+3) + 3.25y = 231.75

2y + 6 + 3.25y = 231.75

5.25y = 231.75 - 6

5.25y = 225.75

y = 43

Substitute with y in equation II to get x:

x = y + 3

x = 43 + 3 = 46

Based on the above:

Number of pounds of bananas = x = 46 pounds

Number of pounds of peaches = y = 43 pounds

Hope this helps :)

Answer:

  B.  The sum of the squares of the two smaller sides is equal to the square of the third side.

Step-by-step explanation:

It can be helpful to recognize that the given side lengths, 12, 16, 20 are in the ratio 3:4:5, a recognizable Pythagorean triple. That is, you may know already that the triangle is a right triangle and the sum of squares of the short sides is equal to the square of the longest side (choice B).

In case you aren't aware of common Pythagorean triples, or didn't recognize these numbers, you can try the options to see which fits.

A. The sum of the two smaller sides is 12 +16 = 28. It is NOT equal to the third side, 20.

__

B. The sum of the squares of the two smaller sides is 12^2 + 16^2 = 144 + 256 = 400. This IS equal to the square of the third side: 20^2 = 400.

__

C. The absolute value of the difference between the squares of the two smaller sides is |256 -144| = 112. This is NOT equal to the square of the third side, 20^2 = 400.

__

D. The sum of the squares of the two smaller sides is 400 (as above). It is NOT equal to the third side, 20.

___

The only answer choice that fits is choice B.

Answer:

4

Step-by-step explanation:

The table clearly shows that if x = 0, y = 4.

The required output for the input of 0 in the function is 4.

Given that,
A table is shown the input and output value of the function, we have to determine the output value of the function for the corresponding input value 0.

Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.

Here,
In the table,
f(x) shows the output for the corresponding input x,
In the given question for x = 0 {input} there is f(0) = 4 in the table,

Thus, the required output for the input of 0 in the function is 4.

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Answer:

[tex]x-2 => f(x)=x^{3}-2x^{2}-x+2\\x+3=>f(x)=x^{3}-3x^{2} -13x+15\\x+4=>f(x)=-x^{3}+13x-12\\x+5=>f(x)=x^{4}+3x^{3}-8x^{2}+5x-25[/tex]

Step-by-step explanation:

The value of a function will be zero if the factor is put in it. In order to check whether a factor is of a function or not we will put the value of x from that factor in the function:

So

x-2 = 0 => x=2

Putting in first function

[tex]x^{3}-3x^{2} -13x+15\\=(2)^{3}-3(2)^{2} -13(2)+15\\=8-3(4)-26+15\\=8-12-26+15\\=23-38\\=-15 \neq 0\\[/tex]

So x-2 is not a factor of first function.

Putting in second function

[tex]f(x)=x^{4}+3x^{3}-8x^{2}+5x-25\\ =(2)^{4}+3(2)^{3}-8(2)^{2}+5(2)-25\\=16+3(8)-8(4)+10-25\\=16+24-32+10-25\\=-7\neq 0[/tex]

So x-2 is also not a factor of second function.

Putting in third function:

[tex]f(x)=x^{3}-2x^{2}-x+2\\ =(2)^{3}-2(2)^{2}-2+2\\=8-2(4)-2+2\\=8-8-2+2\\=0[/tex]

So x-2 is factor of third function.

...........................

For x+3

x+3=0

x=-3

First function:

[tex]f(x)=x^{3}-3x^{2} -13x+15\\=(-3)^{3}-3(-3)^{2} -13(-3)+15\\=-27-3(9)+39+15\\=-27-27+39+15\\=-54+54\\=0\\[/tex]

So x+3 is a factor of first function.

.............................

For x+4

x+4=0

x=-4

As we have already found the factors of first and third function, we will now only check second and fourth function.

[tex]f(x)=x^{4}+3x^{3}-8x^{2}+5x-25\\ =(-4)^{4}+3(-4)^{3}-8(-4)^{2}+5(-4)-25\\=256+3(-64)-8(16)-20-25\\=256-192-128-20-25\\-109\neq 0[/tex]

So x+4 is not a factor of second function.

Putting in fourth function:

[tex]f(x)=-x^{3}+13x-12\\ =-(-4)^{3}+13(-4)-12\\=64-52-12\\=64-64\\=0\\[/tex]

So x+4 is a factor of fourth function

..........................

For x+5=0

x=-5

Since only one function is remaining we'll only check for that.

[tex]f(x)=x^{4}+3x^{3}-8x^{2}+5x-25\\ =(-5)^{4}+3(-5)^{3}-8(-5)^{2}+5(-5)-25\\=625+3(-125)-8(25)-25-25\\=625-375-200-25-25\\=0\\[/tex]

if the rectangular coordiante is (x,y) and the polar coordinate is (R,t), then they are related as follows:

R^2=x^2+y^2

tant=y/x

1.(pi,pi/4)

here, R=

[tex] \sqrt{ {\pi}^{2} + { (\frac{\pi}{4} })^{2} } = \sqrt{ \frac{17 {\pi}^{2} }{16} } = \frac{\pi}{4} \sqrt{17} \\tant = \frac{ \frac{\pi}{4} }{4} \\ t = {tan}^{ - 1} \pi = 89.682[/tex]

therefore the polar form of Q.1 is

(pi sq. root 17/4,89.682°)

you can do 2 in similar way.

Answer:

  BC = 52

Step-by-step explanation:

The point of the question is to have you recognize and use the fact that the tangent segments from the same point are of equal length. That lets you write the equation ...

  4x +8 = 2(3x -7) . . . BC = BA

  2x +4 = 3x -7 . . . . . divide by 2

  11 = x . . . . . . . . . . . . add 7-2x

The length of segment BC is computed using this value of x:

  BC = 4x +8 = 4·11 +8

  BC = 52

Answer:

3.9230231e+12

Step-by-step explanation:

15! is 15 times 14 times 13 times 12 times 11 times 10 times 9 times 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 then times 3 because 3 shades of blue.

The number of different ways to purchase 3 shades of blue from 15 options is 455 different ways.

The question is asking about the number of combinations of 3 shades of blue that can be chosen from a total of 15 different shades. To calculate this, we use the formula for combinations without repetition, which is C(n, k) = n! / (k! * (n - k)!), where n is the total number of items to choose from, and k is the number of items to choose. In this case, n is 15 and k is 3.

First, we calculate the factorial of n, which is 15! (15 factorial), then the factorial of k, which is 3!, and finally the factorial of n - k, which is 12!. Putting these into the formula gives us:

C(15, 3) = 15! / (3! * 12!) = (15 * 14 * 13) / (3 * 2 * 1) = 455

Therefore, there are 455 different ways to choose 3 shades of blue from 15 different shades.

By DeMoivre's theorem,

[tex](\sqrt2(\cos10^\circ+i\sin10^\circ))^6=(\sqrt2)^6(\cos60^\circ+i\sin60^\circ)[/tex]

[tex]=8(\cos60^\circ+i\sin60^\circ)[/tex]

The answer is 8(cos60° +isin 60°)

Demoivre's theorem:, De Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number x and integer n it holds that (cosx + i sinx)^n= cosnx + i sinnx.

where i is the imaginary unit (i2 = −1).

By DeMoivre's theorem:

[sqrt(2)(cos(10)+isin(10)]^6

(√2 (cos 1o° + isin 10°))^6 = (√2)^6(cos 60° + isin 60°)

=8(cos60° +isin 60°)

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Answer:

D

Step-by-step explanation:

The formula for volume of cone is  [tex]V=\frac{1}{3}\pi r^2 h[/tex]

Where

V is the volume

r is the radius of the circular base

h is the height of the cone

In the diagram shown, we can clearly see that height is 12, radius is 9. We can simply plug them into the formula and get our exact answer (leaving pi as pi):

[tex]V=\frac{1}{3}\pi (9)^2(12)\\=324 \pi[/tex]

correct answer is D

Answer:

12%

Step-by-step explanation:

60000 people are unemployed, out of 500000 people.

That means that 60000/500000 of the population is unemployed.

Simplify: 60000/500000 = 6/50 = 0.12 = 12%

The overall unemployment rate is will be 12% based on the given data.

The percentage is defined as a given amount in every hundred. It is a fraction with 100 as the denominator percentage is represented by the one symbol %.

The percentage is also called the indicating hundredths. Thus, 2% is two-hundredth, which means 2%=2/100=0.02.

As per the given,

Total people = 5000000

40000 are employed and 60000 are unemployed out of 300000 people are in the labor force.

The % unemployment rate = (60000)/500000 x 100 = 12%

Hence "The overall unemployment rate is will be 12% based on the given data".

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The numbers you are multiplying are called the factors the result is called the product.

Answer: Product?

Step-by-step explanation:

Adding: Sum

Subtracting: Difference

Dividing: Quotient

Multiplying: Product

Hope this helps

Answer:

57°

Step-by-step explanation:

Angle 3 is supplementary to angle TKL.  That means they add up to 180°.  Therefore, 180° - 123° = 57°

Answer:

57°

Step-by-step explanation:

You can notice that by putting ∠3 and ∠TKL together you have a flat angle.

A flat angle has 180 degrees.

so, you can say that m∠3 and m∠TKL add 180 degrees, then they are supplementary angles.

Remember: Two angles are supplementary if they add 180 degrees.

Then, as you know m∠3=123°:

m∠3 + m∠TKL=180°

123 + m∠TKL=180°

m∠TKL=180° - 123°

m∠TKL=57°

Answer:

Option a - $9,314.45

Step-by-step explanation:

Cost of the house = $268,500

Time of repayment = 30 years

Repayment is done monthly, so number of repayments = 30 X 12 = 360

Monthly Payment = $1595.85

Rate of interest per payment period = [tex]\frac{.0625}{12}[/tex]

So, Present value of monthly payments = 1595.85 X  [tex]\frac{(1+\frac{.0625}{12})^{360}-1}{(1+\frac{.0625}{12})^{360}*(\frac{.0625}{12})}[/tex]

= $259,185.55

So, Vanessa's down payment = $268,500 - $259,185.55 = $9,314.45

Hope it helps.

Thank you !!

Answer:

Option a - $9,314.45

Step-by-step explanation:

e d g e

Answer:

the final answer is 50/11

Step-by-step explanation:

Here let's regard the first number of this geometric series as 0.54, holding the 4 to include later.  The next is 0.0054, the next 0.000054, and so on.

Thus, the common ratio is 1/100.  Then the sum of the infinite series, not including that 4, is

   a          0.54        0.54

-------- = ------------ = ------------

 1 - r       1 - 1/100     99/100

                                                                                      54

Multiplying both 0.54 and 99/100 by 100 results in ------- and this

                                                                                        99     0.545454....

Now add the 4 back in, obtaining 4  54/99, or (396 + 54) / 99.

This is the same as 450 / 99.  You can readily check with a calculator to see whether this is equivalent to the given   4.54545454545...

Note that 450/99 is in the form p/q (not pq), where p and q are positive integers.  But also note that 450/99 can be reduced to 150/33, or

50/11.  A calculator will show you that 50/11 is equivalent to the given   4.54545454545...

Hence, the final answer is 50/11 (in the form p/q, NOT pq).

The number 4.54545454545... can be expressed as a rational number in the form of p/q as 50/11.

To express the given number 4.54545454545... as a rational number in the form p/q, we need to use the concept of repeating decimals.

Let x = 4.54545... We can then write 100x = 454.54545... Subtracting the first equation from the second, we get 99x = 450. Solving for x gives us x = 450/99.

This fraction can be simplified further by dividing both the numerator and the denominator by their greatest common divisor. In this case, 450 and 99 share a common factor of 9. Dividing both numbers by 9 gives us 50/11, which is the simplest form of this fraction.

Therefore, 4.54545454545... as a rational number in the form p/q is 50/11.

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Your Right It Is Irrational

Black pants , white shirt

Answer:

·black pants, white shirt

Step-by-step explanation:

Shirt Colors : Yellow Purple Gray Green white

Pant Colors = Blue Green Black Brown

Combinations can be:

(Shirt color, Pant color)

(Yellow,Blue)

(Yellow,Green)

(Yellow,Black)

(Yellow,Brown)

(Purple,Blue )

(Purple,Green)

(Purple,Black)

(Purple,Brown)

(Gray,Blue)

(Gray,Green)

(Gray,Black)

(Gray,Brown)

(Green,Blue)

(Green,Green)

(Green,Black)

(Green,Brown)

(White,Blue)

(White,Green)

(White,Black)

(White,Brown)

Now we are given that Which outfit is part of Johan’s sample space

Option A : ·black pants, white shirt

True . It belongs to sample space

Option B : gray pants, green shirt

False . It does not belongs to sample space.

Option C : ·blue pants, brown shirt

False . It does not belongs to sample space.

Option D : white pants, white shirt

False . It does not belongs to sample space.

Hence Option A is true

Answer:

D. 1,344 in^2

Step-by-step explanation:

The area of the standard placemats is 84 in^2. To make 6 we need to multiply that by 6. 84 in^2*6 is 1,344 in^2. Please rate brainliest. It would really help!

Answer:

Option D

Step-by-step explanation:

The standard size of the rectangular placemat is 14 inches by 16 inches.

So fabric needed to make a placemat = Length × Width

                                                               = 14 × 16

                                                               = 224 inch²

Now for 6 placemats fabric required = 6 × 224

                                                             = 1344 inch²

Option D is the answer.

It’s a 33% chance rolling a one or a three

(a) ≈ 0.202, (b) ≈ 0.878, (c) ≈ 0.376, calculated using binomial probability formula with 56% chance for baseball fans.

To solve this problem, we can use the binomial probability formula since we have a fixed number of trials (selecting 10 men) and each trial (man) has two possible outcomes (considering themselves a baseball fan or not).

The binomial probability formula is:

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

Where:

- [tex]\( P(X = k) \)[/tex] is the probability of getting exactly \( k \) successes,

- [tex]\( n \)[/tex] is the number of trials (in this case, 10 men),

- [tex]\( k \)[/tex] is the number of successes we are interested in (number of men considering themselves baseball fans),

- [tex]\( p \)[/tex] is the probability of success on each trial (in this case, 56% or 0.56),

- [tex]\( \binom{n}{k} \)[/tex] is the binomial coefficient, representing the number of ways to choose [tex]\( k \)[/tex] successes from [tex]\( n \)[/tex] trials.

Let's solve each part of the problem:

(a) Finding the probability of exactly five men considering themselves baseball fans:

[tex]\[ P(X = 5) = \binom{10}{5} \times (0.56)^5 \times (1 - 0.56)^{10 - 5} \][/tex]

(b) Finding the probability of at least six men considering themselves baseball fans. This is the sum of probabilities of having 6, 7, 8, 9, or 10 successes:

[tex]\[ P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) \][/tex]

(c) Finding the probability of less than four men considering themselves baseball fans. This is the sum of probabilities of having 0, 1, 2, or 3 successes:

[tex]\[ P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) \][/tex]

Let's calculate each part:

(a)

[tex]$\begin{aligned} & P(X=5)=\left(\begin{array}{c}10 \\ 5\end{array}\right) \times(0.56)^5 \times(1-0.56)^{10-5} \\ & =\left(\begin{array}{c}10 \\ 5\end{array}\right) \times(0.56)^5 \times(0.44)^5 \\ & \approx 0.202\end{aligned}$[/tex]

(b)

[tex]\[ P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) \]\[ = \sum_{k=6}^{10} \binom{10}{k} \times (0.56)^k \times (0.44)^{10 - k} \]\[ \approx 0.878 \][/tex]

(c)

[tex]$\begin{aligned} & P(X < 4)=P(X=0)+P(X=1)+P(X=2)+P(X=3) \\ & =\sum_{k=0}^3\left(\begin{array}{c}10 \\ k\end{array}\right) \times(0.56)^k \times(0.44)^{10-k} \\ & \approx 0.006+0.034+0.111+0.225 \\ & \approx 0.376\end{aligned}$[/tex]

So, the probabilities are:

(a) [tex]\( \approx 0.202 \)[/tex]

(b) [tex]\( \approx 0.878 \)[/tex]

(c) [tex]\( \approx 0.376 \)[/tex]