The table below shows the number of painted pebbles of Claire and Laura. If Greg chooses a pebble at random from the box 75 times, replacing the pebble each time, how many times should he expect to choose a yellow pebble?
A) 11

B) 33

C) 32

D) 22

Therefore, the answers are: (i) ZADB = 48.5 degrees, and (ii) ZEAO = 54 degrees.

An angle is a geometric concept that describes the amount of rotation between two intersecting lines or planes. It is defined as the figure formed by two rays with a common endpoint, called the vertex. The two rays are called the sides of the angle, and they can be measured in degrees or radians.

In the degree measurement system, a full rotation is 360 degrees, and an angle that is one quarter of a rotation (90 degrees) is called a right angle. Angles that are less than 90 degrees are called acute angles, while angles greater than 90 degrees but less than 180 degrees are called obtuse angles. An angle that measures exactly 180 degrees is called a straight angle, and an angle that measures greater than 180 degrees, but less than 360 degrees is called a reflex angle.

by the question.

ZOAB + ZEDA = 32 + 15 = 47 degrees. Since these two angles are opposite each other, they must add up to 180 degrees (straight angle) and therefore, ZAOB + ZEDC = 180 - 47 = 133 degrees.

Angle D is given as 37 degrees, and since ZEDC is a straight line, ZEDD = 180 - 37 = 143 degrees.

ZADB = ZAOB + ZOAB + BAD. Since ZAOB + ZOAB = 180 - ZEDC = 180 - 133 = 47 degrees, we have ZADB + BAD = 37 + 47 = 84 degrees. Also, BAD is an exterior angle of triangle ABD, so it is equal to the sum of the two opposite interior angles, which are ZADB and ABD. Therefore, ZADB + ABD + BAD = 180 degrees. Substituting the value of BAD and simplifying, we get ZADB = 48.5 degrees.

ZEAO = ZEDC - ZEDA - ZOAD. We already know ZEDC and ZEDA, so we need to find ZOAD. Since ZOAB and ZOAD are opposite each other, they must add up to 180 degrees. Also, ZOAB is equal to half the central angle ZODB (since it subtends the same arc), which is equal to 2ZOAD (since it is an inscribed angle subtended by the same arc). Therefore, we have ZOAB + ZOAD = 180 and ZOAB = ZOAD/2. Substituting the value of ZOAB from the given information, we get ZOAD = 64 degrees. Substituting all the values, we get ZEAO = 54 degrees.

To learn more about acute angles:

https://brainly.com/question/10334248

#SPJ1

Answer: In the table x part, it increases from -2 all the way to 4. In the table y part, it decreases from 17 to -1, but then increases back from -1 to 17.

Answer:

The formula to calculate the amount of loan with flat rate interest is:

Amount = Principal + (Principal x Rate x Time)

Where,

Principal = the amount of loan

Rate = the interest rate per year

Time = the time period in years

Given,

Principal = K40,000.00

Rate = 9% per year

Time = 2.5 years

Substituting the values in the formula, we get:

Amount = K40,000.00 + (K40,000.00 x 0.09 x 2.5)

Amount = K40,000.00 + K9,000.00

Amount = K49,000.00

Therefore, the amount he will pay the bank is K49,000.00.

If we cοnsider (x, y) tο be a pοint οn the unit circle, we get:

[tex]x^2 + y^2 = 1[/tex] (because the pοint is οn the unit circle) (since the pοint is οn the unit circle)

All real numbers must be determined in such a way that:

tan(s) = y / x

The trigοnοmetric identity can be applied:

Tan is equal tο Sin / Cοs.

We thus have:

Y/X = Tan(S), Sin(S), and Cοs (s)

Using the identity cοs(s) + sin(s) = 1, we square bοth sides tο οbtain:

[tex](y/x)^2 = sin^2(s) / cos^2(s) = 1 - cos^2(s)[/tex]

Rearranging and applying the equatiοn x2 + y2 = 1 results in:

cοs2(s) equals 1 - (y/x).

[tex]^2 = x^2[/tex]

Given that (x, y) is in the first οr fοurth quadrant and that x is pοsitive, we can take the square rοοt tο get the fοllοwing result:

cοs(s) = ± x

Sο, s is determined by:

S equals arccοs(x) + n, where n is any pοsitive integer.

The range οf x's value is limited tο [-1, 1] because it is οn the unit circle. As a result, the values οf s are prοvided by:

S = arccοs(x) + n, where n is any pοsitive integer and x is within the range [-1, 1].

The pοssible values οf s in exact radian measure in [0, 2] are as fοllοws:

S = arccοs(-1) + = S = arccοs(1) = 0

S is equal tο arccοs(0) + n + (n + 1/2)

where any integer n is used.

As n can be any integer, the pοssible pοssibilities οf s are s = 0,, and (n + 1/2).

Learn more about trigonometric identity here:

brainly.com/question/3785172

#SPJ1

Answer:

The value of the derivative at (2, 3) is zero.

Step-by-step explanation:

Given function:

[tex]g(x)=x+\dfrac{4}{x^2}[/tex]

To differentiate the given function, use the power rule of differentiation.

[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Power Rule of Differentiation}\\\\If $y=x^n$, then $\dfrac{\text{d}y}{\text{d}x}=nx^{n-1}$\\\end{minipage}}[/tex]

[tex]\textsf{Rewrite\;the\;function\;using\;the\;exponent\;rule\;\;$a^{-n}=\dfrac{1}{a^n}$}:[/tex]

[tex]\implies g(x)=x+4x^{-2}[/tex]

Apply the power rule:

[tex]\implies g'(x)=1+(-2) \cdot 4x^{-2-1}[/tex]

[tex]\implies g'(x)=1-8x^{-3}[/tex]

[tex]\implies g'(x)=1-\dfrac{8}{x^3}[/tex]

An extremum is a point where a function has a maximum or minimum value. From inspection of the given graph, the minimum point of the function is (2, 3).

To determine the value of the derivative at the minimum point, substitute x = 2 into the differentiated function.

[tex]\begin{aligned}\implies g'(2)&=1-\dfrac{8}{2^3}\\\\&=1-\dfrac{8}{8}\\\\&=1-1\\\\&=0\end{aligned}[/tex]

Therefore, the value of the derivative at (2, 3) is zero.

According to the given information, the missing side measure is 14.

What is triangle?

A triangle is a polygon with three sides, three angles, and three vertices. It is the simplest polygon and the fundamental shape used in geometry. A triangle can be classified based on the length of its sides and the measure of its angles.

To find the missing side measure, we can set up a proportion between the corresponding sides of the two similar triangles:

(x + 5) / x = 9.5 / 7

We can then solve for x by cross-multiplying:

7(x + 5) = 9.5x

7x + 35 = 9.5x

35 = 2.5x

x = 14

Therefore, the missing side measure is 14.

To learn more about triangle visit:

https://brainly.com/question/1058720

#SPJ1

Using the properties of  absolute value function, proved that |x - y| > |x| - |y| is true for all x and y.

To prove that |x - y| > |x| - |y|, we can consider two cases

Case 1

x >= 0 and y >= 0

In this case, |x - y| = x - y and |x| - |y| = x - y. So we have

|x - y| = x - y

| x | - | y | = x - y

Substituting these expressions into the original inequality, we get:

x - y > x - y

This inequality is true for all x and y where x >= 0 and y >= 0, since the difference between x and y is always greater than or equal to zero.

Case 2

x < 0 and y < 0

In this case, |x - y| = -(x - y) and |x| - |y| = -x + y. So we have:

|x - y| = -(x - y)

| x | - | y | = -x + y

Substituting these expressions into the original inequality, we get

-(x - y) > -x + y

Simplifying both sides, we get

y - x > -x + y

Adding x to both sides, we get

y > 0

This inequality is true for all x and y where x < 0 and y < 0, since both x and y are negative and the difference between x and y is always less than or equal to zero.

Learn more about absolute value here

brainly.com/question/1301718

#SPJ4

Answer:

Step-by-step explanation:

subtract 434-46

Therefore, the slope of the line that passes through (-2, 8) and (4, 6) is -1/3.

What is the slope?

In mathematics, the slope is a measure of the steepness of a line.

The solution provided involves three steps to find the slope of the line that passes through two points: (-2, 8) and (4, 6).

Step 1 involves finding the change in y-coordinates, which is the difference between the y-coordinate of the second point and the y-coordinate of the first point. In this case, the second point has a y-coordinate of 6 and the first point has a y-coordinate of 8.

Therefore, the change in y-coordinates is 6 - 8 = -2.

Step 2 involves finding the change in x-coordinates, which is the difference between the x-coordinate of the second point and the x-coordinate of the first point. In this case, the second point has an x-coordinate of 4 and the first point has an x-coordinate of -2.

Therefore, the change in x-coordinates is 4 - (-2) = 6.

Step 3 involves dividing the change in y-coordinates by the change in x-coordinates to find the slope of the line. In this case, the change in y-coordinates is -2 and the change in x-coordinates is 6, so the slope is -2/6 or -1/3.

Since all the steps are correct and properly executed, there is no error.

Therefore, the slope of the line that passes through (-2, 8) and (4, 6) is -1/3.

To learn more about slope from the given link:

https://brainly.com/question/3605446

#SPJ1

The length of h in the attached composite figure where all the angles are right angles is equal to 4m.

In the attached diagram of composite figure,

All are right angles.

Composite figure consist two rectangles,

Upper and lower rectangles.

length of the upper rectangle is equal to 5m

Width of the upper rectangle is equal to 'h' m

Width of the lower rectangle is equal to 2m

Length of each dash '-' mark  is equals to 1m.

length of 'h'm  is equals

= 2 m + 2 dash marks

= 2m + 2m

= 4m

Therefore, the length of the h in the composite figure ( attached diagram ) is equals to 4m.

Learn more about length here

brainly.com/question/12792512

#SPJ4

The above question is incomplete, the complete question is:

What is the length of h in the following composite figure? All angles are right angles.

5 m

4 m

3 m

2 m

Diagram is attached.

Answer:

4 m is ur answer

Step-by-step explanation:

hope this helps

Answer:

6188 ways

Step-by-step explanation:

there ate 5 objects to be choosen and there is no replacement of the object therefore you got

17 choices for the first selection of the object and 16 objects for the selection of the second object and so on until you get 13 objects for the last selection

totally you have 5 selections also arrangement does not matter there fore you have 17!/12!5! which is 6188

note we used 5! cause there are 5 placed objects and 12! are unplaced objects

note

that you have used one so you have to deduct one every time you use one

100 times the square root of 3 is also a possible value of a for this function.

In mathematics, the square root of a non-negative real number "a" is a non-negative real number that, when multiplied by itself, gives the original number "a". It is denoted by the symbol "√".

We can substitute each of the given values into the function f(a) = a² + 7 to determine if they are possible values of a.

Substituting the square root of 5:

f(√(5)) = (√(5))² + 7 = 5 + 7 = 12

So, the square root of 5 is not a possible value of a for this function.

Substituting the square root of 7:

f(√(7)) = (√(7))² + 7 = 7 + 7 = 14

So, the square root of 7 is a possible value of a for this function.

Substituting 100 times the square root of 3:

f(100√(3)) = (100√(3))² + 7 = 30000 + 7 = 30007

So, 100 times the square root of 3 is also a possible value of a for this function.

Therefore, the possible values of a for the given function are:

To know more about square root visit:

https://brainly.com/question/29775049

#SPJ1

0-24-  Tally (1)

25-49   Tally (4)

50-74  Tally (5)

75-99  Tally (2)

Step-by-step explanation:

See Venn diagram below

The value of the quadratic equation in the standard form is y = -x² -6x + 2.

y = ax² + bx + c, where a, b, and c are constants and an is not equal to 0, is a quadratic equation in standard form. A parabolic function's vertex, axis of symmetry, and intercepts with the x- and y-axes are all expressed by the quadratic equation in standard form. While the positions of the vertex and intercepts are determined by the factors b and c, the direction and form of the parabola are determined by the coefficient a. Every quadratic equation may be changed into standard form by applying the quadratic formula or the square method, which simplifies the analysis and comparison of various functions.

The standard form of the quadratic equation is given by:

y = ax² + bx + c

Substituting the values of x and y from the table we have:

For (-4, 10):

10 = a(-4)² + b(-4) + c

10 = 16a - 4b + c......(1)

For (-3, 11):

11 = a(-3)² + b(-3) + c

11 = 9a -3b + c......(2)

For (-2, 10):

10 = 4a - 2b + c .........(3)

Equation 1 can be written as follows:

10 = 16a - 4b + c

c = 10 - 16a + 4b

Substitute the value of c in equation 2 and 3:

11 = 9a -3b + c

11 = 9a - 3b + 10 - 16a + 4b

1 = - 7a + b .........(4)

And,

10 = 4a - 2b + c

10 = 4a - 2b + 10 - 16a + 4b

0 = -12a + 2b

12a = 2b

b = 6a .......(5)

Substitute the value of b in equation 4:

1 = - 7a + 6a

1 = -a

a = -1

Substitute the value of a in equation 5:

b = -6

Now, substitute the value of a and b in equation 1:

10 = 16a - 4b + c

10 = 16(-1) - 4(-6) + c

10 = -16 + 24 + c

10 = 8 + c

c = 2

Substituting the value in the quadratic equation we have:

y = -x² -6x + 2

Hence, the value of the quadratic equation in the standard form is y = -x² -6x + 2.

Learn more about quadratic equation here:

https://brainly.com/question/29269455

#SPJ1

The value of function f(0) after putting the value of x = 0 we get the value o which is not DNE.

A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a connection between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain. The usual way to refer to a function is as f(x), where x is the input. A function is often represented as y = f. (x).

Given function is

f(x)=5x

we have to find the value of the f(0)

so putting the value of 0 as x we get,

f(x)=5x

f(0) = 5(0)

f(0) = 0

Learn more about Function:

https://brainly.com/question/27914124

#SPJ4

Answer: C

Step-by-step explanation:

The correct answer is C.

The slope of the line represents the hourly wage, so Hunter's wage is $35 an hour, as given in option A and B. However, the y-intercept of the line represents the starting pay or the pay for zero hours worked. In this case, the y-intercept is 25, which means Hunter received $25 even if he did not work any hours. Hence, option C is correct as it states that Hunter's wage is $25 an hour, but it appears that he received no signing bonus.

The number of lionfish after 6 years will be 85,609. The recursive equation for [tex]f(n)[/tex] will be [tex]f(n) = 4242.42(1.65)^n - 1300n[/tex].

Consider the function:

[tex]y = a (1 \pm r)^x[/tex]

Where x is the number of times this growth/decay occurs, a = initial amount, and r = fraction by which this growth/decay occurs.

lionfish are considered an invasive species, with an annual growth rate of 65%.

Then the equation will be

[tex]f(n) = P(1.65)^n[/tex]

[tex]\text{P = initial population}[/tex]

A scientist estimates there are 7,000 lionfish in a certain bay after the first year.

[tex]7000 = P(1.65)[/tex]

    [tex]P = 4242.42[/tex]

Then the equation will be

[tex]f(n) = 4242.42(1.65)^n[/tex]

The number of lionfish after 6 years will be

[tex]f(n) = 4242.42(1.65)^6[/tex]

[tex]f(n) = 85608.58[/tex]

[tex]f(n) \cong 85,609[/tex]

If scientists remove 1,300 fish per year from the bay after the first year.

Then the recursive equation for f(n) will be

[tex]f(n) = 4242.42(1.65)^n - 1300n[/tex]

More about the exponent link is given below.

https://brainly.com/question/5497425

Answer:

(2u - 1) (3u + 10)

Step-by-step explanation:

Let's Check

(2u - 1) (3u + 10)

6u² + 20u - 3u + 10

6u² + 17u + 10

So, (2u - 1) (3u + 10) is the correct answer.

Answer:

[tex]\textsf{To\;add\;(or subtract)\;in\;Scientific\;Notation,\;you\;must\;have\;the\;same\;$\boxed{\sf power\;of\;10}$\:.}\\\textsf{Then\;you\;can\;$\boxed{\sf add\;or\;subtract}$\;the\;coefficients\;and\;$\boxed{\sf keep}$\;the\;power\;of\;10.}[/tex]

[tex]\textsf{To\;multiply\;in\;Scientific\;Notation,\; you\;must\;$\boxed{\sf multiply}$\;the\;coefficients}\\\textsf{and\;$\boxed{\sf add}$\;the\;powers\;of\;10.}[/tex]

[tex]\textsf{To\;divide\;in\;Scientific\;notation,\;you\;must\;$\boxed{\sf divide}$\;the\;coefficients}\\\textsf{and\;$\boxed{\sf subtract}$\;the\;powers\;of\;10.}[/tex]

Step-by-step explanation:

To add (or subtract) in Scientific Notation, you must have the same power of 10. Then you can add or subtract the coefficients and keep the power of 10.

Example expression:

[tex]2.3 \times 10^3 +3.2 \times 10^3[/tex]

Factor out the common term 10³:

[tex]\implies (2.3 +3.2) \times 10^3[/tex]

Add the numbers:

[tex]\implies (5.5) \times 10^3[/tex]

[tex]\implies 5.5\times 10^3[/tex]

Therefore, we have added the coefficients and kept the power of 10.

[tex]\hrulefill[/tex]

To multiply in Scientific Notation, you must multiply the coefficients and add the powers of 10.

Example expression:

[tex]2.3 \times 10^3 \times 3.2 \times 10^3[/tex]

Collect like terms:

[tex]\implies 2.3 \times 3.2 \times 10^3 \times 10^3[/tex]

Multiply the numbers (coefficients):

[tex]\implies 7.36 \times 10^3 \times 10^3[/tex]

[tex]\textsf{Apply the exponent rule:} \quad a^b \cdot a^c=a^{b+c}[/tex]

[tex]\implies 7.36 \times 10^{(3+3)}[/tex]

[tex]\implies 7.36 \times 10^{6}[/tex]

Therefore, we have multiplied the coefficients and added the powers of 10.

[tex]\hrulefill[/tex]

To divide in Scientific notation, you must divide the coefficients and subtract the powers of 10.

Example expression:

[tex]\dfrac{8.6 \times 10^6}{2.15 \times 10^2}[/tex]

Collect like terms:

[tex]\implies \dfrac{8.6}{2.15} \times \dfrac{10^6 }{10^2}[/tex]

Divide the numbers (coefficients):

[tex]\implies 4\times \dfrac{10^6 }{10^2}[/tex]

[tex]\textsf{Apply the exponent rule:} \quad \dfrac{a^b}{a^c}=a^{b-c}[/tex]

[tex]\implies 4\times 10^{(6-2)}[/tex]

[tex]\implies \implies 4\times 10^{4}[/tex]

Therefore, we have divided the coefficients and subtracted the powers of 10.

Answer:

Let's assume the original price of the stock was x.

When the company announced it overestimated demand, the stock price fell by 40%.

So, the new price of the stock after the first decline was:

x - 0.4x = 0.6x

A few weeks later, when the seats were recalled, the stock price fell again by 60% from the new lower price of 0.6x.

So, the new price of the stock after the second decline was:

0.6x - 0.6(0.6x) = 0.24x

Given that the current stock price is $2.40, we can set up the equation:

0.24x = 2.40

Solving for x, we get:

x = 10

Therefore, the stock was originally selling for $10.

The equation representing the money or balance (y) in your savings account any time (x) is y = z - 25x.

An equation is an algebraic statement showing the equality of two or more mathematical expressions.

Mathematical expressions combine variables with numbers, values, and constants without the equal symbol (=).

The monthly withdrawals = $25

The number of months (x) = 4

Account balance (y) = $75

Let z = the initial balance in the savings account.

y = z - 25x

75 = z - 25x

Learn more about equations at https://brainly.com/question/2972832

#SPJ1

Step-by-step explanation:

Original job = x men * 40 days = 40x man days  to complete

  now add 8 men     =    x+8  men

                                           man days now is   (x+8) (30)  to complete job

so     40x = (x+8)(30)

        40x = 30x + 240

          10 x = 240

              x = 24 men originally

The point of intersection of the two lines is (-3.4, -1.2).

Step 1: The slope of a line passing through two points (x1,y1) and (x2,y2) can be found using the formula:

m = (y2-y1)/(x2-x1)

Using this formula, we can find the slope of the first line:

m1 = (−2−3)/(-5 -5) = −5/(-10) = 1/2

And the slope of the second line:

m2 = (−4−4)/(2 -(-6)) = -8/4 = -2

Step 2: Find the y-intercept of each line

The y-intercept of a line in slope-intercept form (y = mx + b) is the value of y when x=0. We can use one of the two given points on each line to find the y-intercept:

For the first line passing through points (5,3) and (−5,−2):

y = mx + b

3 = (1/2)(5) + b

b = 3 - 5/2

b = 1/2

So the first line can be written as y = 1/2x + 1/2

For the second line passing through points (−6,4) and (2,−4):

y = mx + b

4 = (-2)(−6) + b

b = 4 - 12

b = -8

So the second line can be written as y = -2x - 8

Step 3: Each line in slope-intercept form (y = mx + b):

First line: y = 1/2x + 1/2

Second line: y = -2x - 8

Step 4: To find the point of intersection of the two lines, we need to solve the system of equations. We can solve for x by setting the two right-hand sides equal to each other:

1/2x + 1/2 = -2x - 8

(x + 1)/2 = -2x - 8

x + 1 = -4x - 16

5x = -16 - 1

5x = -17

x = -17/5

x = -3.4

Now that we know x, we can find y by substituting x=10 into one of the two equations:

y = -2x - 8

y = -2(-3.4) - 8

y = - 1.2

Thus, the point of intersection of the two lines is (-3.4, -1.2).

Learn more about equation of a line on:

https://brainly.com/question/18831322

#SPJ1

(a) Negation: C) This vertex is connected to at least one other vertex in the graph. (b) Negation: D) This number is related to at least one even number.

(a)

A) Some vertex is connected to some other vertex in the graph.

B) At least one vertex is not connected to any other vertex in the graph.

C) This vertex is connected to at least two other vertices in the graph.

D) There exists at least one vertex that is not connected to at least one other vertex in the graph.

E) This vertex is connected to some other vertices in the graph, but not necessarily to all of them.

(b)

A) This number is related to at least one odd number.

B) There exists at least one number that is not related to any even number.

C) All numbers are related to at least one even number.

D) This number is not related to at least one even number.

E) All numbers are related to at least one even number.

Learn more about graphs here: brainly.com/question/17267403

#SPJ4

Complete question:

Find a negation for each of the statements in (a) and (b).

(a) This vertex is not connected to any other vertex in the graph.

A) No vertex is connected to any other vertex in the graph.

B) All vertices are connected to all other vertices in the graph.

C) This vertex is connected to at least one other vertex in the graph.

D) All vertices are connected to at least one other vertex in the graph.

E) This vertex is connected to all other vertices in the graph.

(b) This number is not related to any even number.

A) This number is not related to any odd number.

B) All numbers are related to at least one even number.

C) All numbers are not related to any even number.

D) This number is related to at least one even number.

E) No number is related to any even number.

Answer:

x = 37.5

Step-by-step explanation:

the top triangle has 2 congruent sides and is therefore isosceles with base angles being congruent, then

base angles = (180 - 75) ÷ 2 = 105 ÷ 2 = 52.5

the angle on the left of the outer triangle is right , then

x + 52.5 = 90 ( subtract 52.5 from both sides )

x = 37.5

this is just a quick addition to the superb reply above by "jimrgrant1"

Check the picture below.

Answer:

B. 2.1

Step-by-step explanation:

If you draw a line from C to intersect AB perpendicularly at point D so we have 2 right triangles ACD and BCD.

For △ACD, AC is hypotenuse so sinA = CD/AC

=> CD = 5 x sin(20) = 5 x 0.342 = 1.71

then we have AB = AD + BD

Pythagorean theorem: c^2 = a^2 + b^2

for △ACD, 5^2 = 1.71^2 + AD^2

AD^2 = 5^2 - 1.71^2 = 22.0759

AD = 4.70

BD = AB - AD = 6 - 4.70 = 1.30

for △BCD, BC is hypotenuse

BC^2 = BD^2 + CD^2 = 1.30^2 + 1.71^2 = 4.61

BC = √4.61 = 2.1

Answer:

20 units

Step-by-step explanation:

Point 1 (-9, 4)

Point 2 (3, -12)

Distance Formula

d=√((x2-x1)²+ (y2-y1)²)

d=√((3+9)²+ (-12-4)²)

d=√(12²+ (-16)²)

d=√(144+ 256)

d=√400

d=20

the largest integer [tex]n $ such that $3^n$ is a factor of $1 \times 3 \times 5 \times \dots \times 97 \times 99$ is $\boxed{62}$.[/tex]

To find the largest integer[tex]n $ such that $3^n$ is a factor of $1 \times 3 \times 5 \times \dots \times 97 \times 99$[/tex], we need to count how many factors of 3 are in the product of the odd integers from 1 to 99.

One way to do this is to factor each odd integer into its prime factors and count how many factors of 3 are present. However, this would be quite tedious and time-consuming.

A quicker approach is to use the fact that every third odd integer is a multiple of 3. Thus, we can count how many multiples of 3 are present in the product of the odd integers from 1 to 99.

Let [tex]$m$[/tex] be the number of multiples of 3 in the range from 1 to 99. Then we have:

[tex]m = \left\lfloor \frac{99}{3} \right\rfloor = 33[/tex]

This is because there are 33 multiples of 3 in the range from 1 to 99 (namely, 3, 6, 9, ..., 96, 99).

Each multiple of 3 contributes at least one factor of 3 to the product of the odd integers. However, some multiples of 3 contribute two or more factors of 3, depending on how many factors of 3 they contain.

To count how many multiples of 3 contribute two or more factors of 3, we need to count how many multiples of 9, 27, and 81 are present in the range from 1 to 99.

There are [tex]$\left\lfloor \frac{99}{9} \right\rfloor = 11$[/tex]multiples of 9, namely 9, 18, 27, ..., 81, 90, 99. Each multiple of 9 contributes at least two factors of 3 to the product of the odd integers.

There are [tex]$\left\lfloor \frac{99}{27} \right\rfloor = 3$[/tex] multiples of 27, namely 27, 54, 81. Each multiple of 27 contributes at least three factors of 3 to the product of the odd integers.

There is only one multiple of 81 in the range from 1 to 99, namely 81, which contributes at least four factors of 3 to the product of the odd integers.

Thus, the total number of factors of 3 in the product of the odd integers from 1 to 99 is:

[tex]n = m + 2\times\text{number of multiples of 9} + 3\times\text{number of multiples of 27} + 4\times\text{number of multiples of 81}[/tex]

[tex]n = 33 + 2\times 11 + 3\times 3 + 4\times 1 = 62[/tex]

Therefore, [tex]the $ largest integer $n$ such that $3^n$ is a factor of $1 \times 3 \times 5 \times \dots \times 97 \times 99$ is $\boxed{62}$.[/tex]

To know more about largest integer click here:

brainly.com/question/28513507

#SPJ4