find the value of the missing sides. leave in rationalized and simplified form. show your work.

Answer:

77.7

Step-by-step explanation:

if you know, you know. laso make sure your calculator is on degrees and not radians

Answer:

34. (c) 12

35. (a) -12

36. (a) 51

37. (a) 13

Step-by-step explanation:

34.)

[tex] \implies \: \sf\dfrac{4}{xx + 2} = \dfrac{6}{2xx - 3} \\ \\ \implies \: \sf4(2xx - 3) = 6(xx + 2) \\ \\ \implies \: \sf 8xx - 12 = 6xx + 12 \\ \\ \implies \: \sf 8xx - 6xx = 12 + 12 \sf \\ \\ \implies \: \sf 2xx = 24 \\ \\ \implies \: \sf xx = \dfrac{24}{2} \\ \\ \implies \: \sf xx = 12 \\ [/tex]

Hence, Required answer is option (c) 12.

35.)

[tex] \implies \: \sf \dfrac{xx - 2}{2} = \dfrac{3xx + 8}{4} \\ \\ \implies \: \sf2(3xx + 8) = 4(xx - 2) \\ \\ \sf 6xx + 16 = 4xx - 8 \\ \\ \implies \: \sf 6xx - 4xx = - 8 - 16 \\ \\ \implies \: \sf 2xx = - 24 \\ \\ \implies \: \sf xx = \dfrac{ - 24}{2} \\ \\ \implies \: \sf xx = - 12 \\ [/tex]

Hence, Required answer is option (a) -12.

36.)

[tex] \implies \: \sf\sqrt{xx - 2} = 7 \\ \\ \implies \: \sf xx - 2 = {(7)}^{2} \\ \\ \implies \: \sf xx - 2 = 49 \\ \\ \implies \: \sf xx = 49 + 2 \\ \\ \implies \: \sf xx = 51[/tex]

Hence, Required answer is option (a) 51.

37.)

[tex] \implies \: \sf \sqrt{2xx - 10} = 4 \\ \\ \implies \: \sf 2xx - 10 = {(4)}^{2} \\ \\ \implies \: \sf 2xx - 10 = 16 \\ \\ \implies \: \sf 2xx = 16 + 10 \\ \\ \implies \: \sf 2xx = 26 \\ \\ \implies \: \sf xx = \dfrac{26}{2} \\ \\ \implies \: \sf xx = 13 \\ [/tex]

Hence, Required answer is option (a) 13.

If you need 1 1/4 cups of sugar to make cookies, that is the amount of sugar needed for one batch of cookies. To find out how much sugar is needed to make 16 cookies, we can set up a proportion:

1 1/4 cups sugar is needed to make 1 batch of cookies

x cups sugar is needed to make 16 cookies

We can solve for x by cross-multiplying:

1 1/4 * 16 = x

20/4 = x

5 = x

Therefore, you will need 5 cups of sugar to make 16 cookies.

The matrix A of the linear transformation T(M) with respect to the standard basis for U2×2 is given by:

T([1000]) = [8 0]

[0 0]

T([0010]) = [0 0]

[0 9]

T([0001]) = [0 1]

[0 0]

To find the matrix A of the linear transformation T(M), we need to apply T to each basis vector of U2×2 and express the result as a linear combination of the basis vectors for U2×2. We can then arrange the coefficients of each linear combination as the columns of the matrix A.

Let's begin by finding T([1000]). We have:

T([1000]) = [8097][1000][8097]^-1

= [8 0]

[0 0]

To express this result as a linear combination of the basis vectors for U2×2, we need to solve for the coefficients c1, c2, and c3 such that:

[8 0] = c1[1000] + c2[0010] + c3[0001]

Equating the entries on both sides, we get:

c1 = 8

c2 = 0

c3 = 0

Therefore, the first column of the matrix A is [8 0 0]^T.

Next, we find T([0010]). We have:

T([0010]) = [8097][0010][8097]^-1

= [0 0]

[0 9]

Expressing this as a linear combination of the basis vectors for U2×2, we get:

[0 0] = c1[1000] + c2[0010] + c3[0001]

Equating the entries on both sides, we get:

c1 = 0

c2 = 0

c3 = 0

Therefore, the second column of the matrix A is [0 0 0]^T.

Finally, we find T([0001]). We have:

T([0001]) = [8097][0001][8097]^-1

= [0 1]

[0 0]

Expressing this as a linear combination of the basis vectors for U2×2, we get:

[0 1] = c1[1000] + c2[0010] + c3[0001]

Equating the entries on both sides, we get:

c1 = 0

c2 = 1

c3 = 0

Therefore, the third column of the matrix A is [0 1 0]^T.

Putting all of this together, we have:

A = [8 0 0]

[0 0 1]

[0 0 0]

Therefore, the matrix A of the linear transformation T(M) is:

T([1000]) = [8 0]

[0 0]

T([0010]) = [0 0]

[0 9]

T([0001]) = [0 1]

[0 0]

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Answer:

area of semi circle =pi r^2/2

3.14*6*6/2=56.2

area of rectangle=lb

=20*12=240

240+56.2=296.2

rounding it it will become 300 ft sqr

perimeter of rectangle without including 4th side=20+12+20=52

perimeter of semicircle=pi r+d (d is not needed here)

3.14*6=18.84

so total perimeter=52+18.84=70.84ft

Step-by-step explanation:

the terminal point of the angle determined by sin(0) < 0 and cos(0) > 0 is in the fourth quadrant.

why it is and what is trigonometry?

If sin(0) < 0 and cos(0) > 0, then we know that the angle 0 is in the fourth quadrant of the unit circle.

In the unit circle, the x-coordinate represents cos(θ) and the y-coordinate represents sin(θ). Since cos(0) > 0, we know that the terminal point of the angle is to the right of the origin. And since sin(0) < 0, we know that the terminal point is below the x-axis.

The fourth quadrant is the only quadrant where the x-coordinate is positive and the y-coordinate is negative, so that is the quadrant where the terminal point of the angle lies.

Therefore, the terminal point of the angle determined by sin(0) < 0 and cos(0) > 0 is in the fourth quadrant.

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It focuses on the study of the functions of angles and their applications to triangles, including the measurement of angles, the calculation of lengths and areas of triangles, and the analysis of periodic phenomena.

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Answer:

We can write the diameter and circumferance of base as -

D = 2√(750ρ/πh)

C = 2π√(750ρ/πh)

Step-by-step explanation:

What is function?

A function is a relation between a dependent and independent variable.

Mathematically, we can write → y = f(x) = ax + b.

Given is to find the diameter and height of the tin can.

Assume the density of coffee as {ρ}. We can write the volume of the tin can as -

Volume = mass x density

Volume = 750ρ

We can write -

πr²h = 750ρ

r = √(750ρ/πh)

D = 2r

D = 2√(750ρ/πh)

Now, we can write the circumferance as -

C = 2πr

C = 2π√(750ρ/πh)

Therefore, we can write the diameter and circumferance of base as -

D = 2√(750ρ/πh)

C = 2π√(750ρ/πh)

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Step-by-step explanation:

a probability is always the ratio

desired cases / totally possible cases

(a)

the experimental probability is just using the actual experience to predict any future results.

the total number of cases was 20, and the number of desired cases (yellow) was 12.

so, the experimental probability of landing on yellow is

12/20 = 3/5 = 0.600

(b)

the theoretical probability of a totally fair spinner landing on yellow is 2 out of 5 possibilities, so

2/5 = 0.4000

(c)

the correct statement is the first one.

with a more or less balanced (fair) spinner the experimental numbers should get closer and closer to the theoretical numbers, the more spins we make.

Answer:2

Step-by-step explanation:

Answer:

2 Tons

Step-by-step explanation:

Homer’s car weighs 2 tons because there are 2,000 pounds in a ton and 4,000 divided by 2,000 equals 2

The probability that the average of 49 marathons is between 143 and 147 minutes is 0.5980. The 60th percentile is 148.25 minutes, and the median is 146 minutes.

The average of a sample of 49 marathons will be approximately normally distributed with mean = 146 minutes and standard deviation = 15/sqrt(49) = 15/7.

To find the probability that the average of the sample will be between 143 and 147 minutes, we can standardize the values:

z1 = (143 - 146) / (15/7) = -1.4

z2 = (147 - 146) / (15/7) = 0.4667

Then, using a standard normal distribution table or calculator, we find:

P(-1.4 < Z < 0.4667) = P(Z < 0.4667) - P(Z < -1.4)

= 0.6788 - 0.0808

= 0.5980

So the probability that the average of the sample will be between 143 and 147 minutes is 0.5980.

To find the 60th percentile for the average of these 49 marathons, we need to find the z-score such that the area to the left of the z-score is 0.6. Using a standard normal distribution table or calculator, we find:

P(Z < z) = 0.6

z = 0.25

Then, we can solve for the corresponding value of X:

0.25 = (X - 146) / (15/7)

X = 148.25

So the 60th percentile for the average of these 49 marathons is 148.25 minutes.

To find the median of the average running times, we note that the median of a normal distribution is equal to its mean. Therefore, the median of the average running times is 146 minutes.

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Answer:

Acquired resources

Step-by-step explanation:

Acquired resources

The value for mean, variance, and standard deviation for the given set of data is 63.22, 794.04, and 28.17, respectively.

The method to calculate the various operations are:

Mean:

= (79 + 52 + 64 + 99 + 75 + 48 + 52 + 24 + 76) / 9 = 63.22

Mean is a measure of central tendency found by adding all the observations and dividing the result by the number of frequency or the total number of data set values.

Variance:

= ((79 - 63.22)² + (52 - 63.22)² + (64 - 63.22)² + (99 - 63.22)² + (75 - 63.22)² + (48 - 63.22)² + (52 - 63.22)² + (24 - 63.22)² + (76 - 63.22)² / (9-1) = 794.04

(Here, 63.22 is the mean calculated earlier)

Standard deviation:

= √(Variance)

=√(794.04) = 28.17

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Complete question is:

For each of the given sample data sets below, calculate the mean, variance, and standard deviation. (a) 79, 52, 64, 99, 75, 48, 52, 24, 76 mean =_______

variance = __________

standard deviation =_________

After answering the provided question, we can conclude that As a result, function Mark's annual premium is $1,500.

In mathematics, a function appears to be a connection between two numerical sets in which each individual of the first set (widely recognized as the domain) matches a particular member of the second set (called the range). In other words, a function takes input by one set and provides output from another. The variable x has frequently been used to represent inputs, and the variable y has been used to represent outputs. A formula or a graph can be used to represent a function. For example, the formula y = 2x + 1 represents a linear model whereby each x-value yields a unique value of y.

To determine Mark's total annual premium, add the premiums for liability insurance and collision/comprehensive insurance together.

Mark purchases liability insurance with coverage limits of 50/100/50. This means that his policy will pay up to $50,000 per person for bodily injury, $100,000 for bodily injury per accident, and $50,000 for property damage per accident.

Total Annual Premium: To determine Mark's total annual premium, add together his liability and collision/comprehensive insurance premiums:

Liability Premium + Collision/Comprehensive Premium = Total Annual Premium

Annual premium total = $500 + $1,000

$1,500 is the total annual premium.

As a result, Mark's annual premium is $1,500.

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Answer:

Step-by-step explanation:

We can start by assuming that the relationship between the ticket price and attendance is linear, so we can write the equation for the line that connects the two data points we have:

Point 1: (9, 23000)

Point 2: (7, 30000)

The slope of the line can be calculated as:

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

slope = (30000 - 23000) / (7 - 9)

slope = 3500

So the equation for the line is:

y - y1 = m(x - x1)

y - 23000 = 3500(x - 9)

y = 3500x - 28700

Now we can use this equation to find the attendance for any ticket price. To maximize revenue, we need to find the ticket price that generates the highest revenue. Revenue is simply the product of attendance and ticket price:

R = P*A

R = P(3500P - 28700)

R = 3500P^2 - 28700P

To find the ticket price that maximizes revenue, we need to take the derivative of the revenue equation and set it equal to zero:

dR/dP = 7000P - 28700 = 0

7000P = 28700

P = 4.10

So the ticket price that would maximize revenue is $4.10. However, we need to make sure that this price is within a reasonable range, so we should check that the attendance at this price is between 23,000 and 30,000:

A = 3500(4.10) - 28700

A = 5730

Since 23,000 < 5,730 < 30,000, we can conclude that the ticket price that would maximize revenue is $4.10.

if we make quarterly deposits and invest them at an nominal annual interest rate of 8% compounded quarterly for 40 years, we will have $143,004.54 at the end of the 40 years.

The first step in solving this problem is to calculate the amount of each quarterly deposit. We know that the first deposit is $400, and each subsequent deposit decreases by 1% from the previous deposit. This means that each deposit is 99% of the previous deposit. To calculate the size of each deposit, we can use the following formula:

deposit_ n = deposit_(n-1) * 0.99

Using this formula, we can calculate the size of each quarterly deposit as follows:

deposit_1 = $400

deposit_2 = deposit_1 * 0.99 = $396.00

deposit_3 = deposit_2 * 0.99 = $392.04

deposit_4 = deposit_3 * 0.99 = $388.12

...

We can continue this pattern for 40 years (160 quarters) to find the size of each quarterly deposit.

Next, we need to calculate the future value of these deposits using an nominal annual interest rate of 8% compounded quarterly. We can use the formula for compound interest to calculate the future value:

[tex]FV = PV * (1 + r/n)^(n*t)[/tex]

where FV is the future value, PV is the present value (which is zero since we are starting with deposits), r is the nominal annual interest rate (8%), n is the number of times the interest is compounded per year (4 since we are compounding quarterly), and t is the number of years (40).

We can substitute the values into the formula and solve for FV:

[tex]FV = $400 * (1 + 0.08/4)^(440) + $396.00 * (1 + 0.08/4)^(439) + $392.04 * (1 + 0.08/4)^(4*38) + ... + $1.64 * (1 + 0.08/4)^4[/tex]

After solving this equation, we get a future value of $143,004.54, rounded to two decimal places. This means that if we make quarterly deposits and invest them at an nominal annual interest rate of 8% compounded quarterly for 40 years, we will have $143,004.54 at the end of the 40 years.

This calculation highlights the power of compound interest over long periods of time. By making regular contributions and earning interest on those contributions, our investment grows exponentially over time. It also shows the importance of starting early and consistently contributing to an investment over time in order to achieve long-term financial goals.

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Answer: 1704 square inches

Step-by-step explanation:

The surface area of a square pyramid is given by the formula:

Surface Area = base area + 4 × (1/2 × slant height × base length)

The base area of the pyramid is:

base area = length × width = 24 in × 24 in = 576 in²

The slant height of the pyramid can be found using the Pythagorean theorem:

slant height = sqrt(20² + 12²) = 236/5 in

Therefore, the surface area of the whole pyramid is:

Surface Area = 576 in² + 4 × (1/2 × 236/5 in × 24 in) = 1152 in² + 2256 in² = 3408 in²

Half of this area is black, so the area that is yellow is:

Yellow Area = 1/2 × 3408 in² = 1704 in²

Therefore, the surface area of the toy that is yellow is 1704 square inches.

Answer:

To find the derivative of this function, we'll use the product rule and the chain rule. Let's begin by writing the function in a more readable form using parentheses:

y = e^x * csc(x) * (1 / x) * csc(x)

Now we can apply the product rule, letting u = e^x and v = csc(x) * (1 / x) * csc(x):

y' = u'v + uv'

To find u' and v', we'll need to use the chain rule.

u' = (e^x)' = e^x

v' = (csc(x) * (1 / x) * csc(x))'

= (csc(x))' * (1 / x) * csc(x) + csc(x) * (-1 / x^2) * csc(x) + csc(x) * (1 / x) * (csc(x))'

= -csc(x) * cot(x) * (1 / x) * csc(x) - csc(x) * (1 / x^2) * csc(x) - csc(x) * (1 / x) * csc(x) * cot(x)

= -csc(x) * [cot(x) * (1 / x) + (1 / x^2) + (cot(x) / x)]

Now we can substitute these into the product rule formula:

y' = e^x * csc(x) * (1 / x) * csc(x) * [-cot(x) * (1 / x) - (1 / x^2) - (cot(x) / x)] + e^x * (-csc(x) * cot(x) * (1 / x) * csc(x) - csc(x) * (1 / x^2) * csc(x) - csc(x) * (1 / x) * csc(x) * cot(x))

Next, we can simplify this expression. One way to do this is to factor out common terms:

y' = e^x * csc(x) * (1 / x) * csc(x) * [-cot(x) * (1 / x) - (1 / x^2) - (cot(x) / x)] - e^x * csc(x) * cot(x) * (1 / x) * csc(x) * [1 + (cot(x) / x)]

Now we can simplify further by combining like terms:

y' = e^x * csc(x) * (1 / x) * csc(x) * [-cot(x) * (2 / x) - (1 / x^2)] - e^x * csc(x) * cot(x) * (1 / x) * csc(x) * [1 + (cot(x) / x)]

= e^x * csc(x) * (1 / x) * csc(x) * [-2cot(x) / x - 1 / x^2 - cot(x) / x^2] - e^x * csc(x) * cot(x) * (1 / x) * csc(x) * [1 + cot(x) / x]

At this point, the derivative is simplified as much as possible.

(please could you kindly mark my answer as brainliest)

Answer:

To solve this problem, we can use the formula for compound interest:

A = P(1 + r/n)^(n*t)

Where:

A = the amount of money in the account after the specified time period

P = the initial principal amount (the amount deposited)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the time period in years

In this case:

P = £4100

r = 13.55% = 0.1355

n = 1 (interest is compounded once per year)

t = 4 years

Plugging these values into the formula, we get:

A = £4100(1 + 0.1355/1)^(1*4)

A = £4100(1.1355)^4

A = £4100(1.6398)

A = £6717.58

Therefore, the amount of money in the account after 4 years will be £6717.58.

Answer:

-4(x - 5) + 8x = 9x - 3

Simplifying the left side:

-4x + 20 + 8x = 9x - 3

4x + 20 = 9x - 3

Subtracting 4x from both sides:

20 = 5x - 3

Adding 3 to both sides:

23 = 5x

Dividing both sides by 5:

x = 23/5

Therefore, the equation equivalent to the given equation is:

5x - 23 = 0

Answer:

The derivative does not exist at the extremum (-2, 0).

Step-by-step explanation:

Given function:

[tex]f(x)=(x+2)^{\frac{2}{3}}[/tex]

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

[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Chain Rule of Differentiation}\\\\If $y=f(u)$ and $u=g(x)$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}y}{\text{d}u}\times\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}[/tex]

[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]\begin{aligned}\textsf{Let}\;u &= x+2& \implies f(u) &= u^{\frac{2}{3}}\\\\\implies \dfrac{\text{d}u}{\text{d}{x}}&=1 &\implies \dfrac{\text{d}y}{\text{d}u}&=\dfrac{2}{3}u^{(\frac{2}{3}-1)}=\dfrac{2}{3}u^{-\frac{1}{3}}\end{aligned}[/tex]

Apply the chain rule:

[tex]\implies f'(x) = \dfrac{\text{d}y}{\text{d}{u}} \cdot \dfrac{\text{d}u}{\text{d}{x}}[/tex]

[tex]\implies f'(x) = \dfrac{2}{3}u^{-\frac{1}{3}} \cdot1[/tex]

[tex]\implies f'(x) = \dfrac{2}{3}u^{-\frac{1}{3}}[/tex]

Substitute back in u = x + 2:

[tex]\implies f'(x) = \dfrac{2}{3}(x+2)^{-\frac{1}{3}}[/tex]

[tex]\implies f'(x) = \dfrac{2}{3(x+2)^{\frac{1}{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, 0).

To determine the value of the derivative at (-2, 0), substitute x = -2 into the differentiated function.

[tex]\begin{aligned}\implies f'(-2) &= \dfrac{2}{3(-2+2)^{\frac{1}{3}}}\\\\ &= \dfrac{2}{3(0)^{\frac{1}{3}}}\\\\&=\dfrac{2}{0} \;\;\;\leftarrow \textsf{unde\:\!fined}\end{aligned}[/tex]

As the denominator of the differentiated function at x = -2 is zero, the value of the derivative at (-2, 0) is undefined.  Therefore, the derivative does not exist at the extremum (-2, 0).

Answer:

5 as a percentage of 100 is 5/100 which is 5%

so here we get two outcomes one is 2 and other is 4.

so there is total 2 outcomes.

total no. of possibility is 5

so the probability of waiting between 2 and 4 minutes to use the ATM is 2/5.

Answer:

I'm pretty confident that the answer is E

The bases for the row space and null space of A, we put A into reduced row echelon form and solve for the null space. The dot product of basis vectors shows they are orthogonal.

To find the bases for the row space and null space of A, we perform row operations on A until it is in reduced row echelon form:

[ 1 -1  3 |  5 ]    [ 1 -1  3 |  5 ]

[ 2  1 -5 | -9 ] -> [ 0  3 -11 | -19]

[-1 -1  2 |  2 ]    [ 0  0  0  |  0 ]

[ 1  1 -1 | -1 ]    [ 0  0  0  |  0 ]

The reduced row echelon form of A tells us that there are two pivot columns, corresponding to the first and second columns of A. The third and fourth columns are free variables. Therefore, a basis for the row space of A is given by the first two rows of the reduced row echelon form of A:

[ 1 -1  3 |  5 ]

[ 0  3 -11 | -19]

To find a basis for the null space of A, we solve the system Ax = 0. Since the third and fourth columns of A are free variables, we can express the solution in terms of those variables. Setting s = column 3 and t = column 4, we have:

x1 - x2 + 3x3 + 5x4 = 0

2x1 + x2 - 5x3 - 9x4 = 0

-x1 - x2 + 2x3 + 2x4 = 0

x1 + x2 - x3 - x4 = 0

Solving for x1, x2, x3, and x4 in terms of s and t, we get:

x1 = -3s - 5t

x2 = s + 2t

x3 = s

x4 = t

Therefore, a basis for the null space of A is given by the vectors:

[-3  1  1  0]

[ 5  2  0  1]

To verify that every vector in the row space of A is orthogonal to every vector in the null space of A, we compute the dot product of each basis vector for the row space with each basis vector for the null space:

[ 1 -1  3 |  5 ] dot [-3  1  1  0] = 0

[ 1 -1  3 |  5 ] dot [ 5  2  0  1] = 0

[ 0  3 -11 | -19] dot [-3  1  1  0] = 0

[ 0  3 -11 | -19] dot [ 5  2  0  1] = 0

Since all dot products are equal to zero, we have verified that every vector in the row space of A is orthogonal to every vector in the null space of A.

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_____The given question is incomplete, the complete question is given below:

in excercises 7 and 8 find bases for the row space and null space of a. verify that every vector in the row(a) is orthogonal to every vector in null(a). a = [ 1 -1 3   5 2 1   0 1 -2   -1 -1 1]

Answer:

The answer is 1.5$

Step-by-step explanation:

Let the price of 1 notebook be x$ and 1 elastic book cover be y$

In first case,

2x+5y=$6.75

2x = $6.75-5y

x=($6.75-5y)/2------------- eqn i

In second case,

4x+2y=$7.50

4×($6.75-5y)/2 +2y=$7.50 [From eqn i]

($27-20y)/2 +2y=$7.50

($27-20y+4y)/2=$7.50

($27-16y)/2=$7.50

$27-16y=$7.50×2

$27-16y=$15

$27-$15=16y

$12=16y

y=$12/16

y=$0.75

The price of single elastic book cover is $0.75

Substituting the value of y in eqn i we get

x=($6.75-5y)/2

x=($6.75-5×$0.75)/2

x=($6.75-$3.75)/2

x=$3/2

x=$1.5

Hence, the price of single notebook is $1.5 Ans

Hey mate, please mark me as brainliest if you got the answer.

Here is the stem and leaf plot:

5  4

6   0, 1, 2, 2

7    1, 2, 4, 6, 8

8    2, 3, 4,  6, 7,

9   0, 2, 5, 5, 9

The range is 45.

A stem-and-leaf plot is a table that is used to display a dataset. A stem-and-leaf plot divides a number into a stem and a leaf. The stem is the tens digit and the leaf is the units digit. For example, in the number 54, 5 is the stem and 4 is the leaf.

Range is used to measure the variation of a dataset by finding the difference between the highest number and the lowest number.

Range = highest value - lowest value

99 - 54 = 45

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Set of sophomores taking discrete math = A ∩ B. Set of sophomores not taking discrete math = A - B or A ∩ B^c. Set of students who are sophomores or in discrete math = A ∪ B. Set of students who are not sophomores or not in discrete math = (A ∩ B)^c or A ∪ B^c.

The set of sophomores taking discrete math at your school is the intersection of the set of sophomores A and the set of students in discrete math B. So, it can be expressed as A ∩ B.

The set of sophomores at your school who are not taking discrete math is the difference between the set of sophomores A and the set of students in discrete math B. So, it can be expressed as A - B or A ∩ B^c, where B^c is the complement of B (i.e., the set of students who are not in discrete math).

The set of students at your school who either are sophomores or are taking discrete math is the union of the set of sophomores A and the set of students in discrete math B. So, it can be expressed as A ∪ B.

The set of students at your school who either are not sophomores or are not taking discrete math is the complement of the intersection of the set of sophomores A and the set of students in discrete math B.

This can be expressed as (A ∩ B)^c or as A ∪ B^c, where B^c is the complement of B (i.e., the set of students who are not in discrete math). Note that this set includes all students who are either juniors, seniors, or not enrolled in discrete math.

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