4. The distribution of the heights of students in a large class is roughly Normal. The average height is 68inches, and approximately 99.7% of the heights are between 62 and 74 inches. Thus, the variance of theheight distribution is approximately equal to(A) 2(B) 3(C) 4D 6(E) 9

The simplest form of the radical expression (√2 + √3) / (√2 - √3) is -5 - 2√6.

Given the radical expression in the question;

(√2 + √3) / (√2 - √3)

To simplify the expression, we need to eliminate the radical in the denominator.

We can do this by multiplying both the numerator and denominator by the conjugate of the denominator, which is (√2 + √3):

[(√2 + √3) / (√2 - √3)] × [ (√2 + √3) / (√2 + √3) ]

= (2 + √6 + √6 + 3) / (2 - √9)

= (5 + 2√6) / (2 - 3)

= (5 + 2√6) / (-1)

= -5 - 2√6

Therefore, the simplified form is -5 - 2√6.

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Answer:

y = -8/3x + 8

Step-by-step explanation:

Step 1:  Identify which values we have and need to find in the slope-intercept form:

The general equation of the slope-intercept form of a line is given by:

y = mx + b, where

Since we're told that the y-intercept is 8, this is our b value in the slope-intercept form.

Step 2:  Find m, the slope of the line:

Thus, we can find m, the slope of the line by plugging in (3, 0) for (x, y) and 8 for b:

0 = m(3) + 8

0 = 3m + 8

-8 = 3m

-8/3 = m

Thus, the slope is -8/3.

Therefore, the the equation of the line in slope-intercept form whose x-intercept is 3 and whose y-intercept is 8 is y = -8/3x + 8.

Optional Step 3:  Check the validity of the answer:

Thus, we can check that we've found the correct equation in slope-intercept form by plugging in (3, 0) and (0, 8) for (x, y), -8/3 for m, and 8 for b and seeing if we get the same answer on both sides of the equation when simplifying:

Plugging in (3, 0) for (x, y) along with -8/3 for m and 8 for b:

0 = -8/3(3) + 8

0 = -24/3 + 8

0 = -8 = 8

0 = 0

Plugging in (0, 8) for (x, y) along with -8/3 for m and 8 for b;

8 = -8/3(0) + 8

8 = 0 + 8

8 = 8

Thus, the equation we've found is correct as it contains the points (3, 0) and (0, 8), which are the x and y intercepts.

Answer:

Here's an interesting project idea for mathematics that can relate three different topics:

Topic 1: Fractals

Topic 2: Chaos Theory

Topic 3: Differential Equations

Research Question: Can fractals be used to model chaotic systems described by differential equations?

In this project, you can explore the concept of fractals and their applications in modeling complex systems. You can also delve into chaos theory and differential equations to understand how they are used to describe chaotic systems. By combining these three topics, you can investigate whether fractals can provide a better understanding of chaotic systems by modeling them more accurately.

Your research paper can cover the following areas:

Introduction: Provide an overview of fractals, chaos theory, and differential equations, and explain their relevance to the research question.

Fractals: Discuss the properties of fractals and how they can be used to model complex systems. Provide examples of fractals in nature and technology.

Chaos Theory: Explain the concept of chaos and how it is described by differential equations. Discuss the importance of chaos theory in understanding complex systems.

Differential Equations: Provide an overview of differential equations and their applications in physics, engineering, and other fields. Explain how differential equations are used to model chaotic systems.

Combining the three topics: Explain how fractals can be used to model chaotic systems described by differential equations. Provide examples of fractals used in modeling chaotic systems and compare the results to traditional methods.

Conclusion: Summarize the findings of your research and discuss the implications of using fractals to model chaotic systems.

Overall, this project can be a challenging and rewarding exploration of the interplay between three different mathematical topics.

Step-by-step explanation:

Answer:

x^2 + 3x + 9 = (x + 3)(x + 3)

When x = 2, we have:

(2 + 3) * (2 + 3)

5 * 5

25

When the matrices A and B commute, or when they fulfill AB=BA, this is a significant exception.

A matrix is a rectangular array or table of numbers, symbols, or formulae used to represent mathematical objects or qualities of such things. It is often arranged in rows and columns. For instance, a matrix contains two rows and three columns. a group of integers organized in a rectangular array in rows and columns. Matrix elements or entries refer to numbers. In several disciplines of engineering, physics, economics, statistics, and mathematics, matrices have a wide range of applications.

The eigenvalues and eigenvectors of the sum of two matrices are typically not the same as the sums of the eigenvalues and eigenvectors of the individual matrices. In actuality, there is no way to separate the eigenvalues of A and B from one another.

When a vector v is an eigenvector of both A and B, it is also an eigenvector of A+B, with an eigenvalue equal to the sum of its eigenvalues for A and B, respectively. There is, however, no reason to anticipate that two matrices will share an eigenvector in general. There is no method to create an eigenvector of A+B from two random eigenvectors of A and B alone.

When the matrices A and B commute, or when they fulfill AB=BA, this is a significant exception. In this situation, their invariant spaces coincide, and if both of them are diagonalizable, they are concurrently diagonalizable, which means that their eigenvectors do as well. As was already noted, in this specific instance, the eigenvectors of A+B are likewise the same.

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Answer:

\(\boxed{15 km}\)

Step-by-step explanation:

→ Write out the scale

3 cm = 5 km

→ Multiply everything by 3 to get 9 cm on the left side

9 cm = 15 km

The final result of the integral is:

∫ (log(1 + x²) / (x + 1)²) dx = log(x + 1) - 2 (log(x + 1) / x) - 2Li(x) + C,

where Li(x) is the logarithmic integral function and C is the constant of integration.

We have,

To solve the integral ∫ (log(1 + x²) / (x + 1)²) dx, we can use the method of substitution.

Let's substitute u = x + 1, which implies du = dx. Making this substitution, the integral becomes:

∫ (log(1 + (u-1)²) / u²) du.

Expanding the numerator, we have:

∫ (log(1 + u² - 2u + 1) / u²) du

= ∫ (log(u² - 2u + 2) / u²) du.

Now, let's split the logarithm using the properties of logarithms:

∫ (log(u² - 2u + 2) - log(u²)) / u² du

= ∫ (log(u² - 2u + 2) / u²) du - ∫ (log(u²) / u²) du.

We can simplify the second integral:

∫ (log(u²) / u²) du = ∫ (2 log(u) / u²) du.

Using the power rule for integration, we can integrate both terms:

∫ (log(u² - 2u + 2) / u²) du = log(u² - 2u + 2) / u - 2 ∫ (log(u) / u³) du.

Now, let's focus on the second integral:

∫ (log(u) / u³) du.

This integral does not have a simple closed-form solution in terms of elementary functions.

It can be expressed in terms of a special function called the logarithmic integral, denoted as Li(x).

Therefore,

The final result of the integral is:

∫ (log(1 + x²) / (x + 1)²) dx = log(x + 1) - 2 (log(x + 1) / x) - 2Li(x) + C,

where Li(x) is the logarithmic integral function and C is the constant of integration.

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She deposited $16,000 at 4% and $32,000 at 4.5%

She deposited $16,000 at 4% and $32,000 at 4.5%.

Let's assume Liza deposited an amount x at 4% interest. According to the given information, she deposited twice as much, which is 2x, at 4.5% interest.

To calculate the interest income from the deposits, we'll use the formula: Interest = Principal × Rate × Time.

At 4% interest, the interest earned is 0.04x (4% expressed as a decimal multiplied by x).

At 4.5% interest, the interest earned is 0.045(2x) (4.5% expressed as a decimal multiplied by 2x).

We know that the total annual interest income is $2080. Therefore, we can write the equation:

0.04x + 0.045(2x) = 2080

Simplifying the equation:

0.04x + 0.09x = 2080

0.13x = 2080

x = 2080 / 0.13

x ≈ 16,000

Liza deposited approximately $16,000 at 4% interest, and since she deposited twice as much at 4.5% interest, she deposited approximately $32,000 at 4.5% interest.

Therefore, she deposited $16,000 at 4% and $32,000 at 4.5%.

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Answer:

3.43°

Step-by-step explanation:

tan(angle of inclination of the highway)=6/100

angle=tan^-1(6/100)

angle=3.43°

1/2Base×height

Step-by-step explanation:

1/2×8×3

=4×3

=12

The value of the obtuse angle AOC is 100°.

Obtuse angle is any angle greater than 90° but less than 180°.

To calculate the value of the obtuse angle AOC, use the formula below

Formula:

But,

From the diagram,

Given:

Substitute the value into equation 2

Therefore,

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Level: Grade 7 Mathematics

----

Steps & Solution:

1. The relationship that has constant of proportionality greater than 1

\(\frac{p}{r} =\frac{6}{4} =\frac{3}{2}\) { the calculator of a certain proportionaility }

\(p=\frac{3}{2} r\)

2. The relationship that has constant of proportionaility less than 1

\(\frac{r}{p} =\frac{4}{6} =\frac{2}{3} \\r=\frac{2}{3} p\)

Given:

The values in the table represent the graph of a continuous function.

To find:

The interval that must contain an x-intercept.

Solution:

From the given table it is clear that, initially y-values are negative but after negative values we get y=0.44 at x=-1.9.

This signs of y-value are different, it means the graph of continuous function must contain an x-intercept in the interval -2.3 < x < -1.9.

Therefore, the correct option is B.

Answer:

Its , -2.3 < x < -1.9

Step-by-step explanation:

Edge 2020

Answer:

Step-by-step explanation:

for f(x):

0

2

2√2

4

for g(x):

2

2√2

2√3

2√5

for h(x):

4

6

4+2√2

8

Answer:

X/4=3/6

Start off by cross multiplying the tops and bottoms, so X/4=3/6 turns into 6X=12. Now divide by 6 on both sides.

6X=12      X=2

Answer:

2

Step-by-step explanation:

Cross Multiple and Divide

X. 3

----. = ----

4. 6

So what you want to do is multiply diagonally:

(6)(X) = 6X

(3)(4) = 12

Now that we have our two answers, we put an equal sign between them.

6X = 12

Now to get X alone. Since it's being multiplied by 6, you want to do the opposite and divide.

So remember to divide on both sides to get rid of the 6.

6X divided by 6 is just X.

12 divided by 6 is 2

We got our answers, put the equal sign in-between

X = 2

First, let's find the area of the rectangle.We know that the area of a rectangle is the length times width, or  20⋅32in this example:20⋅32=640So the area of the rectangle is  640ft2.You may not know the area of a semicircle, but that's fine  −we know the area of a circle, or  πr2.Since the area of a semicircle is half the area of a circle, we just do the area of the circle divided by  2:So the area of a semicircle is  πr22.The question asks for  πto just be  3.14, so instead the equation is  A=3.14r22The picture gives the diameter of the circle.To find the radius, or  r, we divide the diameter by  2:202=10Now we can solve for the area of the semicircle:3.14(10)223.14(100)23142157ft2Now that we now the areas of the rectangle and semicircle, we can add them up to find the area of the rose garden:640+157=797ft2

Answer:

STo solve for m in the equation -319.45 = m, we can isolate the variable m by adding 319.45 to both sides of the equation:

-319.45 + 319.45 = m + 319.45

This simplifies to:

0 = m + 319.45

Finally, we can subtract 319.45 from both sides to solve for m:

0 - 319.45 = m + 319.45 - 319.45

-319.45 = m

Therefore, the value of m is -319.45.tep-by-step explanation:

The government agency as a monopolist will produce and sell internet connections up to the point where the marginal cost is 1/2. The price will be set at 1, given the perfectly elastic demand function.

In the scenario where a government agency has a monopoly in the provision of internet connections and the inverse demand function is given by p = 1, we can analyze the market equilibrium and the implications for pricing and quantity.

The inverse demand function, p = 1, implies that the market demand for internet connections is perfectly elastic, meaning consumers are willing to purchase any quantity of internet connections at a price of 1. As a monopolist, the government agency has control over the supply of internet connections and can set the price to maximize its profits.

To determine the optimal pricing and quantity, the monopolist needs to consider the marginal cost of providing internet connections. In this case, the marginal cost is given as 1/2. The monopolist will aim to maximize its profits by equating marginal cost with marginal revenue.

Since the inverse demand function is p = 1, the revenue received by the monopolist for each unit sold is also 1. Therefore, the marginal revenue is also 1. The monopolist will produce up to the point where marginal cost equals marginal revenue, which in this case is 1/2.

As a result, the monopolist will produce and sell internet connections up to the quantity where the marginal cost is 1/2. The monopolist will set the price at 1 since consumers are willing to pay that price.

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Answer:

Given the information you provided, we can model cellular phone usage over time with an exponential growth model. An exponential growth model follows the equation:

`y = a * b^(x - h) + k`

where:

- `y` is the quantity you're interested in (cell phone usage),

- `a` is the initial quantity (34 million in 1995),

- `b` is the growth factor (1.22, representing 22% growth per year),

- `x` is the time (the year),

- `h` is the time at which the initial quantity `a` is given (1995), and

- `k` is the vertical shift of the graph (0 in this case, as we're assuming growth starts from the initial quantity).

So, our specific model becomes:

`y = 34 * 1.22^(x - 1995)`

To find the cellular usage in 2003, we plug 2003 in for x:

`y = 34 * 1.22^(2003 - 1995)`

Calculating this out will give us the cellular usage in 2003.

Let's calculate this:

`y = 34 * 1.22^(2003 - 1995)`

So,

`y = 34 * 1.22^8`

Calculating the above expression gives us:

`y ≈ 97.97` million.

So, the cellular phone usage in 2003, according to this model, is approximately 98 million.

Answer:

0.625

5/8 converted to decimal form is 0.625

what kind of witchcraft is this?

The three statements that are true include the following:

A. The radius of the circle is 3 units.

B. The center of the circle lies on the x-axis.

D. The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

In Mathematics, the standard form of the equation of a circle is represented by this mathematical expression;

(x - h)² + (y - k)² = r²   .......equation 1.

Where:

h and k represents the coordinates at the center of a circle.

r represents the radius of a circle.

Based on the information provided, we have the following equation of a circle:

x² + y² – 2x – 8 = 0      ......equation 2.

In order to determine the true statements, we would rewrite the equation in standard form and then factorize by using completing the square method:

x² – 2x + y² = 8 = 0

x² – 2x + (2/2)² + y² = 8 + (2/2)²

x² – 2x + 1 + y² = 8 + 1

(x – 1)² + (y - 0)² = 9         .......equation 3.

By comparing equation 1 and equation 3, we have the following:

Center (h, k) = (1, 0)

Radius (r) = 3

Additionally, this line and the center of the given circle lies on the x-axis (x-coordinate) because the y-value is equal to zero (0).

(x - 0)² + (y - 0)² = 3²

x² + y² = 9.

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Answer:

ndhduejcijgisjcidjcjzjfjidjicjdici djcojzifcjdjcu

Step-by-step explanation:

no se weydgfds

by the help of calculator

Answer:

1/16

Step-by-step explanation:

Place x=0 and you can see \(\frac{0}{0}\) indefinitness. So you can apply the l'hosptial rule. Its basic you should

\(\lim_{x \to \ 0} \frac{g(x)}{f(x)} = \\\)  \(lim_{x \to \ 0} \frac{g'(x)}{f'(x)}\) so apply the derivative

\(\frac{\frac{1}{x-4} +\frac{1}{\sqrt[4]{(1-x})^3 } }{2x}\) and replace the x=0 and you'll see same answer \(\frac{0}{0}\) re-apply the l'hospital rule and answer

\(\frac{\frac{-1}{(x-4)^2} + \frac{3}{16\sqrt[4]{(1-x)^7} } }{2}\) and replace the 0 you can see the answer \(\frac{1}{16}\)

The x-coordinates at which the integrand is evaluated when using the Midpoint Rule on the interval [a, b] with n subintervals are the midpoints of each subinterval, given by the formulas xᵢ = a + (step size) / 2 + (i - 1) * (step size) for i = 1 to n.

When using the Midpoint Rule to approximate the value of an integral on the interval [a, b] with n subintervals, the x-coordinates at which the integrand is evaluated are the midpoints of each subinterval.

To divide the interval into n subintervals, we use a step size of (b - a) / n. Then, the x-coordinates of the midpoints can be calculated as follows:

First subinterval:

x₁ = a + (step size) / 2

Second subinterval:

x₂ = a + (step size) / 2 + (step size)

Third subinterval:

x₃ = a + (step size) / 2 + 2 * (step size)

And so on, up to the nth subinterval:

xₙ = a + (step size) / 2 + (n - 1) * (step size)

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The x-coordinates at which the integrand is evaluated using the Midpoint Rule on the interval [-1, 23] with n subintervals can be represented by the formula \(x_i = -1 + (24i - 12) / (2n)\).

To determine the x-coordinates at which the integrand is evaluated using the Midpoint Rule on the interval [-1, 23] with n subintervals, we can follow these steps:

Calculate the width of each subinterval, Δx, by dividing the length of the interval [a, b] by the number of subintervals, n:

Δx = (b - a) / n

In this case, a = -1 and b = 23.

Substitute the values of a, b, and n;

Δx = (23 - (-1)) / n

Δx = 24 / n

Determine the x-coordinate of the midpoint for each subinterval using the formula:

\(x_i\) = a + (i - 1/2)  Δx

where i ranges from 1 to n.

\(x_i\) = -1 + (i - 1/2) (24 / n)

\(x_i\)  = -1 + (24i - 12) / (2n)

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Answer:

Solutions are x=7 square 2

The solutions of the quadratic equation 98 - x² = 0 are ±7√2.

Quadratic expressions are polynomial equations of second degree.

The general form of a quadratic equation is ax² + b x + c = 0.

Given quadratic equation is,

98 - x² = 0

This can be written as,

98 = x², or

x² = 98

Now, taking the square root on both sides,

√x² = √98

x = √98

Write 98 as the product of perfect square.

98 = 49 × 2

√98 = √(49 × 2) = √49 × √2 = ±7√2

Hence the solutions are ±7√2.

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The probability of drawing a king from a standard deck of 52 cards is 1/13.

In a standard deck of 52 playing cards, there are four kings: the king of hearts, the king of diamonds, the king of clubs, and the king of spades.

To find the probability of drawing a king, we need to determine the ratio of favorable outcomes (drawing a king) to the total number of possible outcomes (drawing any card from the deck).

The total number of possible outcomes is 52 because there are 52 cards in the deck.

The favorable outcomes, in this case, are the four kings.

Therefore, the probability of drawing a king is given by:

Probability = (Number of favorable outcomes) / (Number of possible outcomes)

= 4 / 52

= 1 / 13

Thus, the probability of drawing a king from a standard deck of 52 cards is 1/13.

This means that out of every 13 cards drawn, on average, one of them will be a king.

It is important to note that the probability of drawing a king remains the same regardless of any previous cards that have been drawn or any other factors.

Each draw is independent, and the probability of drawing a king is constant.

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