In the past month, Salma rented 1 video game and 7 DVDs. The rental price for the video game was $3.30. The rental price fo each DVD was $4.60. What is the total amount that Salma spent on video game and DVD rentals in the past month?

The probability that a seventh grader chosen at random will play the bass is approximately 27 percent.      

To find the probability that a seventh grader chosen at random will play the bass, we need to calculate the total number of seventh graders who play the bass and divide it by the total number of seventh graders.

According to the data given, there are 12 seventh grade students who play the bass, and the total number of seventh grade students is:

Total number of seventh grade students = 13 + 12 + 14 + 5 = 44

Therefore, the probability that a seventh grader chosen at random will play the bass is:

Probability = Number of seventh graders who play bass / Total number of seventh grade students

Probability = 12 / 44

To express this probability as a percentage, we need to multiply it by 100:

Probability as a percentage = (12 / 44) x 100

Probability as a percentage = 27.3

Rounding this to the nearest whole number gives:

Probability as a percentage ≈ 27

Therefore, the probability that a seventh grader chosen at random will play the bass is approximately 27 percent.

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Answer:

Step-by-step explanation:

the actual answer is 68%

After addressing the issue at hand, we can state that Hence, the sphere's linear equatiοn vοlume is rοughly 904.32 cm3.

The algebraic equatiοn y = mx + b is knοwn as a linear equatiοn. M serves as the y-intercept, and B serves as the slοpe. The previοus clause has twο variables, y and x, and is sοmetimes referred tο as a "linear equatiοn with twο variables". Bivariate linear equatiοns are thοse with twο independent variables. The fοllοwing are a few examples οf linear equatiοns: 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, and 3x - y + z = 3. When an equatiοn takes the fοrm y = mx + b, with m denοting the slοpe and b denοting the y-intercept, it is referred tο as being linear. The term "linear" refers tο an equatiοn having the fοrm y=mx+b, where m stands fοr the slοpe and b fοr the y-intercept.

The fοllοwing equatiοn determines a sphere's vοlume:

V = (4/3)πR³

where R is the sphere's radius and is a cοnstant that rοughly equates tο 3.14.

R = 5.6 cm and = 3.14 are substituted intο the equatiοns tο yield:

V = (4/3) x 3.14 x (5.6)³

V = 4.19 x (5.6) x (5.6) x (5.6)³

V= 904.32 cm³ (rοunded tο twο decimal places)

Hence, the sphere's vοlume is rοughly 904.32 cm³.

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The answer of the question based on the container is in the shape of cube answer is, the volume of the container is (1/512)t⁹u¹², not 1/24t⁹u¹² as Parker claimed.

A volume is  measure of  amount of space occupied by  three-dimensional object. It is typically measured in a cubic units, like  cubic meters, cubic centimeters, or cubic feet.

Parker is not correct.

The volume of a cube is calculated by multiplying the length, width, and height of the cube. In this case, the length of the container is given as 1/8t³u⁴, but we do not have any information about the width and height. Assuming that the width and height are also 1/8t³u⁴, the volume of the cube would be:

V = (1/8t³u⁴) x (1/8t³u⁴) x (1/8t³u⁴)

V = (1/512)t⁹u¹²

Therefore, the volume of the container is (1/512)t⁹u¹², not 1/24t⁹u¹² as Parker claimed.

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18.43°

Given:A 10ft pole casts a 30ft shadow.To find:The angle of elevation of the sun.Diagram:Let AC be the height of the pole and BC be its shadow cast on the ground by the pole.Now, AB represents the ray of the sun. Angle ABC is the angle of elevation of the sun.Measurement:We have AC = 10ft, BC = 30ft.Let x be the angle of elevation of the sun. Then we have, tan(x) = opposite side/adjacent side = AC/BC = 10/30 = 1/3.To find the value of x, we have to take an inverse tangent of 1/3 on the calculator. We have,x = tan-1(1/3)x = 18.43°Therefore, the angle of elevation of the sun is 18.43°.

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Answer:

Step-by-step explanation:

To find the equation of the line passing through the points (0, 10) and (2, 14) in the form ax + y = c, we can use the point-slope form of the equation of a line:

y - y1 = m(x - x1)

where m is the slope of the line and (x1, y1) is a point on the line. We can use the point (0, 10) as our reference point, so x1 = 0 and y1 = 10. We need to find the slope of the line:

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

where (x2, y2) is the other point on the line. Using (0, 10) and (2, 14), we get:

m = (14 - 10)/(2 - 0) = 2

So the slope of the line is 2. Now we can use the point-slope form of the equation to find the equation of the line:

y - 10 = 2(x - 0)

Simplifying, we get:

y - 10 = 2x

Adding 10 to both sides, we get:

y = 2x + 10

This is the equation of the line in slope-intercept form. To write it in the form ax + y = c, we can rearrange the terms:

-2x + y = 10

Therefore, the value of a is -2.

The equation of the line passing through the points (0, 10) and (2, 14) in the form ax + y = c is 2x - y = -10. The value of a is 2.

To find the equation of the line passing through the points (0, 10) and (2, 14), we need to use the slope-intercept form of the equation: y = mx + b. First, let's find the slope (m) using the formula: m = (y2 - y1) / (x2 - x1). Plugging in the coordinates, we get m = (14 - 10) / (2 - 0) = 2. Now, we can use one of the points and the slope to find the y-intercept (b). Let's use the point (0, 10). Since y = mx + b, we can rearrange the equation to solve for b: b = y - mx. Plugging in the values, we get b = 10 - (2 * 0) = 10. So the equation of the line is y = 2x + 10. In the form ax + y = c, we can rewrite the equation as 2x - y = -10. Therefore, the value of a is 2.

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Yes, the unbiased estimator for $\sigma^2$, $S^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^{2}$, does attain the Cramer-Rao Lower Bound (CRLB) for $\sigma^2$. The CRLB for $\sigma^2$ is given by the equation $Var(\hat{\sigma^2})=\frac{2\sigma^4}{n-1}$.

The Maximum Likelihood Estimator (MLE) for $\sigma^2$ is given by $\hat{\sigma^2}=\frac{1}{n}\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}$. If $S^2$ attains the CRLB, then the MLE $\hat{\sigma^2}$ is the Unbiased Minimum Variance Unbiased Estimator (UMVUE) of $\sigma^2$.

This can be shown by first noting that the density of $X_1,...,X_n$ is given by $f(x)=\frac{1}{\sigma{\sqrt{2\pi}}}e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}$, $\forall x \epsilon (-\infty,\infty)$. Then, since $S^2$ is an unbiased estimator of $\sigma^2$ with variance $Var(S^2)=\frac{2\sigma^4}{n-1}$, it attains the CRLB for $\sigma^2$. Therefore, the MLE for $\sigma^2$, $\hat{\sigma^2}$, is the UMVUE for $\sigma^2$.

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Suppose we have a random sample X1, X2, ..., Xn from a normal distribution with mean and variance σ^2, and density function f(x) = (1/(σ√(2π))) * e^(-(x-μ)^2/(2σ^2)), where x is in the range of (-∞, ∞). We are interested in two questions:

Does the sample variance S^2 attain the Cramer-Rao lower bound (CRLB) for σ^2?

If the maximum likelihood estimator (MLE) for σ^2, denoted by \widehat{σ^2} = (1/n) * ∑(Xi- MEAN)^2, is attained by the CRLB, is it also the uniformly minimum variance unbiased estimator (UMVUE)?

It is known that the unbiased estimator for σ^2 is S^2 = (1/(n-1)) * ∑(Xi- MEAN)^2, and it has variance Var(S^2) = (2σ^4)/(n-1).

Yοur friend is nοt cοrrect. Since a similarity transfοrmatiοn can οnly transfοrm οne shape intο anοther shape that is similar tο it.

Similarity transfοrmatiοn:  

A similarity transfοrmatiοn is a type οf geοmetric transfοrmatiοn that preserves the shape οf a geοmetric figure, while pοssibly changing its size, οrientatiοn, and pοsitiοn in the plane οr in space.

Specifically, a similarity transfοrmatiοn is a cοmpοsitiοn οf a dilatiοn (οr a unifοrm scaling), fοllοwed by a rigid transfοrmatiοn (a rοtatiοn, reflectiοn, οr translatiοn).

Transfοrming rhοmbus intο a square using a similarity transfοrmatiοn:  

A rhοmbus is a quadrilateral with all sides οf equal length, while a square is a special type οf rhοmbus with all angles equal tο 90°.

Althοugh a square is a rhοmbus, nοt every rhοmbus is a square, and it is nοt pοssible tο transfοrm every rhοmbus intο a square using a similarity transfοrmatiοn.

Tο transfοrm a rhοmbus intο a square, yοu wοuld need tο change the length οf at least οne οf its sides, which is nοt allοwed in a similarity transfοrmatiοn.

Instead, yοu wοuld need tο use οther types οf transfοrmatiοns, such as a shear οr a cοmbinatiοn οf rοtatiοns and translatiοns.

Therefοre,

Yοur friend is nοt cοrrect. Since a similarity transfοrmatiοn can οnly transfοrm οne shape intο anοther shape that is similar tο it.

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Answer:

9 hours

Step-by-step explanation:

The x-coordinates of the vertices are at the same distance away from the x-axis but in the opposite direction of the triangle.

A triangle is a geometric shape consisting of three straight sides and three angles. It is a polygon with three sides. Triangles are one of the simplest shapes in geometry, and they can be classified based on their side lengths and angle measures.

According to given information :

B. The x-coordinates of the vertices are the same distance away from the x-axis but in the opposite direction.

When an object is reflected across the x-axis, the x-coordinates of its vertices remain the same, but their signs are flipped.

So, if the original coordinates of the vertices of ABC are (x₁, y₁), (x₂, y₂), and (x₃, y₃), then the coordinates of the reflected image A'B'C' are (x₁, -y₁), (x₂, -y₂), and (x₃,-y₃).

The x-coordinates remain the same, but the y-coordinates change in sign.

Therefore, the x-coordinates of the vertices are the same distance away from the x-axis but in the opposite direction.

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There are 40,320 different ways to arrange the letters of the word "notebook".

The word "notebook" contains 8 letters.

To calculate the number of different ways the letters can be arranged,

we can use the formula for permutations,

which is: n! / (n-r)!

where, n is the total number of items and r is the number of items selected.

For "notebook",

We want to find the number of permutations of all 8 letters,

So, n = 8 and r = 8.

Plugging these values into the formula,

we get: 8! / (8-8)! = 8!
Simplifying, 8! is equal to: 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320

Therefore, there are 40,320 different ways to arrange the letters of the word "notebook".

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Step-by-step explanation:

inversely proportional to x² means

y = k/x²

directly proportional would have meant

y = kx²

so, based on

y = k/x²

we need to find k out of the given data points, so that we can calculate autobahn data points.

let's start with x = 1

16 = k/1² = k

let's verify : x = 2

4 = 16/2² = 16/4 = 4 correct

x = 3

16/9 = 16/3² = 16/9 correct

x = 4

1 = 16/4² = 16/16 correct

a)

y = 16/x²

b)

y = 25

25 = 16/x²

25x² = 16

let's pull the square root on both sides

5x = 4

x = 4/5

FYI : x could have been also -4/5 (remember, any square root has a positive and negative number solution, as the square is the same). but the request was for the positive number.

Being Respectful, Practicing Empathy, Being Open-Minded, Remaining Patient

Attitude is an individual's way of perceiving and responding to their environment. It is a state of mind that influences how people think, feel, and behave. Attitudes are shaped by experiences, beliefs, and values, and they can be positive or negative, depending on the situation.

1. Being Respectful: A positive attitude helps to foster respect between yourself and those you meet. Respect is the foundation of any successful relationship, and having a positive attitude can help ensure that respect is maintained in the face of any potential conflicts.

2. Practicing Empathy: Having a positive attitude allows you to practice empathy and understanding of the people around you. Taking the time to understand the perspectives of others can help to prevent misunderstandings and conflicts.

3. Being Open-Minded: A positive attitude allows you to remain open-minded and flexible in any situation. This can help to stave off interpersonal conflicts by allowing you to better understand the needs and wants of those around you.

4. Remaining Patient: A positive attitude can help you remain patient and understanding of people in your life. Being patient in the face of any potential conflicts can help to resolve issues in a peaceful manner and prevent them from escalating.

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The correct answer is (C) They call 200 randomly selected town residents.

If the equation j=0. 239c, the food that contains 200 calories provides 47.8 joules of energy.

To convert the energy in calories to joules using the equation j = 0.239c, we can simply substitute the given value of c (200 calories) into the equation and solve for j:

j = 0.239c

j = 0.239 x 200

j = 47.8 joules

It's important to note that joules are the standard unit of energy in the International System of Units (SI), while calories are more commonly used in nutrition and food science.

The conversion factor between joules and calories is not exact, as it depends on the definition of the calorie used.

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Complete question is:

Joules and calories are two different units of energy. The equation j=0. 239c relates the measures, where c stands for calories and j stands for joules. If a food contains 200 calories, how many joules of energy does it provide?

Answer:

$40

Step-by-step explanation:

We can represent the price of an item on sale for [tex]x\%[/tex] off as:

[tex]S=P \cdot \dfrac{100-x}{100}[/tex],

where P is the product's original price and S is the sale price.

Applying this to the problem at hand:

[tex]\$16=P \cdot \dfrac{100-60}{100}[/tex]

↓ simplifying the fraction

[tex]\$16=P \cdot \dfrac{2}{5}[/tex]

To solve for P in this equation (the original price), we have to multiply both sides of the equation by the reciprocal of its coefficient.

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Remember that any fraction multiplied by its reciprocal is 1:

[tex]\dfrac{2}{5} \cdot \dfrac{5}{2} = \dfrac{10}{10} = 1[/tex]

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[tex]\dfrac{5}{2} \left( \$16\right)= \left(P \cdot \dfrac{2}{5}\right)\dfrac{5}{2}[/tex]

[tex]\boxed{\$40 = P}[/tex]

So, the original price of the shirt was $40.

A. Unbiased estimators behave with reliable results, where as biased estimators are inconsistent.

Unbiased estimators are preferred over biased estimators because they provide accurate estimates of population parameters. An estimator is said to be unbiased if, on average, it gives an estimate that is equal to the true population parameter being estimated.

Biased estimators, on the other hand, have a tendency to systematically overestimate or underestimate the population parameter, which can lead to incorrect conclusions. While biased estimators may sometimes have lower variability or require smaller sample sizes, they cannot be relied upon to provide accurate estimates in the long run. Unbiased estimators, on the other hand, retain the original distribution of the data and are consistent, making them more reliable in providing accurate estimates of population parameters.

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A radical equation is √(x + 2) = 4 - x and the extraneous solution is x = 7

From the question, we have the following parameters that can be used in our computation:

In steps 4 and 5, we have

x = ∛5 * 5 * 5 * 5

x = 5∛5¬ * 5¬ * 5¬ * 5

The above steps are carried out because it would simplify the radical expression to its simplest form

The reason it is done is to have the value of x in its simplest form

Here's an example of a radical equation:

√(x + 2) = 4 - x

Square both sides

x + 2 = 16 - 8x + x^2

Evaluate the like terms

x^2 - 9x + 14 = 0

Factor the quadratic equation

(x - 2)(x - 7) = 0

Solve for x:

x = 2 or x = 7

For x = 2:

√(2 + 2) = 4 - 2

√4 = 2

2 = 2

This is true, so x = 2 is a valid solution.

For x = 7:

√(7 + 2) = 4 - 7

√9 = -3

3 = -3

This is not true, so x = 7 is not a valid solution.

So, there is an extraneous solution

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Answer:

Step-by-step explanation:

[tex]sin\theta=\frac{x}{y}[/tex]

[tex]sin45=\frac{x}{9}[/tex]

[tex]x=9sin45[/tex]

  [tex]=7.7cm[/tex] (to 1 decimal place)

Answer:3.5 ish

Step-by-step explanation:

By two column proof, we get:

∠2 = 60°, ∠3 = 120°, ∠4 = 60°, ∠5 = 120°, ∠6 = 60°, ∠7 = 120° and ∠8 = 60

Two column Proof:

A two-column geometric proof consists of a list of statements and the reasons why we know these statements are true. The complaints are listed in the left column and the reasons for the complaint are listed in the right column.

According to the Question:

l II m and p is their transversal and 1 = 120°

∠1 + ∠2 = 180 °(Straight line)

120° + ∠2 = 180° > ∠2 = 180° - 120° = 60°

∠2 = 60°

But ∠1 = ∠3 (Vertically opposite angles)

∠3 = ∠1 = 120°

Similarly ∠4 = ∠2

By Vertically opposite angles,

∠4 = 60°

∠5 = ∠1 (Corresponding angles)

∠5 = 120°

Similarly, by corresponding angles:

∠6 = ∠2

∠6 = 60°

∠7 = ∠5 (Vertically opposite angles)

∠7 = 120°

And,

∠8 = ∠6 (Vertically opposite angles)

∠8 = 60°

Hence ∠2 = 60°, ∠3 = 120°, ∠4 = 60°, ∠5 = 120°, ∠6 = 60°, ∠7 = 120° and ∠8 = 60°

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We can conduct a hypothesis test to determine the scale 1's mean weight is less than the weight of scale 2.

H0: μ1 ≥ μ2 (the mean weight on Scale 1 is greater than or equal to the mean weight on Scale 2)

Ha: μ1 < μ2 (the mean weight on Scale 1 is less than the mean weight on Scale 2)

We will use a significance level of α = 0.10.

First, we calculate the sample means and the difference between the means:

Sample mean weight on Scale 1, μ1 = (12.35 + 15.08 + 9.00 + 11.55 + 24.36 + 9.88 + 15.36 + 7.05 + 13.39)/10 = 12.43

Sample mean weight on Scale 2, μ2 = (12.51 + 14.99 + 9.10 + 11.47 + 24.35 + 10.00 + 15.55 + 7.04 + 13.57)/10 = 13.15

Difference between the means, ud = μ1 - μ2 = -0.72

s.d.(ud) = sqrt((s1^2/n1) + (s2^2/n2))

s1 and s2 are sample standard deviations n1 n1 and n 2

s1 = sqrt(((12.35 - 12.43)^2 + (15.08 - 12.43)^2 + ... + (13.39 - 12.43)^2)/9) = 3.153

s2 = sqrt(((12.51 - 13.15)^2 + (14.99 - 13.15)^2 + ... + (13.57 - 13.15)^2)/9) = 2.114

n1 = n2 = 10

The test statistic can now be calculated as:

t = (ud - 0) / (s.d.(ud)/sqrt(n)) = (-0.72 - 0) / (1.109/sqrt(10)) = -2.05

Using a t-distribution table with 9 degrees of freedom and a one-sided test at α = 0.10, we find the critical value to be -1.383. As the calculated value so we will not consider null hypothesis.

The mean weight of scale 1 can be less than the scale 2 mean weight at 0.10 level.

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The expected number of students who have registered for the course on a pass/fail basis is 1.5.

To answer the given question, we can use the formula for expected value:
Expected Value = (Probability of Event Occurring) x (Number of Trials)
Here, the event is students taking the course on a pass/fail basis, and the number of trials is the total number of students registered for the course.

We are given that the probability of a student taking the course on a pass/fail basis is 3%, or 0.03.
Using this information,

we can substitute the values into the formula and calculate the expected number of students taking the course on a pass/fail basis:
Expected Value = (0.03) x (50)
Expected Value = 1.5.

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Answer: C.17/2

Step-by-step explanation: because 2(1/2+3/4)+6

So first (1/2+3/4)= 5/4

Then 2(5/4)=5/2

Finally 5/2+6=17/2

There are 126,749,601,088,000 different seating arrangements.

The number of different seating arrangements for a class of 30 students seated in a classroom with 6 rows of 5 desks each can be found by using the permutation formula.

To seat 30 students in 30 desks, there are 30 choices for the first seat, 29 choices for the second seat, 28 choices for the third seat, and so on until there is only 1 choice for the 30th seat. This can be expressed mathematically as:

30 × 29 × 28 × ... × 2 × 1

However, since the desks are arranged in rows, the order in which the students are seated within each row does not matter. So we need to divide the above expression by the number of ways in which 5 students can be arranged in a row. This can be calculated as:

5 × 4 × 3 × 2 × 1

So the total number of different seating arrangements is:

(30 × 29 × 28 × ... × 2 × 1) / (5 × 4 × 3 × 2 × 1)

which simplifies to 30! / (5!)^6

Using a calculator, this expression evaluates to:

126, 749, 601, 088, 000

Therefore, by Permutation formula there are 126,749,601,088,000 different seating arrangements that a teacher can make for a class of 30 seated in a classroom with 6 rows of 5 desks each.

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The percentage of returns that were greater than 30 percent is 16% and the percentage of returns that were below -24 percent is 2.5%..

The historical returns on a portfolio had an average return of 12 percent and a standard deviation of 18 percent. We can use the Empirical Rule to answer the questions about the percentage of returns that were greater than 30 percent and below -24 percent. The Empirical Rule states that for a bell-shaped distribution:

- 68% of the data falls within 1 standard deviation of the mean
- 95% of the data falls within 2 standard deviations of the mean
- 99.7% of the data falls within 3 standard deviations of the mean

a. What percentage of returns were greater than 30 percent?

30 percent is 1 standard deviation above the mean (12 percent + 18 percent = 30 percent). Therefore, according to the Empirical Rule, 68% of the data falls within 1 standard deviation of the mean, or between -6 percent and 30 percent.

This means that 32% of the data falls outside of this range, with 16% above 30 percent and 16% below -6 percent. So the percentage of returns that were greater than 30 percent is 16%.

b. What percentage of returns were below -24 percent?

-24 percent is 2 standard deviations below the mean (12 percent - 2*18 percent = -24 percent). Therefore, according to the Empirical Rule, 95% of the data falls within 2 standard deviations of the mean, or between -24 percent and 48 percent.

This means that 5% of the data falls outside of this range, with 2.5% above 48 percent and 2.5% below -24 percent. So the percentage of returns that were below -24 percent is 2.5%.

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So, the probability that a randomly selected light bulb will last between 750 and 900 hours is 40.82%, or 0.4082 as a decimal.

Probability refers to the measure or quantification of the likelihood of an event or outcome occurring. It is a numerical value between 0 and 1, where 0 represents an impossible event (never occurring) and 1 represents a certain event (always occurring).

Probability can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, if we flip a fair coin, the probability of getting heads is 0.5, because there is one favorable outcome (heads) out of two possible outcomes (heads or tails).

by the question.

Using the given mean and standard deviation, we can standardize the range of 750 to 900 hours as follows:

[tex]z1 = (750 - 750) / 75 = 0[/tex]

[tex]z2 = (900 - 750) / 75 = 1.33[/tex]

We can then use the 68-95-99.7 rule to find the probability that a randomly selected light bulb will last between 750 and 900 hours:

[tex]P(750 \leq X \leq 900) = P(0 \leq Z \leq 1.33)[/tex]

From the table of standard normal probabilities, we can find that the area to the left of 1.33 is 0.9082, and the area to the left of 0 is 0.5. Therefore:

[tex]P(0 \leq Z \leq 1.33) = 0.9082 - 0.5 = 0.4082[/tex]

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To simplify (9x^3+2x^2-5x+4)-(5x^3-7x+4), we can start by combining like terms.

First, we need to distribute the negative sign to the second set of parentheses:

(9x^3+2x^2-5x+4) - 1(5x^3-7x-4)

= 9x^3 + 2x^2 - 5x + 4 - 5x^3 + 7x - 4 (distributing the negative sign)

= (9x^3 - 5x^3) + 2x^2 - 5x + (4 - 4 + 7x) (grouping like terms)

= 4x^3 + 2x^2 + 2x (combining like terms)

Therefore, the simplified expression is 4x^3 + 2x^2 + 2x.

We cannot factor this expression further as there are no common factors between the terms.

1) Remove parentheses.

[tex]9x^{3}+ 2x^{2} -5x+4-5x^{3} +7x-4[/tex]

2) Collect like terms.

[tex](9x^{3}-5x^{3})+2x^{2} +(-5x+7x)+(4-4)[/tex]

3) Simplify.

[tex]4x^{3}+2x^{2}+2x[/tex]

Answer:

[tex]y\:=\:-\frac{3}{5}x\:+1[/tex]

Step-by-step explanation:

Linear approximation to g(x) at x=0:

For this, we will require to compute g(0), g'(0), and use the formula g(x) ≈ g(0) + g'(0)x.

Let's first find g(0):

g(0) = 3(0) + 4cos(1/2(0)) = 4

Next, let's find g'(x):

g(x) = 3x + 4cos(1/2 x)

g'(x) = 3 - 2sin(1/2 x)

Evaluate g'(0):

g'(0) = 3 - 2sin(1/2(0)) = 3

So we have the linear approximation around x=0:

g(x) ≈ 4 + 3x.

Approximating 3(0.2) + 4cos(0.1) using linear approximation:

We will use the result from the first part:

g(x) ≈ 4 + 3x, so that g(0.2) ≈ 4 + 3(0.2) = 4.6

Now we will approximate 4cos(0.1) by using the linear approximation of cos(x) at x=0, which is:

cos(x) ≈ 1 - x

So that cos(0.1) ≈ 1 - 0.1 = 0.9

Now we use these approximations to approximate 3(0.2) + 4cos(0.1):

3(0.2) + 4cos(0.1) ≈ 3(0.2) + 4(0.9) = 1.8 + 3.6 = 5.4

The approximation is 5.4.

A particle moves in such a way that at times t, its velocity is given by v(t) = 4e^(t/4) - 4 (cm/s). What is the displacement of the particle during the first 2 seconds?

The velocity is given by v(t) = 4e^(t/4) - 4. This is a continuous function, so we can calculate the displacement by finding the antiderivative of the velocity function and evaluating it between the limits of 0 and 2.

We can also notice that the velocity function is the derivative of the displacement function, so the displacement function is simply the antiderivative of the velocity function: s(t) = ∫v(t) dt.

Let's compute the displacement for the first 2 seconds by evaluating the antiderivative between 0 and 2. We can use the formula for the antiderivative of an exponential function ∫e^x dx = e^x + C.

s(t) = ∫v(t) dt = 4e^(t/4) - 4t + C

The constant of integration is arbitrary and we will determine it by using the initial condition that s(0) = 0:

s(0) = 4e^(0/4) - 4(0) + C = 4 - 0 + C = 4 => C = 0

Now we have:

s(t) = 4e^(t/4) - 4t

The displacement for the first 2 seconds is:

s(2) - s(0) = (4e^(2/4) - 4(2)) - (4e^(0/4) - 4(0)) = (4e^(1/2) - 8) - 4 = 4

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The polynomial fraction division (1/x + 4) ÷ (x - 3/x² + 7x + 12) = (x + 3)/(x - 3)

A polynomial is a mathematical expression in which the least power of the unknown is 2.

Since we are given the polynomial expression

(1/x + 4) ÷ (x - 3/x² + 7x + 12).

We proceed to divide and simplify as follows.

(1/x + 4) ÷ (x - 3/x² + 7x + 12) = (1/x + 4) × (x² + 7x + 12)/(x - 3)

= (x² + 7x + 12)/(x - 3)(x + 4)

To simplify further, we factorize the numerator. So, we have that

x² + 7x + 12 = x² + 4x + 3x + 12

Factorizing, we have that

x² + 4x + 3x + 12 = x(x + 4) + 3(x + 4)

= (x + 3)(x + 4)

So, we have that

x² + 7x + 12 = (x + 3)(x + 4)

So, substituting this into the fraction, we have that

(x² + 7x + 12)/(x - 3)(x + 4) = (x + 3)(x + 4)/(x - 3)(x + 4)

Cancelling out (x + 4), we have that

(x² + 7x + 12)/(x - 3)(x + 4) = (x + 3)/(x - 3)

So, we have (1/x + 4) ÷ (x - 3/x² + 7x + 12) = (x + 3)/(x - 3)

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