George was given 11 grams of medicine, but the full dose is supposed to be 25 grams. What percent of his full dose did George receive? Solve with percent table and equivalent fractions

Answer:

should be 53 if im right

[tex]\sin(3x+13)=\cos(4x)\\\sin(3x+13)=\cos(90-4x)\\3x+13=90-4x\\7x=77\\x=11[/tex]

Answer:

15.7 years

Step-by-step explanation:

we know that

The deforestation is a exponential function of the form

[tex]y=a(b)^{x}[/tex]

where

y ----> the number of trees still remaining in the forest

x ----> the number of years

a is the initial value (a=500,000 threes)

b is the base

b=100%-4.7%=95.3%=95.3/100=0.953

substitute

[tex]y=500,000(0.953)^{x}[/tex]

The linear equation of planting threes in the region is equal to

[tex]y=15,000x[/tex]

using a graphing tool

Solve the system of equations

The intersection point is (15.7,235,110)

see the attached figure

therefore

For x=15.7 years

The number of trees they have planted will be equal to the number of trees still remaining in the forest

Answer: There is only one solution, and it is viable.

Step-by-step explanation:

Intersecting chords theorem:

[tex]7\cdot7=10x[/tex]

i.e. when two chords intersect, the products of the resulting line segments' lengths are equal. Then

[tex]10x=49\implies x=\dfrac{49}{10}=\boxed{4.9}[/tex]

Answer:

4.9

Step-by-step explanation:

got it right

Answer: There are 22 whole cheese sandwiches that can be prepared.

Step-by-step explanation:

Since we have given that

Number of slices of bread = 44

Number of slices of cheese = 75

According to question, a cheese sandwich consists of 2 slices of bread and 3 slices of cheese.

So, we need to find the number of whole cheese sandwiches that can be prepared.

Number of sandwich containing only slice of bread is given by

[tex]\dfrac{44}{2}=22[/tex]

Number of sandwich containing only slice of cheese is given by

[tex]\dfrac{75}{3}=25[/tex]

As we know that each sandwich should contain both slice of bread and slice of cheese.

So, Least of (22, 25) = 22

Hence, there are 22 whole cheese sandwiches that can be prepared.

Answer:

1) x = 17,  line a parallel to line b

2) x = 18,  line a parallel to line b

Step-by-step explanation:

If line a parallel to line b then (10x - 40) + 50 = 180

Solve for x

10x - 40 + 50 = 180

Combine like terms

10x + 10 = 180

10x = 170

  x = 17

x = 17,  line a parallel to line b

-------------------------------------------------

If line a parallel to line b then 5x - 16 = 74

Solve for x

5x - 16 = 74

5x = 90

  x = 18

x = 18,  line a parallel to line b

The value of x which makes line a parallel to line b can be found by equating the slopes of the two lines and solving for x.

In mathematics, two lines a and b are parallel if and only if their slopes are equal. When we are given the equations of the lines and are asked to find the value of x that make the lines parallel, we start by setting the slopes of the two lines equal to each other. Let's assume now that line a is represented by y = mx + b1 and line b by y = nx + b2. In order for line a to be parallel to line b, m must be equal to n. Therefore, you solve for x from the equation m=n.

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Answer:

1. Y

2. N

3. N

4. N

Step-by-step explanation:

Let's use the second equation, since it seems to be easier to use.

To check if an ordered pair is a solution, plug it in to the equation.

1. [tex]-10+18=8[/tex] --> Y

2. [tex]25-12=13[/tex] --> N

3. [tex]0-9=-9[/tex] --> N

4. [tex]35-27=8[/tex] --> N

Edit : The 4th equation doesn't work for the first equation, whereas the first one still does.

Answer:

(-2,-6)

Step-by-step explanation:

-9x +2y = 6

5x - 3y = 8

1) Make one of the coefficients the same - y.

-9x +2y = 6 * 3

5x - 3y = 8 * 3

-27x +6y = 18

10x - 6y = 16

2) Add the new equations.

(-27x +6y = 18) + (10x - 6y = 16)

(-27x +6y) + (10x - 6y) = 18 + 16

-17x = 34

3) Divide to find the value of x

-17x = 34

x = 34/-17

x = -2

4) Substitute x into either equation to find the value of y.

-9(-2) +2y = 6

18 +2y = 6

2y = -12

y = -12/2

y = -6

5(-2) - 3y = 8

-10 - 3y = 8

-3y = 18

y = 18/-3

y = -6

Your answer is (-2,-6).

Answer: Hence, Option 'B' is correct.

Step-by-step explanation:

Since we have given that

Number of births = 1400

Number of birth of girls = 1378

Number of birth of boys is given by

[tex]1400-1378\\\\=22[/tex]

so, the number of girls is significantly higher than the number of boys.

So, the number of births of girls is significantly high.

Hence, Option 'B' is correct.

B. The number of girls is significantly high.

When evaluating whether the number of girl births in a sample is significantly high or low, we can reference the expected natural ratio of girls to boys, which is typically 100:105. Given that in the scenario 1378 out of 1400 births were girls, this significantly deviates from the expected natural ratio. For comparison, an article in Newsweek states that the natural ratio is 100:105, and in China, it is 100:114, which equals 46.7 percent girls. If we consider a sample where out of 150 births, there are 60 girls (or 40%), this is lower than the expected percentage based on China's statistics but not implausible. However, in the case of the scenario with 1378 girls out of 1400 births, the proportion of girls is approximately 98.43%, which seems very unlikely given the natural ratio, suggesting an unusual or non-random process may be involved.

Therefore, based on subjective judgment and without applying more precise statistical tests, the number of girls being 1378 out of 1400 births is significantly high compared to natural birth rates or the stated birth rate in China. This leads us to select the correct answer: B. The number of girls is significantly high.

Answer:

The average rate of change of f(t) from t = 3 seconds to t = 5 seconds is __-80___feet per second.

Step-by-step explanation:

The average change rate m is calculated using the following formula

[tex]m=\frac{f(t_2)-f(t_1)}{t_2-t_1}[/tex]

In this case [tex]f(t) = -16t^2 + 48t + 100[/tex],  [tex]t_2 = 5\ s\ \ , t_1=3\ s[/tex]

Then

[tex]f(t_2) = f(5) =-16(5)^2 + 48(5) + 100[/tex]

[tex]f(t_2) = -60[/tex]

[tex]f(t_1) = f(3) =-16(3)^2 + 48(3) + 100[/tex]

[tex]f(t_1) = 100[/tex]

Finally

[tex]m=\frac{(-60)-100}{5-3}[/tex]

[tex]m=-80[/tex]

Answer:

  66.2

Step-by-step explanation:

We know that the middle 68% of the distribution is between -1 and +1 standard deviations from the mean. -1 standard deviation is a score of ...

  69.9 - 3.7 = 66.2

_____

Comment on the question

The way the question is worded, "A" can be any value less than 68.1, down to -∞. The question does does not require the A-B range to be symmetrical.

Answer: a) 30% and b) 45%

Step-by-step explanation:

Since we have given  that

Probability that adults regularly consume coffee P(C) = 45% = 0.45

Probability that adults regularly consume carbonated soda P(S) = 40% = 0.40

Probability that adults regularly consume atleast one of these two products P(C∪S) = 55% = 0.55

a) What is the probability that a randomly selected adult regularly consumes both coffee and soda?

As we know that

P(C∪S ) = P(C) +P(S)-P(C∩S)

[tex]0.55=0.45+0.40-P(C\cap S)\\\\0.55=0.85-P(C\cap S)\\\\0.55-0.85=-P(C\cap S)\\\\-0.30=-P(C\cap S)\\\\P(C\cap S)=0.30=30\%[/tex]

b) What is the probability that a randomly selected adult doesn't regularly consume at least one of these two products?

P(C∪S)'=n(U)-P(C∪S)

[tex]\\P(C\cup S)'=100-55=45\%[/tex]

Hence, a) 30% and b) 45%

For this case we must solve the following equation:

[tex]4 ^ {x + 1} = 21[/tex]

We find Neperian logarithm on both sides:

[tex]ln (4 ^ {x + 1}) = ln (21)[/tex]

According to the rules of Neperian logarithm we have:

[tex](x + 1) ln (4) = ln (21)[/tex]

We apply distributive property:

[tex]xln (4) + ln (4) = ln (21)[/tex]

We subtract ln (4) on both sides:

[tex]xln (4) = ln (21) -ln (4)[/tex]

We divide between ln (4) on both sides:

[tex]x = \frac {ln (21)} {ln (4)} - \frac {ln (4)} {ln (4)}\\x = \frac {ln (21)} {ln (4)} - 1\\x = 1,19615871[/tex]

Rounding:

[tex]x = 1.1962[/tex]

Answer:

x = 1.1962

Answer: [tex]x[/tex]≈[tex]1.196[/tex]

Step-by-step explanation:

Given the equation [tex]4^{(x + 1)} = 21[/tex] you need to solve for the variable "x".

Remember that according to the logarithm properties:

[tex]log_b(b)=1[/tex]

[tex]log(a)^n=nlog(a)[/tex]

Then, you can apply  [tex]log_4[/tex] on both sides of the equation:

[tex]log_4(4)^{(x + 1)} = log_4(21)\\\\(x + 1)log_4(4) = log_4(21)\\\(x + 1) = log_4(21)[/tex]

Apply the Change of base formula:

 [tex]log_b(x) = \frac{log_a( x)}{log_a(b)}[/tex]

Then you get:

[tex]x =\frac{log(21)}{log(4)}-1[/tex]

[tex]x[/tex]≈[tex]1.196[/tex]

This question involves operations on sets to identify specific members based on conditions. Part a) resolves to D. {2,4}, while part b) finds the solution to be B. {2,4,6,7,8,9,10,11}, highlighting the application of intersection, union, and complement operations in set theory.

To solve these problems, we need to understand the operations on sets such as intersection (A∩B), union (A∪B), and the complement of a set (Bc). For part a), we identify set A as {2,4,6,8,10}, B as {1,3,5}, and C as {1,2,3,4,5}. A∩(B∪C) means we're looking for the intersection of A with the union of B and C. Since B∪C = {1,2,3,4,5}, intersecting this with A gives us D. {2,4} as the answer.

For part b), (A∩Bc)∪(B∩C)c means we're looking at elements in A but not in B, combined with elements not in both B and C. Since Bc = {6,7,8,9,10,11} and (B∩C)c = {6,7,8,9,10,11}, union these two gives us answer B. {2,4,6,7,8,9,10,11}, by including A∩Bc = {2,4,6,8,10} and excluding duplicates when union with (B∩C)c.

Answer:

$1314.37

Step-by-step explanation:

We have to calculate final value i.e. balance to earn $100,000 annually from interest.

= [tex]\frac{100,000}{0.08}[/tex] = $1,250,000

Now, N = n × y  = 12 × 25 = 300

         I  = 8% =  APR = 0.08

        PV = 0  = PMT = 0

        FV = 1,250,000 = A

[tex]A=\frac{PMT\times [(1+\frac{apr}{n})^{ny}-1]}{\frac{apr}{n}}[/tex]

[tex]PMT=\frac{A\times (\frac{APR}{n})}{[(1+\frac{APR}{n})^{ny}-1]}[/tex]

[tex]PMT=\frac{1,250,000\times (\frac{0.08}{12})}{[(1+\frac{0.08}{12})^{12\times 25}-1]}[/tex]

[tex]PMT=\frac{1,250,000\times (0.006667)}{[(1+\frac{0.08}{12})^{12\times 25}-1]}[/tex]

[tex]PMT=\frac{1,250,000\times (0.006667)}{[(1+0.006667)^{300}-1]}[/tex]

[tex]PMT=\frac{\frac{25000}{3}}{[1.006667^{300}-1]}[/tex]

[tex]PMT=\frac{\frac{25000}{3}}{6.340176}[/tex]

Monthly payment (PMT) = $1314.369409 ≈ $1314.37

$1314.37 is required monthly payment in order to $100,000 interest.

Answer:

take 47.99 x .20 = 9.598

$9.60 off

then take 47.99 - 9.60 = $ 38.39

take 38.39 x .06 = 2.3034

$ 2.30 (tax)

add 38.39 + 2.30 = $40.69 or $40.70 is the final purchase price

(the two amounts depends on your choice answer or how it is rounded)

Step-by-step explanation:

In this question (https://brainly.com/question/12792658) I derived the Taylor series for [tex]\mathrm{sinc}\,x[/tex] about [tex]x=0[/tex]:

[tex]\mathrm{sinc}\,x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}[/tex]

Then the Taylor series for

[tex]f(x)=\displaystyle\int_0^x\mathrm{sinc}\,t\,\mathrm dt[/tex]

is obtained by integrating the series above:

[tex]f(x)=\displaystyle\int\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}\,\mathrm dx=C+\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}[/tex]

We have [tex]f(0)=0[/tex], so [tex]C=0[/tex] and so

[tex]f(x)=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}[/tex]

which converges by the ratio test if the following limit is less than 1:

[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(-1)^{n+1}x^{2n+3}}{(2n+3)^2(2n+2)!}}{\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}}\right|=|x^2|\lim_{n\to\infty}\frac{(2n+1)^2(2n)!}{(2n+3)^2(2n+2)!}[/tex]

Like in the linked problem, the limit is 0 so the series for [tex]f(x)[/tex] converges everywhere.

The Taylor series for the function f(x) = ∫ sinc(t)dt based at 0 is derived from the Taylor series of sinc(x) by integrating it term by term, given in summation notation as ∑ (-1)ⁿ * xⁿ⁺¹ / (n+1)! for n=0 to n=∞. The series converges for all real numbers (-∞, ∞).

In order to find the Taylor series for the function f(x) = ∫ sinc(t)dt based at 0, one can use the Taylor series for sinc(x) and integrate term by term. We know the Taylor series for sinc(x) is x - x³/3! + x⁵/5! - ..., so the Taylor series for f(x) can be written as x²/2 - x⁴/4*3! + x⁶/6*5! - ... . In summation notation, this is ∑ (-1)ⁿ * xⁿ⁺¹ / (n+1)! for n=0 to n=∞.

The Taylor series for any function converges to the function itself within a certain interval called the radius of convergence. For the Taylor series of sinc(x), due to the nature of sine being bounded between -1 and 1, the series will converge for all real numbers (-∞, ∞).

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Answer:

  f(x) = x^4 +x^3 -2x^2 -8x +9

Step-by-step explanation:

You know that the anitderivative of ax^b is ax^(b+1)/(b+1). The first antiderivative is ...

  f'(x) = 4x^3 +3x^2 -4x +p . . . . . where p is some constant

The second antiderivative is ...

  f(x) = x^4 +x^3 -2x^2 +px +q . . . . where q is also some constant

Then the constants can be found from ...

  f(0) = q = 9

  f(1) = 1 + 1 - 2 +p + 9 = 1

  p = -8

The solution is ...

  f(x) = x^4 +x^3 -2x^2 -8x +9

_____

The graphs verify the results. The second derivative is plotted against the given quadratic, and they are seen to overlap. The function values at x=0 and x=1 are the ones specified by the problem.

Final answer:

To find f(x) given f''(x) = 12x² + 6x − 4, one must integrate twice and use the initial conditions f(0) = 9 and f(1) = 1 to solve for the constants. The final function is f(x) = x⁴ + x³ - 2x² - 8x + 9.

Explanation:

The question asks to find the antiderivative f(x) given its second derivative f''(x) =  12x² + 6x − 4, and two initial conditions, f(0) = 9, and f(1) = 1. To solve for f(x), we first integrate the second derivative twice to get the original function.

Integrating f''(x), we get:

f'(x) = ∫( 12x² + 6x - 4)dx = 4x³ + 3x² - 4x + C

We then integrate f'(x) to find f(x):

f(x) = ∫(4x³ + 3x² - 4x + C)dx = x⁴ + x³ - 2x² + Cx + D

Using the initial conditions:

Therefore, the original function is f(x) =  x⁴ + x³ - 2x² - 8x + 9.

Answer:

Step-by-step explanation:

A ratio is defined as a comparison of two amounts by division.It is of the form of [tex]\frac{p}{q}[/tex] where p and q are the quantities.

A. 1580 people/square mile in division form

1580[tex]\frac{people}{\text{square mile}}[/tex]

This satisfies ratio definition as this compares two quantities People and Square mile and is of the form   [tex]\frac{p}{q}[/tex]

Where,

p: People, q:  Square mile

Example,

[tex]\frac{1580\times people}{1\times squaremile}[/tex]

expresses that per 1 square mile there are 1580 people. Thus rate satisfies the ratio definition.

B. 360 kilowatt-hour/4 months

in division form [tex]\frac{\text{360 kilowatt-hour}}{\text{4months}}[/tex]

This satisfies ratio definition as this compares two quantities kilowatt-hour and months and is of the form   [tex]\frac{p}{q}[/tex]

where, p: kilowatt-hour and q: people

Example,

[tex]\frac{\360\times kilowatt-hour}{4\times months}[/tex]

expresses that per 4 months there are 360 kilowatt-hour. Thus, rate satisfies the ratio definition.

C. 450 people/year

In division form 450[tex]\frac{people}{year}[/tex]

This satisfies ratio definition as this compares two quantities people and year and is of the form   [tex]\frac{p}{q}[/tex]

where, p: people and q: year

Example,

[tex]\frac{450\times people}{1\times\text{year}}[/tex]

expresses that per year there are 450 people. Thus, rate satisfies the ratio definition.

D. 355 calories/6 ounces

In division form [tex]\frac{355clories}{6ounces}[/tex]

This satisfies ratio definition as this compares two quantities calories and ounces and is of the form   [tex]\frac{p}{q}[/tex]

where p: calories and q: ounces

Example,

[tex]\frac{355\times calories}{6\times\text{ounces}}[/tex]

expresses that per 6 ounces there are 355 calories. Thus, rate satisfies the ratio definition.

Answer:(-6,312), (6,312)

Step by Step explanation:

10x^2-y=48

y=-48+10x^2

2(-48+10x^2)=16x^2+48

Solve the equation for x.

x=-6

x=6

y=-48+10(-6)^2

y=-48+10×6^2

y=312

y=312

Answer:

ab + 3a + 8b + 24

Step-by-step explanation:

(a + 8)(b + 3)

a(b + 3) + 8(b + 3)

ab + 3a + 8b + 24

Try this solution:

the rule: if f1>f2, then E1>E2, where f1;f2 - frequency, E1;E2 - energy of light.

The formula is L=c/f, where L - the wavelength, c - 3*10⁸ m/s, f - frequency.

Frequency for the wavelength 519 nm. is:

[tex]f=\frac{c}{L}=\frac{3*10^8}{519*10^{-9}}=\frac{3*10^17}{519}=578034682080924=5.78*10^{14}( \frac{1}{sec})[/tex]

Answer: the energy of light of wavelength 519 nm.

Answer:

Two lines are parallel when they share the same slope.

Step-by-step explanation:

Two lines are parallel when they share the same slope.

The slope-intercept form of the equation of a line is: y=mx + b, where 'm' is the slope and 'b' the y-intercept.

If two equations have the same value for 'm', then those lines are parallel, for example:

y = 3x + 8 (Red line)

y = 3x + 5 (Blue line)

y = 3x - 10 (Green line)

All the equations stated above are parallel, to show that, I'm attaching the graph of the equations :).

Answer: 640

Step-by-step explanation:

Since the two triangles are similar we can simply multiply the lesser triangle's area by a constant to get our answer.

Polygon FGHIJ is ABCDE with a scale change of 4

For the reason that we are dealing with area, we will multiply 40 by 4² in stead of just 4.

40 * 16 = 640

Answer:

B. 640

Step-by-step explanation:

got it right 2021

(a+3)(a-2)

Multiply the two brackets together

a^2-2a+3a+3*-2

a^2+a-6

Answer is a^2+a-6

ANSWER

[tex]{a}^{2} + a - 6[/tex]

EXPLANATION

The given expression is

[tex](a + 3)(a - 2)[/tex]

We expand using the distributive property to obtain:

[tex]a(a - 2) + 3(a - 2)[/tex]

We multiply out the parenthesis to get:

[tex] {a}^{2} - 2a + 3a - 6[/tex]

Let us now simplify by combining the middle terms to obtain;

[tex]{a}^{2} + a - 6[/tex]

Answer:

the answer is a

Step-by-step explanation:

i just know

Final answer:

The probability of choosing a green ball from the three boxes, given their individual selection probabilities and color distributions, is calculated using the law of total probability. The overall probability of selecting a green ball is found to be 29/80, or roughly 36.25%.

Explanation:

The question asks for the probability of choosing a green ball from three different boxes, given their individual probabilities of being chosen and the distribution of red and green balls in each box. To solve this, we employ the law of total probability which combines the probability of each event (selecting a box) with the conditional probability of finding a green ball within that selected box.

Box 1: Probability of green ball = 5 green balls / (5 red + 5 green) = 1/2

Box 2: Probability of green ball = 3 green balls / (7 red + 3 green) = 3/10

Box 3: Probability of green ball = 4 green balls / (6 red + 4 green) = 2/5

The overall probability is calculated as: P(Green) = P(Box 1) * P(Green|Box 1) + P(Box 2) * P(Green|Box 2) + P(Box 3) * P(Green|Box 3) = (1/4) * (1/2) + (1/2) * (3/10) + (1/4) * (2/5) = 1/8 + 3/20 + 1/10 = 29/80.

Therefore, the probability that the ball chosen is green is 29/80 or approximately 36.25%.

[tex]\bf 12~~,~~\stackrel{12+3}{15}~~,~~\stackrel{15+3}{18}~~,~~\stackrel{18+3}{21}~\hspace{10em}\stackrel{\textit{common difference}}{d=3} \\\\[-0.35em] ~\dotfill\\\\ n^{th}\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} a_n=n^{th}\ term\\ n=\textit{term position}\\ a_1=\textit{first term}\\ d=\textit{common difference}\\ \cline{1-1} a_1=12\\ d=3 \end{cases} \\\\\\ a_n=12+(n-1)3\implies a_n=12+3n-3\implies a_n=3n+9[/tex]

Answer:

tₙ = 3(3 + n)

Step-by-step explanation:

Points to remember

nth term of an AP

tₙ = a + (n - 1)d

Where a - first term of AP

d - Common difference of AP

To find the nth term  

The given series is,

12,15,18,21 .....

Here a = 12 and d = 15 - 12 = 3

tₙ = a + (n - 1)d

  = 12 + (n - 1)3

  =12 + 3n - 3

  = 9 + 3n

  = 3(3 + n)

Therefore tₙ = 3(3 + n)

Answer:

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

Step-by-step explanation:

We are given with Second-order derivative of function f(x).

[tex]f''(x)=9-\frac{3}{x}[/tex]

We need to find Third-order derivative of the function f(x).

[tex]f''(x)=9-\frac{3}{x}=9-3x^{-1}[/tex]

We know that,

f'''(x) = (f''(x))'

So,

[tex]f'''(x)=\frac{\mathrm{d}\,f''(x)}{\mathrm{d} x}[/tex]

[tex]f'''(x)=\frac{\mathrm{d}\,(9-3x^{-1})}{\mathrm{d} x}[/tex]

[tex]f'''(x)=\frac{\mathrm{d}\,9}{\mathrm{d} x}-\frac{\mathrm{d}\,3x^{-1}}{\mathrm{d} x}[/tex]

[tex]f'''(x)=0-3\frac{\mathrm{d}\,x^{-1}}{\mathrm{d} x}[/tex]

[tex]f'''(x)=-3(-1)x^{-1-1}[/tex]

[tex]f'''(x)=3x^{-2}[/tex]

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

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

Answer:

Yes they do.

And yes they do follow a normal distribution.

Percentages are close to 68-95-99.7%, we can declare that yes, the 68-95-99.7% rule is roughly followed and yes data appear to follow a normal distribution.

It's the probability curve of a continuous distribution that's most likely symmetric around the mean. On the Z curve, at Z=0, the chance is 50-50. A bell-shaped curve is another name for it.

We have a data of final exam scores of 20 Introductory.

a) Range of 1 standard deviation:

(77.7 – 8.44, 77.7 + 8.44)                [69.3, 86.1]

Range of 2 standard deviation:

(77.7 – 2(8.44), 77.7 + 2(8.44))            [60.8, 94.6]

Range of 3 standard deviation:

(77.7 – 3(8.44), 77.7 + 3(8.44))           [52.4, 103.0]

Number of data points lie within 1 standard deviation = 14

Percent of data points lie within 1 SD = (14/20)×100 = 70%

Number of data points lie within 2 SD = 19

Percent of data points lie within 1 SD = (19/20)×100 = 95%

Number of data points lie within 3 SD = 20

Percent of data points lie within 1 SD = (20/20)×100 = 100%

Because these percentages are close to 68-95-99.7%, we can declare that yes, the 68-95-99.7% rule is roughly followed.

b)

Because the histogram in the graph is symmetric, and the normal probability plot reveals that the points are very close to a straight line, the data appears to follow a normal distribution.

Thus, percentages are close to 68-95-99.7%, we can declare that yes, the 68-95-99.7% rule is roughly followed and yes data appear to follow a normal distribution.

Learn more about the normal distribution here:

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The work is equal to the line integral of [tex]\vec F[/tex] over each line segment.

Parameterize the paths

The work done by [tex]\vec F[/tex] over each segment (call them [tex]C_1,\ldots,C_4[/tex]) is

[tex]\displaystyle\int_{C_1}\vec F\cdot\mathrm d\vec r_1=\int_0^2\vec0\cdot\vec\imath\,\mathrm dt=0[/tex]

[tex]\displaystyle\int_{C_2}\vec F\cdot\mathrm d\vec r_2=\int_0^1(t^2\,\vec\imath+24t\,\vec\jmath+32t^2\,\vec k)\cdot(4\,\vec\jmath+\vec k)\,\mathrm dt=\int_0^1(96t+32t^2)\,\mathrm dt=\frac{176}3[/tex]

[tex]\displaystyle\int_{C_3}\vec F\cdot\mathrm d\vec r_3=\int_0^2(\vec\imath+(24-12t)\,\vec\jmath+32\,\vec k)\cdot(-\vec\imath)\,\mathrm dr=-\int_0^2\mathrm dt=-2[/tex]

[tex]\displaystyle\int_{C_4}\vec F\cdot\mathrm d\vec r_4=\int_0^1((1-t)^2\,\vec\imath+2(4-4t)^2\,\vec k)\cdot(-4\,\vec\jmath-\vec k)\,\mathrm dt=-2\int_0^1(4-4t)^2\,\mathrm dt=-\frac{32}3[/tex]

Then the total work done by [tex]\vec F[/tex] over the particle's path is 46.

The total work done by the force field as the particle follows this path is 4/3 units of work.

To find the total work done by the force field as the particle moves along the path, we'll calculate the line integrals for each segment and then sum them up:

For the first segment (from the origin to (2, 0, 0)):

Parameterization: r1(t) = (2t, 0, 0), t from 0 to 1.

Work on this segment: ∫[0,1] F(r1(t)) · r1'(t) dt

F(r1(t)) = (0, 0, 0), as z = 0.

r1'(t) = (2, 0, 0).

∫[0,1] F(r1(t)) · r1'(t) dt = ∫[0,1] (0) dt = 0.

For the second segment (from (2, 0, 0) to (2, 4, 1)):

Parameterization: r2(t) = (2, 4t, t), t from 0 to 1.

Work on this segment: ∫[0,1] F(r2(t)) · r2'(t) dt

F(r2(t)) = (t^2, 0, 8t^2).

r2'(t) = (0, 4, 1).

∫[0,1] F(r2(t)) · r2'(t) dt = ∫[0,1] (0 + 0 + 8t^2) dt = ∫[0,1] 8t^2 dt = [8t^3/3] from 0 to 1 = (8/3).

For the third segment (from (2, 4, 1) to (0, 4, 1)):

Parameterization: r3(t) = (2 - 2t, 4, 1), t from 0 to 1.

Work on this segment: ∫[0,1] F(r3(t)) · r3'(t) dt

F(r3(t)) = (1, 0, 8).

r3'(t) = (-2, 0, 0).

∫[0,1] F(r3(t)) · r3'(t) dt = ∫[0,1] (-2) dt = [-2t] from 0 to 1 = -2.

For the fourth segment (from (0, 4, 1) back to the origin):

Parameterization: r4(t) = (0, 4 - 4t, 1), t from 0 to 1.

Work on this segment: ∫[0,1] F(r4(t)) · r4'(t) dt

F(r4(t)) = (1, 0, 32 - 8t).

r4'(t) = (0, -4, 0).

∫[0,1] F(r4(t)) · r4'(t) dt = ∫[0,1] (0 + 0 + 0) dt = 0.

Now, sum up the work done on each segment:

Total work = 0 + (8/3) - 2 + 0 = 4/3.

So, the total work done by the force field as the particle follows this path is 4/3 units of work.

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