Six years ago my son was one-third my age at that time.

Six years from now he will be one-half my age at that time.

How old is my son?

Step-by-step explanation:

1.Assuming the same sample size and considering the same value for the errors ( not taking into consideration the type of music, the volume of the sound and cow familiar the runner is with that type of stimuli, age group, time of the day/ number of days, running conditions like wether and equipment, distance) one can state:

A. Music has no influence over the running speed ( when jogging) in Washington DC

B.When listening to music, people ( in Washington DC) run faster while jogging

The mean running speed is a simple, ponderate or other type of mean ( that takes into consideration the variations of speed at the beginning and by the end of the race?

Answer: 297 miles

Step-by-step explanation:

The drive from Waterton to Middleton is 76 miles, from Middleton to Oak Hill is 87 miles, and from Oak Hill directly to Waterton it is 134 miles.

Then: [tex]76+87+134=297[/tex]

So Juan drives 297 miles.

The general solution of this differential equation is y(t) = c1 cos(5t) + c2 sin(5t), where c1 and c2 are constants determined by the initial conditions.

Differentiating the second equation with respect to t, we get: d^2y/dt^2 = -5 dx/dt, Substituting dx/dt from the first equation, we get: d^2y/dt^2 = -5(-5y) = 25y.

This is a second order differential equation in y. The general solution of this differential equation is y(t) = c1 cos(5t) + c2 sin(5t), where c1 and c2 are constants determined by the initial conditions.

To find x as a function of t, we can substitute y(t) into the first equation and solve for x: dx/dt = -5y = -5(c1 cos(5t) + c2 sin(5t)) , Integrating both sides with respect to t, we get: x(t) = -c1 sin(5t) + c2 cos(5t) + k

where k is a constant of integration. Using the initial conditions x(0) = 4 and y(0) = 1, we can solve for the constants c1, c2, and k: x(0) = -c1 sin(0) + c2 cos(0) + k = c2 + k = 4, y(0) = c1 cos(0) + c2 sin(0) = c1 = 1

Substituting c1 = 1 and c2 + k = 4 into the equation for x, we get:

x(t) = -sin(5t) + 4

So the solution to the system of differential equations with initial conditions x(0) = 4 and y(0) = 1 is x(t) = -sin(5t) + 4 and y(t) = cos(5t).

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Answer:

Step-by-step explanation: manej

Final answer:

In algebra, a tenth of a number is algebraically represented by multiplying the number by [tex]10^{-1}[/tex], which is equivalent to dividing the number by 10. This application of negative exponents simplifies expressions, especially in scientific notation, making it easier to work with large and small quantities.

Explanation:

In algebra, when we refer to a tenth of a number, we are usually dealing with fractions or exponential notation. A tenth of a number can be represented algebraically as the number divided by 10, which is the same as multiplying the number by [tex]10^{-1}[/tex]. This is because negative exponents indicate the reciprocal of a number; in other words, 10-1 equals 1/10 or 0.1.

This concept relates to the powers of ten and how each power of 10 affects the size of a number. For instance, 102 is 100, and 101 is 10, which is ten times smaller than 100. Conversely, 100 is 1, which is ten times smaller than 10, and thus, logically, [tex]10^{-1}[/tex] is 0.1, which is ten times smaller still. In expressing measurements in scientific work, especially for very small numbers, we frequently use this exponential form.

Thus, a tenth of an algebraic expression would mean multiplying the expression by [tex]10^{-1}[/tex] or dividing the expression by 10. This process is a form of simplification and re-scaling of numbers that are commonly used in scientific notation, which includes both positive and negative exponents. By understanding these principles, one can efficiently work with both large and small quantities in scientific and mathematical contexts.

Answer:

  ∠ABC = 84°

  ∠BEF = 64°

Step-by-step explanation:

∠ABC is supplementary to the 96° angle shown, so is 180° -96° = 84°.

__

∠ABD, marked as (x+y)°, is a vertical angle with ∠EBC, so has the same measure, 96°. ∠BEF, marked as y°, is a vertical angle with the one marked 2x°.

These relationships can be expressed as two equations:

Using the second of these equations to substitute for y in the first equation, we have ...

  x + 2x = 96

  x = 96/3 = 32

  y = 2x = 2·32 = 64 . . . . . . substitute the value of x into the second equation

Then ∠BEF = 64°.

Answer: There are 17576000 ways to generate different license plates.

Step-by-step explanation:

Since we have given that

Numbers are given = 0 to 9 = 10 numbers

Number of letters = 26

We need to generate the license plate numbers.

Since there are repetition allowed.

We would use "Fundamental theorem of counting".

So, the number of different license numbers may be created as given as

[tex]26\times 26\times 26\times 10\times 10\times 10\\\\=26^3\times 10^3\\\\=17576\times 1000\\\\=17576000[/tex]

Hence, there are 17576000 ways to generate different license plates.

Answer:

p(F)=0.73

Step-by-step explanation:

we have by identity

[tex]p(A\cup B)=p(A)+p(B)-p(A\cap B)[/tex]

Thus for given events E and F

we have[tex]p(E\cup F)=p(E)+p(F)-p(E\cap F)[/tex]

Applying values we get

[tex]p(F)=p(E\cup F)+p(E\cap F)-p(E)[/tex]

Thus

p(F) = 0.82+0.09-0.18

p(F) = 0.73

Answer with explanation:

Given two sets

          A={2, 4, 6, 8} and B={2, 3, 5, 7, 9}

⇒n(S)=Cardinality of a set

Means the number of distinct elements in a Set is called it's cardinal number.

→In Set A,total number of distinct elements is 4.

n(A)=4

Answer:

i). [tex]\$ 7500[/tex]

ii).[tex]\$ 67500[/tex]

Step-by-step explanation:

Given in the question-

Saving rate is s(t)= 3000(t+2)

We know that savings in the 1st year can be calculated as

    [tex]\int_{0}^{1}3000(t+2).dt[/tex]

    [tex]3000\left [ \frac{t^{2}}{2}+2t \right ]_0^1[/tex]

    [tex]3000\left [ \frac{1}{2}+2 \right ][/tex]

 = [tex]\$ 7500[/tex]

So savings in the first 5 years can be calculated as

     [tex]\int_{0}^{5}3000(t+2).dt[/tex]

    [tex]3000\left [ \frac{t^{2}}{2}+2t \right ]_0^5[/tex]

    [tex]3000\left [ \frac{25}{2}+5 \right ][/tex]

 = [tex]\$ 67500[/tex]

Answer:

There are 26 investors which contributed 6,000

And 34 investors which contributed 3,000

Step-by-step explanation:

You need to set up a two equation problem

[tex]\left \{ {{258,000=6,000y+3,000x} \atop {60=y+x}} \right.[/tex]

Now you have to clear "x" or "y" from the second equation:

[tex]y = 60 - x[/tex]

And replace on the first equation:

[tex]258,000 = 6,000 (60 - x) + 3,000x\\258,000 = 360,000 - 6,000x + 3,000x\\3,000x = 360,000 - 258,000\\x = 102,000/3,000\\x = 34[/tex]

And now you use this "x" vale on the second equation

[tex]60 = y + x\\60 = y + 34\\60 - 34 = y\\y = 26[/tex]

There are 26 investors which contributed 6,000

And 34 investors which contributed 3,000

Answer:

A) (-17+5k,17-4k)

B)  (-4+3k,4-2k)

C) No integer pairs.

Step-by-step explanation:

To do this, I'm going to use Euclidean's Algorithm.

4x+5y=17

5=4(1)+1

4=1(4)

So going backwards through those equations:

5-4(1)=1

-4(1)+5(1)=1

Multiply both sides by 17:

4(-17)+5(17)=17

So one integer pair satisfying 4x+5y=17 is (-17,17).

What is the slope for this equation?

Let's put it in slope-intercept form:

4x+5y=17

Subtract 4x on both sides:

     5y=-4x+17

Divide both sides by 5:

      y=(-4/5)x+(17/5)

The slope is down 4 and right 5.

So let's show more solutions other than (-17,17) by using the slope.

All integer pairs satisfying this equation is (-17+5k,17-4k).

Let's check:

4(-17+5k)+5(17-4k)

-68+20k+85-20k

-68+85

17

That was exactly what we wanted since we were looking for integer pairs that satisfy 4x+5y=17.

Onward to the next problem.

6x+9y=12

9=6(1)+3

6=3(2)

Now backwards through the equations:

9-6(1)=3

9(1)-6(1)=3

Multiply both sides by 4:

9(4)-6(4)=12

-6(4)+9(4)=12

6(-4)+9(4)=12

So one integer pair satisfying 6x+9y=12 is (-4,4).

Let's find the slope of 6x+9y=12.

6x+9y=12

Subtract 6x on both sides:

      9y=-6x+12

Divide both sides by 9:

       y=(-6/9)x+(12/9)

Reduce:

       y=(-2/3)x+(4/3)

The slope is down 2 right 3.

So all the integer pairs are (-4+3k,4-2k).

Let's check:

6(-4+3k)+9(4-2k)

-24+18k+36-18k

-24+36

12

That checks out since we wanted integer pairs that made 6x+9y=12.

Onward to the last problem.

4x+10y=9

10=4(2)+2

4=2(2)

So the gcd(4,10)=2 which means this one doesn't have any solutions because there is no integer k such that 2k=9.

Answer:  C. 25

Step-by-step explanation:

Given : The average rat (of this strain) can learn to run this type of maze in a box without any special coloring : [tex]\mu=25[/tex]

The mean number of trials to learn the maze, for the rats tested with the colorful wallpaper= [tex]M= 11[/tex]

We know that the sampling distribution D is given by :-

[tex]\mu_D=\mu[/tex]

Similarly the mean of the distribution M in the given situation is given by :_

[tex]\mu_M=\mu=25[/tex]

The mean of the distribution M in the given situation is 25. Then the correct option is C.

It is also called the Gaussian Distribution. It is the most important continuous probability distribution. The curve looks like a bell, so it is also called a bell curve.

An experimental psychologist is interested in whether the color of an animal's surroundings affects the learning rate.

He tests 16 rats in a box with colorful wallpaper.

The average rate (of this strain) can learn to run this type of maze in a box without any special coloring in an average of 25 trials, with a variance of 64, and a normal distribution.

The mean number of trials to learn the maze, for the rats tested with the colorful wallpaper, is 11.

We know that the sampling distribution D is given by

μD = μ

Similarly, the mean of the distribution M in the given situation is given by

μD = μ = 25

More about the normal distribution link is given below.

https://brainly.com/question/12421652

Answer:

slope -2/5

y-intercept -3/5

Step-by-step explanation:

Slope-intercept form of a line is y=mx+b where m is the slope and b is the y-intercept.

Our goal is to write 2x+5y=-3 into y=mx+b to determine the slope and y-intercept.

So we need to isolate y.

2x+5y=-3

Subtract 2x on both sides:

   5y=-2x-3

Divide both side by 5:

   [tex]y=\frac{-2}{5}x-\frac{3}{5}[/tex]

Compare this to y=mx+b.

You should see m is -2/5 and b is -3/5

so the slope is -2/5 and the y-intercept is -3/5

Answer:

The slope is: [tex]-\frac{2}{5}[/tex] or [tex]-0.4[/tex]

The y-intercept is: [tex]-\frac{3}{5}[/tex] or [tex]-0.6[/tex]

Step-by-step explanation:

The equation of the line in Slope-Intercept form is:

[tex]y=mx+b[/tex]

Where "m" is the slope of the line and "b" is the y-intercept.

To write the given equation in this form, we need to solve for "y":

[tex]2x + 5y = -3\\\\5y=-2x-3\\\\y=-\frac{2}{5}x-\frac{3}{5}[/tex]

Therefore, you can identify that the slope of this line is:

[tex]m=-\frac{2}{5}=0.4[/tex]

And the y-intercept is:

[tex]b=-\frac{3}{5}=-0.6[/tex]

Answer:

Given,

The jewelry business orders $ 1320 in supplies each month.

That is, the invested amount = $ 1320,

Also, the selling price of each earring/ring set = $ 55,

Part 1 : Let x be the number of set that have been sold each month to break even ( in which revenue and invested amount are equal )

So, Total revenue ( the cost of x sets) = 55x,

⇒ 55x = 1320

Divide both sides by 55,

We get,

x = 24

Hence, 24 sets are needed to sell each month to break even.

Part 2 : Let y be the number of sets in which the profit is $ 715,

Total revenue = 55x

Profit = Total revenue - invested amount

⇒ 55x - 1320 = 715,

⇒ 55x = 715 + 1320

⇒ 55x = 2035

⇒ x = 37

Hence, 37 sets are sold for the total profit of $715.

Part 3 :

Revenue in selling 7 sets = 55 × 7 = $ 385

Profit = $ 385 - $ 1320 = - $ 935

Hence, the profit is - $ 935 after selling 7 sets.

Answer: a)  0.6141

b) 0.9772

Step-by-step explanation:

Given : Mean : [tex]\mu= 62.3\text{ in}[/tex]

Standard deviation : [tex]\sigma = \text{2.4 in}[/tex]

The formula for z -score :

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

a) Sample size = 1

For x= 63 in. ,

[tex]z=\dfrac{63-62.3}{\dfrac{2.4}{\sqrt{1}}}=0.29[/tex]

The p-value = [tex]P(z<0.29)=[/tex]

[tex]0.6140918\approx0.6141[/tex]

Thus, the probability is approximately = 0.6141

b)  Sample size = 47

For x= 63 ,

[tex]z=\dfrac{63-62.3}{\dfrac{2.4}{\sqrt{47}}}\approx2.0[/tex]

The p-value = [tex]P(z<2.0)[/tex]

[tex]=0.9772498\approx0.9772[/tex]

Thus , the probability that they have a mean height less than 63 in =0.9772.

The generating function is [tex]f(x)[/tex] where

[tex]f(x)=\displaystyle\sum_{k=0}^\infty a_kx^k[/tex]

with [tex]a_k=k^3[/tex] for [tex]k\ge0[/tex].

Recall that for [tex]|x|<1[/tex], we have

[tex]g(x)=\dfrac1{1-x}=\displaystyle\sum_{k=0}^\infty x^k[/tex]

Taking the derivative gives

[tex]g'(x)=\dfrac1{(1-x)^2}=\displaystyle\sum_{k=1}^\infty kx^{k-1}=\sum_{k=0}^\infty(k+1)x^k[/tex]

[tex]\implies g'(x)-g(x)=\dfrac x{(1-x)^2}=\displaystyle\sum_{k=0}^\infty kx^k[/tex]

Taking the derivative again, we get

[tex]g''(x)=\dfrac2{(1-x)^3}=\displaystyle\sum_{k=2}^\infty k(k-1)x^{k-2}=\sum_{k=0}^\infty(k^2+3k+2)x^k[/tex]

[tex]\implies g''(x)-3g'(x)+g(x)=\dfrac{x^2+x}{(1-x)^3}=\displaystyle\sum_{k=0}^\infty k^2x^k[/tex]

Take the derivative one last time to get

[tex]g'''(x)=\dfrac6{(1-x)^4}=\displaystyle\sum_{k=3}^\infty k(k-1)(k-2)x^{k-3}=\sum_{k=0}^\infty(k^3+6k^2+11k+6)x^k[/tex]

[tex]\implies g'''(x)-6g''(x)+7g'(x)-g(x)=\dfrac{x^3+4x^2+x}{(1-x)^4}=\displaystyle\sum_{k=0}^\infty k^3x^k[/tex]

So the generating function is

[tex]\boxed{f(x)=\dfrac{x^3+4x^2+x}{(1-x)^4}}[/tex]

Answer: Odds against rolling a sum of 3 = 17:1

Step-by-step explanation:

On rolling a pair of dice,

Total number of outcomes = 6 × 6 = 36

Outcomes with a sum of 3:

there is only 2 outcomes whose sum is 3, that is, (1,2) and (2,1)

∴ Favorable outcome = 2

Unfavorable outcome = 34

Odds against refers to the ratio of unfavorable outcomes to the favorable outcomes

so,

odds against rolling a sum of 3 = [tex]\frac{unfavorable\ outcomes}{favorable\ outcomes}[/tex]

= [tex]\frac{34}{2}[/tex]

= 17:1

Answer:

The mathematical expression is false

Step-by-step explanation:

* Lets use the figure to answer the question

- There are four triangles in the figure

- Δ ROB and Δ PTA appear congruent because:

# The side RO appears equal the side PT

∴ RO ≅ PT

# The side OB appears equal the side TA

∴ OB ≅ TA

# The side RB appears equal the side PA

∴ RB ≅ PA ⇒ SSS

∴ Δ ROB ≅ Δ PTA

- Δ DEF and Δ YXW appear congruent because:

# The side DE appears equal the side YX

∴ DE ≅ YX

# The side EF appears equal the side XW

∴ EF ≅ XW

# The side DF appears equal the side YW

∴ DF ≅ YW

∴ Δ DEF ≅ Δ YXW ⇒ SSS

- Δ ROB and Δ DEF have different shapes and sizes

∵ Δ ROB not appear congruent to Δ DEF

∴ Δ ROB ≠ Δ DEF

∴ The mathematical expression is false

Answer : The mathematical expression is false.

Step-by-step explanation :

As we are given 4 triangles in which ΔROB & ΔPTA and ΔDEF & ΔYXW are appears congruent.

First we have to show that ΔROB and ΔPTA appear congruent.

Side RO appears equal to Side PT

Side OB appears equal to Side TA

Side RB appears equal to Side PA

∴ ΔROB ≅ ΔPTA   (by SSS)

Now we have to show that ΔDEF and ΔYXW appear congruent.

Side DE appears equal to Side YX

Side EF appears equal to Side XW

Side DF appears equal to Side YW

∴ ΔDEF ≅ ΔYXW   (by SSS)

According to given expression, ΔROB and ΔDEF have different shapes and sizes.

So, ΔROB not appear congruent to ΔDEF

Therefore, the mathematical expression is false.

[tex]y''+2y'-3y=0 [/tex]

Second order linear homogeneous differential equation with constant coefficients, ODE has a form of,

[tex]ay''+by'+cy=0[/tex]

From here we assume that for any equation of that form has a solution of the form, [tex]e^{yt}[/tex]

Now the equation looks like this,

[tex]((e^{yt}))''+2((e^{yt}))'-3e^{yt}=0[/tex]

Now simplify to,

[tex]e^{yt}(y^2+2y-3)=0[/tex]

You can solve the simplified equation using quadratic equation since,

[tex]e^{yt}(y^2+2y-3)=0\Longleftrightarrow y^2+2y-3=0[/tex]

Using the QE we result with,

[tex]\underline{y_1=1}, \underline{y_2=-3}[/tex]

So,

For two real roots [tex]y_1\neq y_2[/tex] the general solution takes the form of,

[tex]y=c_1e^{y_1t}+c_2e^{y_2t}[/tex]

Or simply,

[tex]\boxed{y=c_1e^t+c_2e^{-3t}}[/tex]

Hope this helps.

r3t40

Step-by-step explanation:

If p is the number of physics books and s is the number of sociology books, then:

p + s = 440

s = p - 54

Substituting:

p + (p - 54) = 440

2p - 54 = 440

2p = 494

p = 247

Solving for s:

s = p - 54

s = 247 - 54

s = 193

The store sold 247 physics books and 193 sociology books.

Answer:

303.6 feet

Step-by-step explanation:

Given,

The average Ferris wheel rotates at 6.9 miles per hour.

So, the speed of the wheel = 6.9 miles per hour,

We know that,

Distance = Speed × Time

So, the distance covered by the wheel in 5 minutes ( or 1/12 hours because 1 hour = 60 minutes ) ride = [tex]6.9\times \frac{1}{12}[/tex]

[tex]=\frac{6.9}{12}[/tex]

[tex]=0.575\text{ miles}[/tex]

Since, 1 mile = 5280 feet,

Hence, the distance covered by the wheel in 5 minutes = 0.575 × 528 = 303.6 feet.

The average Ferris wheel covers a circular distance of 3024 feet in a 5-minute ride.

To calculate the circular distance covered by the average Ferris wheel in a 5-minute ride, we need to convert the speed from miles per hour to feet per minute. There are 5,280 feet in a mile and 60 minutes in an hour, so we can convert 6.9 miles per hour to feet per minute using the formula:

6.9 miles/hour x 5,280 feet/mile x 1 hour/60 minutes = 604.8 feet/minute

Now that we know the Ferris wheel covers 604.8 feet in 1 minute, we can calculate the circular distance covered in 5 minutes by multiplying the feet per minute by the number of minutes:

604.8 feet/minute x 5 minutes = 3024 feet

Therefore, the average Ferris wheel covers a circular distance of 3024 feet in a 5-minute ride.

Answer:  [tex]\hat{p}=0.5625[/tex]

Step-by-step explanation:

Given : Sample size : [tex]n=4000[/tex]

The number of people who are in favor of gun control legislation =2250

The proportion of people who are in favor of gun control legislation will be :-

[tex]p_0=\dfrac{2250}{4000}=0.5625[/tex]

We assume that the the given situation is normally distributed.

Then , the point estimate for the proportion [tex]\hat{p}[/tex] of citizens who are in favor of gun control legislation is equals to the sample proportion.

i.e.  [tex]\hat{p}=0.5625[/tex]

Answer:

Domain : [-4, ∞)

Range : [ -2, ∞)

Step-by-step explanation:

The given function is y = [tex]\sqrt{(x+4)}-2[/tex]

Domain of the given function will be

(x + 4) ≥ 0

[ Since square root of numbers less than 0 is not possible ]

Domain : x ≥ (-4)

Or [-4, ∞) will be the domain

Now range of the function will be x ≥ -2

[ -2, ∞) will be the range of the given function.

There are 36 total possible outcomes.

Rolling a sum of 11 or higher, there are 3 possible rolls, to make a 3/36 = 1/12 probability.

Rolling a sum of 4 there are also 3 possibilities, so the chance would be the same.

Rolling a sum of 9, there are 4 possibilities, which is a better chance.

Rolling a sum less than 5, there is 6 possibilities, which is a better chance.

Rolling greater than 5 but less than 7 means rolling a sum of 6, there are 5 chances, which is a better chance.

Rolling greater than 9 but less than 11, means rolling a 10, there are 3 possibilities, which is the same.

Rolling greater than 2 and less than 4 means rolling a 3, there are 2 possibilities, which is less.

The answers would be:

Rolling a sum of 9,

Rolling a sum less than 5

Rolling greater than 5 but less than 7

Answer:

B, C, D

Step-by-step explanation:

I got it right on Edg

Answer: 0.1880

Step-by-step explanation:

The binomial distribution formula is given by :-

[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex]

where P(x) is the probability of x successes out of n trials, p is the probability of success on a particular trial.

Given : The probability of new vehicles require warranty service within the first year : p =0.13.

Number of trials  : n= 12

Now, the required probability will be :

[tex]P(x=0)=^{12}C_0(0.13)^{0}(1-0.13)^{12-0}\\\\=(1)(0.13)^{0}(0.87)^{12}=0.188031682201\approx0.1880[/tex]

Thus,  the probability that none of these vehicles requires warranty service = 0.1880

Answer:

A participant in a cognitive psychology study is given 50 words to remember and she recalls 17 words.  

So, here variable will be the number of words the participant can remember, out of 50.

The possible values can be the whole numbers 0, 1, 2, 3, 4, upto... 50.

And the score is 17. That is the score she remembers out of 50.

Step-by-step explanation:

I just found the answer and I hope that this helps :)!!

The rate at which the distance between the ships is changing at 4 PM depends on their velocities.

To find the rate at which the distance between the ships is changing, we can use the concept of relative velocity. Let's consider ship B as the reference point. Ship A is moving west at 20 knots (which is equivalent to 20 nautical miles per hour), and ship B is moving north at 21 knots. The distance between the ships can be considered as the hypotenuse of a right triangle, with the velocities of the ships representing the triangle's sides.

Using the Pythagorean theorem, we can write the equation: d^2 = x^2 + y^2, where d is the distance between the ships, x is the velocity of ship A, and y is the velocity of ship B. We need to find the rate of change of d with respect to time (dt).

Taking the derivative on both sides of the equation with respect to time, we get: 2d * (dd/dt) = 2x * (dx/dt) + 2y * (dy/dt).

Substituting the given values, x = -20 knots (negative because ship A is moving west), y = 21 knots, and dx/dt = dy/dt = 0 (since ship B is not changing its velocity), we can solve for dd/dt, which represents the rate at which the distance between the ships is changing.

Therefore, dd/dt = 2x * (dx/dt) + 2y * (dy/dt) = 2 * -20 knots * 0 + 2 * 21 knots * 0 = 0.

Thus, the distance between the ships is not changing at 4 PM.

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Answer: 0.1587

Step-by-step explanation:

Given : The foreman of a bottling plant has observed that the amount of soda in each 16-ounce bottle is actually a normally distributed random variable, with

[tex]\mu=15.9\text{ ounces}[/tex]

Standard deviation : [tex]\sigma=0.1\text{ ounce}[/tex]

Let x be the amount of soda in a randomly selected bottle.

Z-score : [tex]\dfrac{x-\mu}{\sigma}[/tex]

[tex]z=\dfrac{16-15.9}{0.1}=1[/tex]

The probability that the bottle will contain more than 16 ounces using standardized normal distribution table  :

[tex]P(x>16)=P(z>1)=1-P(z<1)\\\\=1-0.8413447=0.1586553\approx0.1587[/tex]    

Hence, the probability that the bottle will contain more than 16 = 0.1587

Find the number of miles he drove by subtracting the odometer readings:

38735.5 - 38320.8 = 414.7 miles.

Now divide the number of miles driven by the number of gallons:

414.7 / 14.5 = 28.6 miles per gallon.

Answer:

211 people

Step-by-step explanation:

1.  This is a least common multiple question, otherwise known as an LCM question. We know this because if the one extra person had not shown up to the party, all groups would have been formed evenly. This means the amount of people at the party is 1 more than a number 2, 3, 5, and 7 can go into.

2. We also know this is an LCM question because we are being asked for the smallest amount of people who could have possibly attended the party.

3.  From this we know the amount of people at the party must be an odd number. If the number were even, there would have been no left over when groups of 2 were formed.

4. The amount of people at the party must end in a 1. This is because all multiples of 5 always end in a 0 or 5. Because there is on extra person, we must add 1 to all multiples of 5 we check. However, 5 + 1 is 6. This is a problem because 6 is an even number, and as we already established, the amount of people at the party must end in an odd number. So, we now know the smallest amount of people at the party will end in a 1.

5. Because we know the largest teams attempted to be formed with 1 left over is teams of 7 people, we only need to check multiples of 7. It is the largest number, so doing this will save us time.

6. Since we know the amount of people at the party must end in a 1, and we are only checking multiples of 7, we only need to check multiples of 7 that end in a 0. This is because any multiple of 7, 7 will go into evenly without a remainder. So we must add 1 to every multiple we check in order to make a remainder of 1. The only number we can add 1 to in order to get 1 is 0, so we only need to check multiples of 7 that end in 0. The only multiples of 7 that end in a 0 are when 7 is multiplied by a ten, ex: 10, 20, 30, 40, 50, ect.

7. Only searching for odd numbers, numbers that end in 1, and multiples of 7 means we only have to check if any possible answer when divided by 3 has a remainder of 1. We only have to check by the number 3 because any number ending in 1 will automatically have a remainder of one for 2 and 5, and because we are using multiple of 7 we don't need to check through 7.

8. Now that we know all our rules, all we need to do is list multiples of 7 that end in 0. Then we will add 1 to them and check to see if they have a remainder of 1 when divided by 3.

Using these rules will narrow our search drastically.

Applicable multiples of 7

(7 × 10) = 70 → 70 + 1 = 71 → 71/3 = 23 R2 → not possible

(7 × 20) = 140 → 140 + 1 = 141 → 141/3 = 47 → not possible

(7 × 30) = 210 → 210 +1 = 211 → 211/3 = 70 R1 → possible

The smallest possible amount of people at the party is 211.

Final answer:

The smallest amount of people who could have been at the party is 210.

Explanation:

To find the smallest amount of people who could have been at the party, we need to find the least common multiple (LCM) of the numbers 2, 3, 5, and 7. The LCM is the smallest number that is divisible by all of the given numbers without leaving a remainder.

Therefore, the smallest amount of people who could have been at the party is 210.