The spread of a virus can be modeled by exponential growth, but its growth is limited by the number of individuals that can be infected. For such situations, the function P(t) = ((Kpe)^rt)/K+p(e^rt - 1) can be used, where P(t) is the infected population t days after the first infection, p is the initial infected population, K is the total population that can be infected, and r is the rate the virus spreads, written as a decimal.

a. A town of 10,000 people starts with 2 infected people and a virus growth rate of 20%. When will the growth of the infected population start to level off, and how many people will be infected at that point? Explain your reasoning, and include any graphs you draw, with or without technology.

b. When will the infected population equal to the uninfected population?

Answer:

The simplest form of [tex]\sqrt[3]{27a^{3}b^{7}}[/tex] is

3ab²(∛b)

Step-by-step explanation:

The given term is:

[tex]\sqrt[3]{27a^{3}b^{7}}[/tex]

To convert it into its simplest form, we will apply simple mathematical rules to simplify the power of individual terms.

[tex]\sqrt[3]{27a^{3}b^{7}}\\= \sqrt[3]{3^{3} a^{3}b^{7}}\\= \sqrt[3]{3^{3}a^{3}b^{6}b}\\= 3^{3/3} a^{3/3}b^{6/3}b^{1/3}}\\= 3ab^{2}(\sqrt[3]{b})[/tex]

While simplifying the term, we basically took the cube root of individual terms. The powers cancelled out cube root for some terms. In the end, we were left with the simplest form of the expression.

Answer:

46,189

Step-by-step explanation:

The prime numbers that are less than 20 are :

1,2,3,5,7,11,13,17,19

to get the greatest value, we multiply the four numbers with the largest values i.e

11 x 13 x 17 x 19 = 46,189

The greatest possible product of four different prime numbers each less than 20 is found by multiplying the four largest primes in that range: 19, 17, 13, and 11, which equals 46,189.

To find the greatest possible product of four different prime numbers each less than 20, we should choose the four largest prime numbers in that range. The largest primes less than 20 are 19, 17, 13, and 11. Multiplying these together gives us:

19 \times 17 \times 13 \times 11 = 46,189.

Thus, the greatest possible result when multiplying four different prime numbers, each less than 20, is 46,189, which matches option 'd'.

The probability that a defective component came from shipment II is:

                   [tex]0.7143\ or\ 71.43\%[/tex]

Let A denote the event that the defective component was from shipment I

Also, P(A)=2%=0.02

and B denote the event that the defective component was from shipment II.

i.e. P(B)=5%=0.05

Also, P(shipment I is chosen)=1/2=0.5

and P(shipment II is chosen)=1/2=0.5

The  probability that a defective component came from shipment II is calculated by Baye's rule as follows:

[tex]=\dfrac{\dfrac{1}{2}\times 0.05}{\dfrac{1}{2}\times 0.02+\dfrac{1}{2}\times 0.05}}\\\\\\=\dfrac{0.05}{0.07}\\\\=\dfrac{5}{7}\\\\=0.7143\ or\ 71.43\%[/tex]

Hence, the answer is:

                        [tex]0.7143\ or\ 71.43\%[/tex]

By applying Bayes' Rule, we can compute the probability that a defective component came from shipment II as approximately 71.4%.

Given that there are two shipments of components both containing defective parts, we can apply the Bayes' Rule to answer your question.

Let's assume that D is the event that a component is defective and I and II are events that the component came from shipment I and shipment II respectively. Since the defective component can come from either shipment with equal probability, P(I) = P(II) = 0.5. Also, it's given that the component is defective, so P(D) = 1.

The probability that a component from shipment I is defective, P(D/I), is 2% or 0.02 and from shipment II is 5% or 0.05. We want to find the probability that a defective component came from shipment II, or P(II/D).

To do this, we use Bayes' Rule: P(II/D) = [P(D/II) * P(II)] / P(D).

Substituting the values in, we get: P(II/D) = [0.05 * 0.5] / [0.5 * (0.02 + 0.05)] = 0.0625 / 0.035 = ~0.714. So the probability that a defective component came from shipment II is approximately 0.714 or 71.4%.

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Answer:

C.94

Step-by-step explanation:

The alternate interior angle of angle c is angle 6. So from there you just subtract the 84 from the 180 to get the 94 degrees.

Answer:

  A) The annual multiplier was 1.0339; the annual increase was 0.0339 of the value.

  B) 3.39% per year

  C) $182,000

Step-by-step explanation:

A) Let's let t represent years since 1987. Then we can fill in the numbers and solve for r.

  165000 = 100000(1 +r)^15

  1.65^(1/15) = 1 +r . . . . . divide by 100,000; take the 15th root

  1.03394855265 -1 = r ≈ 0.0339

The value was multiplied by about 1.0339 each year.

__

B) The value increased by about 3.39% per year.

__

C) S = $100,000(1.03394855265)^18 ≈ $182,000

Answer:

The correct answer is option C.  1/12

Step-by-step explanation:

It is given that, two dies are rolled.

The outcomes of tossing two dies are,

(1,1), (1,2), (1,3), (1,4), (15), (1,6)

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

(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

Number of possible outcomes = 36

To find the probability

The possible outcomes are getting sum 10 which are,

(4,6), (5, 5) and (6,4)

Number of possible outcomes = 3

Therefore probability of getting sum 10 = 3/36 = 1/12

The correct answer is option C.  1/12

Answer:

In the account that paid 6% Susan invest [tex]\$6,000[/tex]

In the account that paid 5% Susan invest [tex]\$4,000[/tex]

Step-by-step explanation:

we know that

The simple interest formula is equal to

[tex]I=P(rt)[/tex]

where

I is the Final Interest Value

P is the Principal amount of money to be invested

r is the rate of interest  

t is Number of Time Periods

Part a) account that paid 6% simple interest per year

in this problem we have

[tex]t=1\ years\\ P=\$x\\r=0.06[/tex]

substitute in the formula above

[tex]I1=x(0.06*1)[/tex]

[tex]I1=0.06x[/tex]

Part b) account that paid 5% simple interest per year

in this problem we have

[tex]t=1\ years\\ P=\$10,000-\$x\\r=0.05[/tex]

substitute in the formula above

[tex]I2=(10,000-x)(0.05*1)[/tex]

[tex]I2=500-0.05x[/tex]

we know that

[tex]I1+I2=\$560[/tex]

substitute and solve for x

[tex]0.06x+500-0.05x=560[/tex]

[tex]0.01x=560-500[/tex]

[tex]0.01x=60[/tex]

[tex]x=\$6.000[/tex]

therefore

In the account that paid 6% Susan invest [tex]\$6,000[/tex]

In the account that paid 5% Susan invest [tex]\$4,000[/tex]

Susan invested $6,000 at 6% and the remainder, $4,000, at 5% interest.

Susan invested $10,000 in two different accounts, one with a 6% simple interest and the other with a 5% simple interest. After one year, she received a total of $560 in interest. We need to find out how much she invested in each account.

Let's denote x as the amount invested at 6% and (10,000 - x) as the amount invested at 5%. Using the formula for simple interest, interest = principal × rate × time, we can set up two equations based on the given information:

The interest from the account with 6% interest: 0.06 × x

The interest from the account with 5% interest: 0.05 × (10,000 - x)

The sum of these interests is $560, so the equation is:

0.06x + 0.05(10,000 - x) = 560

Now we solve for x:

0.06x + 500 - 0.05x = 560

0.01x = 60

x = 60 / 0.01

x = $6,000

Therefore, Susan invested $6,000 at 6% and the remainder, $4,000, at 5% interest.

Answer:

The correct answer is second option

4a²b²c²∛b)

Step-by-step explanation:

It is given an expression, ∛(64a⁶b⁷c⁹)

Points to remember

Identities

ⁿ√x = x¹/ⁿ

To find the equivalent expression

We have,  ∛(64a⁶b⁷c⁹)

∛(64a⁶b⁷c⁹) =  (64a⁶b⁷c⁹)1/3

 = (4³/³ a⁶/³ b⁷/³ c⁹/³)          [Since 64 = 4³]

 = 4a² b² b¹/³ c³

 = 4a²b²c³(b¹/³)

 = 4a²b²c³ (∛b)

Therefore the correct answer is second option

4a²b²c³(∛b)

Answer Do it

Step-by-step explanation 165 divided by 100 =x times 43

Answer:

In mixed number format: 6 1/2

Step-by-step explanation:

To solve the following equation: 7(x - 2) = 3(x + 4), first we need to apply the distributive property:

7(x - 2) = 3(x + 4) → 7x -14 = 3x + 12

Solving for 'x' → 4x = 26 → x = 6.5

→ In mixed number format: 6 1/2

For this case we must solve the following equation:

[tex]7 (x-2) = 3 (x + 4)[/tex]

Applying distributive property to the terms within the parenthesis we have:

[tex]7x-14 = 3x + 12[/tex]

We subtract 3x on both sides of the equation:

[tex]7x-3x-14 = 12\\4x-14 = 12[/tex]

Adding 14 to both sides of the equation:

[tex]4x = 12 + 14\\4x = 26[/tex]

Dividing between 4 on both sides of the equation:

[tex]x = \frac {26} {4} = \frac {13} {2}[/tex]

ANswer:

[tex]x = \frac {13} {2}\\x = 6 \frac {1} {2}[/tex]

1.052 is equal to 105.2 percent.

Given that a decimal number 1.052, we need to write 1.052 as a percent,

To express a decimal number as a percent, you need to multiply it by 100.

Let's calculate 1.052 as a percent:

1.052 x 100 = 105.2

To understand this, let's break it down:

The number 1.052 represents 105.2% because it is greater than 1 (100%). By multiplying it by 100, we shift the decimal point two places to the right, resulting in 105.2.

In percentage terms, 105.2% means that 1.052 is 105.2 parts out of 100. This can also be interpreted as 105.2 per hundred or simply 105.2 out of every 100 units.

Therefore, 1.052 can be written as 105.2%.

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For this case we have that by definition, the GCF or (Greatest Common Factor) is given by the greatest common factor that divides both terms without leaving a residue.

15: 1,3,5,15

3: 1.3

Then we have the GCF of the expression is:

[tex]3t ^ 2s (t ^ 3-5s ^ 2)[/tex]

ANswer:

Option C

1 pencil and 4 pens = 5 total

Picking the pencil would be 1/5 ( 1 pencil out of 5 total items)

5 color pencils  + 5 crayons = 10 total items.

Picking a color pencil would be 5/10 which reduces to 1/2

To find the probability of both happening, multiply them together:

1/5 x 1/2 = 1/10

The probability is 1/10

Answer:

Step-by-step explanation:

For equations like these, where the coefficient of one of the variables is 1, it is convenient to choose values for the other variable. Values of 0 and 1 are usually easy to work with.

In the first equation, ...

  for x=0, y = 2 . . . . . . . . . point (0, 2)

  for x=1, y = -2+2 = 0 . . . point (1, 0)

In the second equation, ...

  for y=0, x = -4 . . . . . . . . . point (-4, 0)

  for y=1, x +3 = -4, so x = -7 . . . . point (-7, 1)

Answer:

Answer is contained in the explanation

Step-by-step explanation:

[tex]g(x)=5-x\\g'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\\g'(x)=\lim_{h \rightarrow 0} \frac{[5-(x+h)]-[5-x]}{h}\\g'(x)=\lim_{h \rightarrow 0} \frac{5-x-h-5+x}{h}\\g'(x)=\lim_{h \rightarrow 0} \frac{-h}{h}\\g'(x)=\lim_{h \rightarrow 0} -1\\g'(x)=-1[/tex]

g(x)=5-x has domain all real numbers (you can plug an a number and always get a number back)

So in interval notation this is [tex](-\infty, \infty)[/tex]

g'(x)=-1 has domain all real numbers (the original function had domain issues... and no matter the number you plug in you do get a number, that number being -1)

So in interval notation this is [tex](-\infty, \infty)[/tex]

The derivative of given function g(x) is

g'(x)=-1

Domain of function g(x) is (-∞,∞)

Domain of derivative is  (-∞,∞)

Given :

[tex]g(x) = 5 - x[/tex]

Lets find derivative using definition of derivative

[tex]\lim_{h \to 0} \frac{g(x+h)-g(x)}{h} \\g(x)=5-x\\g(x+h)=5-(x+h)\\g(x+h)=5-x-h\\\lim_{h \to 0} \frac{5-x-h-(5-x)}{h} \\\\\lim_{h \to 0} \frac{5-x-h-5+x}{h} \\\\\lim_{h \to 0} \frac{-h}{h} \\\\-1[/tex]

Derivative g'(x)=-1

g(x) is a linear function . for all linear function the domain is set of all real numbers

Domain of function g(x) is (-∞,∞)

Derivative function g'(x) =-1. For all values of x  the value of y is -1

So domain is set of all real numbers

Domain of derivative is  (-∞,∞)

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Answer:

624

Step-by-step explanation:

The sequence is 49, 47, 45,...., 7, 5, 3.  This is an arithmetic sequence, because the difference between terms is the same.

The sum of the first n terms of an arithmetic sequence is:

S = n/2 (a₁ + an)

where a₁ is the first term and an is the nth term.

Here, we know that a₁ = 49 and an = 3.  But we need to find what n is.  To do that, we use definition of an arithmetic sequence:

an = a₁ + (n-1) d

where d is the common difference (in this case, -2)

3 = 49 + (n-1) (-2)

2(n-1) = 46

n - 1 = 23

n = 24

So there are 24 terms in the sequence.

The sum is:

S = 24/2 (49 + 3)

S = 12 (52)

S = 624

There are 624 bricks in the wall.

The total number of bricks in the wall is 624. This is a math problem that involves arithmetic sequence, where each term is obtained from the previous one by subtracting a fixed number (2, in this case), and concepts from algebra (equations).

The problem describes a scenario where each row of a brick wall has two fewer bricks than the row below it, which characterizes a sequence in mathematics. More specifically, this is an arithmetic sequence, which is characterized by a common difference between terms, in this case, the difference is -2.

To solve the problem, we need to find the sum of an arithmetic sequence. The formula of the sum is given by:

S = n/2 * (a1 + an)

Where S is the sum, n the number of terms, a1 the first term, and an the last term. Here, a1 is 49 and an is 3. To find n, we use the formula n = (a1 - an) / d + 1, with d being the common difference which is -2. Solving the equation we find that n = 24.

We now plug these values into the sum formula and find that the sum S, which represents the total number of bricks in the wall is

S = 24/2 * (49 + 3) = 12 * 52 = 624.

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Answer:

53.27

Step-by-step explanation:

To begin, we want to figure out what percent of products are defective and non defective, and of which type, so we can figure out probabilities. So, we start with that 5% of chairs are defective. We know that 35% of product sales are chairs, so 5% of 35% is 35%*0.05 (a percent can be divided by 100 to convert to decimals)= 1.75%. For tables, 12% of 55 is 55*0.12=6.6 percent. For beds, we first must figure out what percent of sales are beds. For this, we must take our total (100%) and subtract everything that is not beds, which is tables and chairs. This, 100-35-55=10, which is our percent of beds. Then, 8% of that is 0.8%. So, we know that the probability that an item is defective is 0.8+6.6+1.75=9.15%. The item pulled was not defective, so we want to figure out the probability of that, which would be the total-defective=100-9.15=90.85%. We then need to figure out which of that 90.85 is divided by tables, as we want to figure out what the probability of a table is. We know that 12% of tables are defective, so 100-12=88% are not. 88% of 55% is 55*0.88=48.4, so there is a 48.4% chance that if you picked out anything, it would be a non defective table. However, we are only picking things out from nondefective items, or the 90.85%. We know that 48.4 is 48.4% of 100, but we want to figure out what percent 48.4 is of 90.85. To find this, we do (48.4/90.85) * 100, which is 53.27 rounded. Feel free to ask further questions!

Answer:

Speed of boat x = 84 km/hr

Speed of current = 12 km/hr

Step-by-step explanation:

Let 'x' be the speed of boat  and 'y' be the speed of still water

Upstream speed = x - y  and

Downstream speed = x + y

It is given that, A motorboat travels 180 km in 3 hours going upstream and 504 in 6 hours going downstream

Upstream speed = x - y = 180/3 = 60 km/hr

Downstream speed = x + y =  504/6 = 84 km/hr

To find the value of x and y

x + y = 84  ----(1)

x - y = 60   ----(2)

(1) + (2) ⇒

x + y = 84  ----(1)

x - y = 60  ----(2)

2x  + 0  = 144

x = 144/2 = 72

x + y = 84

y = 84 - 72 = 12

Therefore speed of boat x = 84 km/hr

Speed of current = 12 km/hr

Answer: $5,224.74

Step-by-step explanation:

You need to calculate the 6% of  $4,929.00.

Convert the percentage to decimal form:

[tex]\frac{6}{100}=0.06[/tex]

Now multiply  $4,929.00 by 0.06:

[tex](\$4,929.00)(0.06)=\$295.74[/tex] (This is the 6% of $4,929.00)

Finally, you need to add $295.74 to $4,929.00. Then you get:

[tex]\$4,929.00+\$295.74=\$5,224.74[/tex]

Therefore, the 6% sales tax on $4,929.00 is: $5,224.74

The future value of ​$510 per year for 8 years compounded annually at 9 ​percent is $1,016.21.

The investment's future value refers to the compounded value of the present cash flows in the future, using an interest rate.

The future value can be determined using the future value table or formula.

We can also determine the future value using an online finance calculator as below.

N (# of periods) = 8 years

I/Y (Interest per year) = 9%

PV (Present Value) = $510

PMT (Periodic Payment) = $0

Results:

FV = $1,016.21 ($510 + $506.21)

Total Interest = $506.21

Thus, the future value of ​$510 per year for 8 years compounded annually at 9 ​percent is $1,016.21.

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Answer:

Step-by-step explanation:

I'm making the assumption you are looking for a graph of y=log_3(x)

So 3^0=1          which means log_3(1)=0              graph (1,0)

     3^1=3          which means log_3(3)=1              graph (3,1)

     3^2=9         which means log_3(9)=2             graph (9,2)

     3^3=27       which means log_3(27)=3           graph (27,3)

Can you find a graph that fits these points?

Answer: July

Step-by-step explanation:

Formula of z score :

[tex]z=\dfrac{X-\mu}{\sigma}[/tex]

Given: The mean high temperature in January = [tex]\mu_1=36^{\circ} F[/tex]

Standard deviation : [tex]\sigma_1=8^{\circ}F[/tex]

For X = [tex]57^{\circ}F[/tex]

[tex]z=\dfrac{57-36}{8}=2.625[/tex]

The mean high temperature in July = [tex]\mu_1=72^{\circ} F[/tex]

Standard deviation : [tex]\sigma_1=9^{\circ}F[/tex]

[tex]z=\dfrac{57-72}{8}=-1.875[/tex]

⇒ 57° F is about 2.6 standard deviations above the mean of January high temperatures, and  57° F is about 1.9 standard deviations below the mean of July’s high temperatures.

A general rule says that z-scores lower than -1.96 or higher than 1.96 are considered unusual .

Hence, the 57˚F is  more unusual in January.

Final answer:

A high temperature of 57 degrees is more unusual in January than in July, as it is 2.625 standard deviations above the January mean, compared to 1.667 standard deviations below the July mean.

Explanation:

To determine in which month it is more unusual to have a high temperature of 57 degrees Fahrenheit, we can calculate the z-score for each month. The z-score tells us how many standard deviations away from the mean a particular value is.

For January, the z-score is calculated as follows:

Z = (57 - 36) / 8 = 21 / 8 = 2.625

This means that a temperature of 57 degrees in January is 2.625 standard deviations above the January mean.

For July, the z-score is calculated as follows:

Z = (57 - 72) / 9 = -15 / 9 = -1.667

This means that a temperature of 57 degrees in July is 1.667 standard deviations below the July mean.

Since the absolute value of the January z-score (2.625) is higher than the absolute value of the July z-score (-1.667), a high temperature of 57 degrees is more unusual in January than in July.

Answer:

The solutions of the system of equations are (-1,8) and (5,2)

Step-by-step explanation:

[tex]y=-x+7[/tex] -------> equation A (equation of a line)

[tex]y=0.5(x-3)^{2}[/tex] ----> equation B (vertical parabola open upward)

Solve the system of equations by graphing

Remember that the solution is the intersection points both graphs

using a graphing tool

The intersection points are (-1,8) and (5,2)

see the attached figure

therefore

The solutions of the system of equations are (-1,8) and (5,2)

Answer: (-1,8) and (5,2)

Step-by-step explanation: The person above me is correct. Give him five stars and a thanks!

Answer:

-∞ < x < ∞

Explanation:

x³ is the inverse of ∛x and x³ has range of all real numbers and is one to one function, so its inverse will have domain of all real numbers.

Answer:

Option 1 negative (-) infinity < X < infinity

Step-by-step explanation:

⇒ The sum of the slopes of line b and line c is 0.

⇒   False ⇒ NOT true

Answer:

The sum of the slopes of line b and line c is 0.

Step-by-step explanation:

Remember that the product of the slopes of two parallel lines is -1, so in order to be -1 you have to multiply M*-1/m=-1 so since to add them up you would do it like this m+(-1/m) taht wouldn´t get as result 0, so that would be the option that is not correct, remember that parallel lines have the same slope, so that also eliminates all of the other options.

Answer:

p = 0.718 and n = 20

Step-by-step explanation:

p is the probability of success and n is the number of trials.

Here, p = 0.718 and n = 20.

Answer:

There is a 29.50% probability of winning more than 15 times.

Step-by-step explanation:

For each time you play the arcade game, there are only two possible outcomes. Either you win, or you lose. This means that we can solve this problem using the binomial probability distribution.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinatios of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

In this problem we have that:

The probability of winning a game is 0.718. So [tex]p = 0.718[/tex].

The game is going to be played 20 times, so [tex]n = 20[/tex].

If you play the arcade game 20 times, we want to know the probability of winning more than 15 times.

This is

[tex]P(X > 15) = P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20) = 0.2950[/tex].

There is a 29.50% probability of winning more than 15 times.

[tex]h(x)=(5-2x^6)^3+\dfrac1{5-2x^6}[/tex]

Let [tex]g(x)=5-2x^6[/tex] and [tex]f(x)=x^3+\dfrac1x[/tex]. Then [tex]h(x)=f(g(x))[/tex].

Set [tex]u=5-2x^6[/tex]. By the chain rule,

[tex]\dfrac{\mathrm dh}{\mathrm dx}=\dfrac{\mathrm dh}{\mathrm du}\cdot\dfrac{\mathrm du}{\mathrm dx}[/tex]

Since [tex]h(u)=u^3+\dfrac1u[/tex] and [tex]u(x)=5-2x^6[/tex], we have

[tex]\dfrac{\mathrm dh}{\mathrm du}=3u^2-\dfrac1{u^2}[/tex]

[tex]\dfrac{\mathrm du}{\mathrm dx}=-12x^5[/tex]

Then

[tex]\dfrac{\mathrm dh}{\mathrm dx}=\left(3u^2-\dfrac1{u^2}\right)(-12x^5)=\boxed{-12x^5\left(3(5-2x^6)^2-\dfrac1{(5-2x^6)^2}\right)}[/tex]

which we could rewrite slightly as

[tex]\dfrac{\mathrm dh}{\mathrm dx}=-\dfrac{12x^5(3(5-2x^6)^4-1)}{(5-2x^6)^2}[/tex]

To differentiate the given function using the chain rule, we need to identify the functions f(x) and g(x), then calculate their derivatives. Once we have the derivatives, we can apply the chain rule to find the derivative of the composite function h(x).

Chain Rule

To differentiate the function h(x) = (5 - 2x^6)³ + 1/(5 - 2x^6) using the chain rule, we can write it as the composite function h(x) = f(g(x)).

Let's identify the functions f(x) and g(x):

f(x) = x³, g(x) = (5 - 2x^6)

Next, let's calculate the derivatives of f(x) and g(x):

f'(x) = 3x², g'(x) = -12x^5

Finally, we can apply the chain rule to differentiate h(x):

h'(x) = f'(g(x)) * g'(x) = (3(5 - 2x^6)²) * (-12x^5)

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Answer:

  11

Step-by-step explanation:

Suitable software can generate these collections. They are ...

{13,1,1,1,1,1}, {11,3,1,1,1,1}, {9,5,1,1,1,1}, {9,3,3,1,1,1},

{7,7,1,1,1,1}, {7,5,3,1,1,1}, {7,3,3,3,1,1}, {5,5,5,1,1,1},

{5,5,3,3,1,1}, {5,3,3,3,3,1}, {3,3,3,3,3,3}}

The number of distinct collections of six positive odd integers that sum to 18 is indeed:[tex]\[\boxed{11}\][/tex]

[tex]\[2(k_1 + k_2 + k_3 + k_4 + k_5 + k_6) + 6 = 18\]\[k_1 + k_2 + k_3 + k_4 + k_5 + k_6 = 6\][/tex]

Here are the possible combinations of non-negative integers (up to permutations) that sum to 6:

[tex]1. \( (6, 0, 0, 0, 0, 0) \)2. \( (5, 1, 0, 0, 0, 0) \)3. \( (4, 2, 0, 0, 0, 0) \)4. \( (4, 1, 1, 0, 0, 0) \)5. \( (3, 3, 0, 0, 0, 0) \)6. \( (3, 2, 1, 0, 0, 0) \)7. \( (3, 1, 1, 1, 0, 0) \)8. \( (2, 2, 2, 0, 0, 0) \)9. \( (2, 2, 1, 1, 0, 0) \)10. \( (2, 1, 1, 1, 1, 0) \)11. \( (1, 1, 1, 1, 1, 1) \)[/tex]

Now let's map these back to the odd integers using [tex]\(a_i = 2k_i + 1\):[/tex]

[tex]1. \( (13, 1, 1, 1, 1, 1) \)2. \( (11, 3, 1, 1, 1, 1) \)3. \( (9, 5, 1, 1, 1, 1) \)4. \( (9, 3, 3, 1, 1, 1) \)5. \( (7, 7, 1, 1, 1, 1) \)6. \( (7, 5, 3, 1, 1, 1) \)7. \( (7, 3, 3, 3, 1, 1) \)8. \( (5, 5, 5, 1, 1, 1) \)9. \( (5, 5, 3, 3, 1, 1) \)10. \( (5, 3, 3, 3, 3, 1) \)11. \( (3, 3, 3, 3, 3, 3) \)[/tex]

These are all distinct collections (up to permutations) of positive odd integers that sum to 18.

Thus, the number of distinct collections of six positive odd integers that sum to 18 is indeed:[tex]\[\boxed{11}\][/tex]

Answer: 0.8664

Step-by-step explanation:

Given : Mean : [tex]\mu =  60,000\text{ miles}[/tex]

Standard deviation : [tex]\sigma = 4,000\text{ miles}[/tex]

Sample size : [tex]n=40[/tex]

The formula to calculate the z-score :-

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

For x=  59,050

[tex]z=\dfrac{59050-60000}{\dfrac{4000}{\sqrt{40}}}\approx-1.50[/tex]

For x= 60,950

[tex]z=\dfrac{60950-60000}{\dfrac{4000}{\sqrt{40}}}\approx1.50[/tex]

The P-value : [tex]P(-1.5<z<1.5)=P(z<1.5)-P(z<-1.5)[/tex]

[tex]=0.9331927-0.0668072=0.8663855\approx0.8664[/tex]

Hence, the likelihood of finding that the sample mean is between 59,050 and 60,950=0.8664

The likelihood of finding that the sample mean is between 59,050 and 60,950 miles, according to the given normal distribution, is approximately 86.64%.

To solve this problem, we consider that the population mean is 60,000 and the standard deviation is 4,000. If we choose a sample of 40 tires, the standard deviation of the sample mean (standard error) is the standard deviation divided by the square root of the sample size (σ/√n).

This gives us 4,000/√40 = 633. The z-scores for the lower and upper bounds of our interval (59,050 and 60,950) are calculated by subtracting the population mean from these values, and dividing by the standard error. For 59,050: (59,050 - 60,000)/633 = -1.5 and for 60,950: (60,950 - 60,000)/633 = 1.5.

Using standard normal distribution tables, we know that the probability associated with a z-value of 1.5 is 0.9332. Since the normal distribution is symmetric, the probability associated with -1.5 is also 0.9332. Therefore, the probability that the sample mean lies between 59,050 and 60,950 is 0.9332 - (1 - 0.9332) = 0.8664 or approximately 86.64%.

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Answer: There is probability of 96.4% that  a day in March will be foggy if it is a cloudy.

Step-by-step explanation:

Since we have given that

Probability of the days in March are cloudy = 57%

Probability of the cloudy days in March are foggy = 55%

Let A be the event of cloudy days in March.

Let B be the event of foggy days in March.

So, here,

P(A) = 0.57

P(A∩B) = 0.55

We need to find the probability that days are foggy given that it is cloudy.

We would use "Conditional probability":

[tex]P(B\mid A)=\dfrac{P(A\cap B)}{P(A)}=\dfrac{0.55}{0.57}=0.964=96.4\%[/tex]

Hence, There is probability of 96.4% that  a day in March will be foggy if it is a cloudy.