If records indicate that 15 houses out of 1000 are expected to be damaged by fire in any year, what is the probability that a woman who owns 14 houses will have fire damage in 2 of them in a year? (Round your answer to five decimal places.)

Answer:

  the graph is shown below

Step-by-step explanation:

The solution region is that quadruply-shaded area above the line y=-2 and to the left of the line x=1. It is further bounded above by the line y = -1/2x +3.5.

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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⇒ 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:

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:

x = -5 sin (2t)

Step-by-step explanation:

k is the spring stiffness.  The unstretched length of the spring is L.

When the mass is added, the spring stretches to an equilibrium position of L+s, where mg = ks.  When the mass is displaced a distance x (where x is positive if the displacement is down and negative if it's up), the spring is stretched a total distance s + x.

There are two forces on the mass: weight and force from the spring.  Sum of the forces in the downward direction:

∑F = ma

mg − k(s + x) = ma

mg − ks − kx = ma

Since mg = ks:

-kx = ma

Acceleration is second derivative of position, so:

-kx = m d²x/dt²

Let's find k:

F = kx

400 = 2k

k = 200

We know that m = 50.  Substituting:

-200x = 50 d²x/dt²

-4x = d²x/dt²

d²x/dt² + 4x = 0

This is a linear second order differential equation of the form:

x" + ω² x = 0

The solution to this is:

x = A cos (ωt) + B sin (ωt)

Here, ω² = 4, so ω = 2.

x = A cos (2t) + B sin (2t)

We're given initial conditions that x(0) = 0 and x'(0) = -10 (remember that down is positive and up is negative).

Finding x'(t):

x' = -2A sin (2t) + 2B cos (2t)

Plugging in the initial conditions:

0 = A

-10 = 2B

Therefore:

x = -5 sin (2t)

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?

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: 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:

gragh

Step-by-step explanation:

its one of the few ways to analyze data and compare it to find the best choice

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 Do it

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

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:

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:  B) 67%

Step-by-step explanation:

Find the Area of the Bullseye and Middle ring

A = π r²

A (inside) = π(8)² = 64π

Find the Area of the entire Target

A (target) = π (14)² = 196π

Find the Area of the Outer ring

A (outer ring) = A (target) - A(inside)

                      =   196 π    -    64π

                      =    132 π

The last step is to find the probability of landing on the outer ring:

[tex]P=\dfrac{success (area\ of\ outer\ ring)}{total\ possible\ outcomes(area\ of\ target)}=\dfrac{132\pi}{196\pi}=0.673=\large\boxed{67\%}[/tex]

Answer:

67%

Step-by-step explanation:

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.

Answer:

Part A ⇒ n>-140/p and p≠0  

Part B. ⇒ d=-30-2a/5

Step-by-step explanation:

Part A. -np-80<60

First, add by 80 both sides of equation.

-np-80+80<60+80

Simplify.

60+80=140

-np<140

Then, multiply by -1 both sides of equation.

(-np)(-1)>140(-1)

Simplify.

np>-140

Divide by p both sides of equation.

np/p>-140/p; p≠0

Simplify to find the answer.

n>-140/p; p≠0 is the correct answer from part a.

___________________________________

Part B. 2a-5d=30

First add by 2a from both sides of equation.

2a-5d+2a=30+2a

Then, simplify.

-5d=30-2a

Divide by -5 from both sides of equation.

-5d/-5=30/-5-2a/5

Simplify, to find the answer.

d=-30-2a/5 is the correct answer from part b.

Part A:

For this case we have the following inequality:

[tex]-np-80 <60[/tex]

We add 80 to both sides of the inequality:

[tex]-np <60+80\\-np <140[/tex]

Dividing between -p on both sides, having to change the inequality sign:

[tex]n> - \frac {140} {p}[/tex]

Part B:

For this case we have the following equation:

[tex]2a-5d = 30[/tex]

Subtracting 2a on both sides:

[tex]-5d = 30-2a[/tex]

Dividing between -5 on both sides:

[tex]d = \frac {30-2a} {- 5}\\d = \frac {-30+2a} {5}\\d = - \frac {30} {5} +\frac {2a} {5}\\d = -6+ \frac {2a} {5}[/tex]

Answer:

[tex]n> - \frac {140} {p}\\d = -6+\frac {2a} {5}[/tex]

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.

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:

-∞ < 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:

Answer:

honesty you cant realy use a calculator because you need to meadian mode and range them first

Step-by-step explanation:

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 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:

  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]

The problem is to calculate the number of additional men needed to build an airstrip in less time given two men already took five hours. The additional man required is one after solving the equation, but this is not an option provided, indicating a potential error in the question.

The question involves calculating the number of additional men required to build an airstrip in less time. If it took 2 men 5 hours to build an airstrip, we can say they have a combined work rate of 1 airstrip per 5 hours, or ​​(1/5) airstrip per hour. To finish the job in 4 hours, which is 1 hour less than the original time, we would need a work rate of 1 airstrip per 4 hours.

So, if we let the number of additional men be X, we can set up the equation as follows:

Since we cannot hire half a person, we round up to the nearest whole number. Hence, one additional man would be sufficient to complete the work in 1 hour less time. However, none of the options given (A) Two, (B) Three, (C) Four, or (D) Six, are correct. Therefore, the answer is not provided in the given options and this represents a possible error in the question itself.

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'.

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: The value of n is  374. The value of p is 0.21. The value of q is 0.79.

Step-by-step explanation:

Given : In a clinical trial of a cholesterol drug, 374 subjects were given a placebo, and 21% of them developed headaches.

∴ Sample size : [tex]n=374[/tex]

The probability that cholesterol drug developed headaches :[tex]p=0.21[/tex]

Then , the probability that cholesterol drug did not develop headaches :[tex]q=1-p=1-0.21=0.79[/tex]

Hence, The value of n is  374.

The value of p is 0.21.

The value of q is 0.79.

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

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.

[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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