Im stuck on this question plz help

Step-by-step explanation:

[tex]6.\\\\\dfrac{\tan x}{\cot x}\qquad\text{use}\ \cot x=\dfrac{1}{\tan x}\\\\=\dfrac{\tan x}{\frac{1}{\tan x}}=\tan x\cdot\dfrac{\tan x}{1}=\tan x\cdot\tan x=\left(\tan x\right)^2\qquad\text{use}\ \tan x=\dfrac{\sin x}{\cos x}\\\\=\left(\dfrac{\sin x}{\cos x}\right)^2\qquad\text{use}\ \left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}\\\\=\dfrac{\sin^2x}{\cos^2x}\qquad\text{use}\ \sin^2x+\cos^2x=1\to\cos^2x=1-\sin^2x\\\\=\dfrac{\sin^2x}{1-\sin^2x}[/tex]

[tex]7.\\\\\cos x\cdot\cot x+\sin x\qquad\text{use}\ \cot x=\dfrac{\cos x}{\sin x}\\\\=\cos x\cdot\dfrac{\cos x}{\sin x}+\sin x=\dfrac{\cos^2x}{\sin x}+\dfrac{\sin^2x}{\sin x}=\dfrac{\cos^2x+\sin^2x}{\sin x}=\dfrac{1}{\sin x}[/tex]

Answer:

[tex]5x+2 , 2x^2+5x-2,\frac{x^2+5x}{2-x^2}[/tex]

Step-by-step explanation:

We are given f(x) and g(x)

1. (f+g)(x)

(f+g)(x) = f(x) + g(x)

           = [tex]x^2+5x+2-x^2[/tex]

           = [tex]5x+2[/tex]

Domain : All real numbers as it there exists a value of (f+g)(x) f every x .

2. (f-g)(x)

(f-g)(x) = f(x)-g(x)

          = [tex]x^2+5x-2+x^2[/tex]

          =[tex]2x^2+5x-2[/tex]

Domain : All real numbers as it there exists a value of (f-g)(x) f every x .

Part 3 .

[tex](\frac{f}{g})(x)\\(\frac{f}{g})(x) = \frac{f(x)}{g(x)}\\=\frac{x^2+5x}{2-x^2}[/tex]

Domain : In this case we see that the function is not defined for values of x for which the denominator becomes 0 or less than zero . Hence only those values of x are defined for which

[tex]2-x^2>0[/tex]

or [tex]2>x^2[/tex]

   Hence taking square roots on both sides and solving inequality we get.

[tex]-\sqrt{2} <x<\sqrt{2}[/tex]

Answer:

See below.

Step-by-step explanation:

C1  = {0, 1, 2} and C2 = {2, 3, 4}.

C1 ∪ C2 = {0, 1, 2, 3, 4}       (Note: the 2 is not repeated in the result}.

C1 ∩ C2 = {2}.

Answer:

  Pablo and Elena will be 38 miles apart in 19/2 hrs.

Step-by-step explanation:

Their separation speed is 13 mph - 9 mph = 4 mph. Using the relation ...

  time = distance/speed

we can find the time from ...

  time = (38 mi)/(4 mi/h) = 38/4 h = 19/2 h

For this case we have the following expression:

[tex](\frac {-27x ^ 0 * y ^ {- 2}} {54x ^ {- 5} * y ^ {- 4}}) ^ {- 2} =[/tex]

By definition we have to:

[tex]a^0 = 1[/tex]

So:

[tex](\frac {-27y ^ {- 2}} {54x ^ {- 5} * y ^ {- 4}}) ^ {- 2} =[/tex]

Simplifying:

[tex](\frac {-y ^ {- 2}} {2x ^ {- 5} * y ^ {- 4}}) ^ {- 2} =[/tex]

By definition of power properties we have to:

[tex](a ^ n) ^ m = a ^ {n * m}[/tex]

So, rewriting the expression we have:

[tex](-1)^{-2}\frac{-y^{-2*-2}}{2^{-2}*x^{-5*-2}*y^{-4*-2}}=\\\frac{1}{(-1)^2}*\frac{-y^{4}}{2^{-2}x^{10}*y^{8}}=[/tex]

SImplifying:

[tex]+1*\frac{y^{4-8}}{2^{-2}x^{10}}=\\\frac{y^{-4}}{2^{-2}x^{10}}=\\\frac{2^2}{x^{10}y^{4}}[/tex]

Answer:

[tex]\frac{4}{x^{10}y^{4}}[/tex]

Answer:

  5.655 m³/min

Step-by-step explanation:

Halfway from the bottom of the tank, the radius is half that at the top, so is 1 m. That means the surface area of the water at that point is ...

  A = πr² = π(1 m)² = π m²

The rate of flow of water into the tank is the product of this area and the rate of change of depth:

  flow rate = area × (depth rate of change)

  = (π m²) × (1.8 m/min) = 1.8π m³/min

  flow rate ≈ 5.655 m³/min

Expressing the radius in terms of the height allows for finding the relationship between the changing volume and the rising water level.

We must use the concept of related rates, which involves finding the relationship between different rates of change within a geometrical context, particularly using the volume formula for a cone V = (1/3)\pi r^2 h and the fact that dv/dt represents the rate of change of volume with respect to time, which is equivalent to the water flow rate we are looking for.

Given the dimensions of the conical tank (height 3 m and radius 2 m), and the current water level at 1.5 m from the bottom, we know that the water level is rising at 1.8 m/min. By expressing the tank's radius as a function of its height, we can relate the changing volume to the changing height of the water level using the chain rule, eventually finding the rate at which water is flowing into the tank in m³/min, which is the same as the rate at which volume is increasing.

Answer:

  (x, y, z) = (0, 0, 0) +t(1, 5, 4)

Step-by-step explanation:

Such a parametric equation can be written in the form ...

  (first point) + t×(change in point values)

where the change in point values is the difference between the point coordinates.

Since the first point is (0, 0, 0), the change is easy to find. It is exactly equal to the second point's coordinates. Hence our equation is ...

  (x, y, z) = (0, 0, 0) +t(1, 5, 4)

The sum can be "simplified" to ...

  (x, y, z) = (t, 5t, 4t)

_____

Comment on answer forms

I like the first form for many uses, but the second form can preferred in some circumstances. Use the one consistent with your reference material (textbook or teacher preference).

Answer: 0.2

Step-by-step explanation:

Given: The commuting time on a particular train is uniformly distributed over the interval (42,52).

∴ The probability density function of X will be :-

[tex]f(x)=\dfrac{1}{b-a}\\\\=\dfrac{1}{52-42}=\dfrac{1}{10}, 42<x<52[/tex]

Thus, the required probability :-

[tex]P(X<44)=\int^{44}_{42}f(x)\ dx\\\\=\int^{44}_{42}\dfrac{1}{10}\ dx\\\\=\dfrac{1}{10}[x]^{44}_{42}=\dfrac{1}{10}(44-42)=\dfrac{1}{5}=0.2[/tex]

Hence, the  probability that the commuting time will be less than 44 ​minutes= 0.2

the probability that the commuting time will be less than 44 minutes is 0.2 or 20%.

To calculate the probability that the commuting time will be less than 44 minutes, given that it is uniformly distributed between 42 and 52 minutes, we use the formula for the probability of a continuous uniform distribution:

P(a < X < b) = (b - a) / (max - min)

Here, min is 42, max is 52, a is 42 (since that's the minimum time), and b is 44 (the time we are interested in). So the probability that the commuting time is less than 44 minutes will be:

P(42 < X < 44) = (44 - 42) / (52 - 42) = 2 / 10

This simplifies to 1/5 or 0.2.

Therefore, the probability that the commuting time will be less than 44 minutes is 0.2 or 20%.

Answer:

8 is the number.

Step-by-step explanation:

We are given the following expression in words which we are to translate into mathematical expression and tell the number:

'eight times the sum of 5 and some number is 104'

Assuming the number to be [tex]x[/tex], we can write it as:

[tex] 8 ( 5 + x ) = 1 0 4 [/tex]

[tex] 5 + x = \frac { 1 0 4 } { 8 } [/tex]

[tex] 5 + x = 1 3 [/tex]

[tex]x=13-5[/tex]

x = 8

ANSWER

[tex]8[/tex]

EXPLANATION

Let the number be y.

Eight times the sum of the number and 5 is written as:

[tex]8(5 + y)[/tex]

From the question, this expression must give us 104.

This implies that:

[tex]8(5 + y) = 104[/tex]

Expand the parenthesis to get;

[tex]40 + 8y = 104[/tex]

Group similar terms to get:

[tex]8y = 104 - 40[/tex]

Simplify:

[tex]8y = 64[/tex]

[tex]y = \frac{64}{8} [/tex]

This finally evaluates to

[tex]y=8[/tex]

Hence the number is 8

f(x) = 3x^2 - 5x +2

Let me show you the picture below and the answer is (0 , 2)

Answer:

(0,2)

Step-by-step explanation:

the y-intercept of the function, f(x) = 3x² -5x + 2 when : x = 0

f(0) = 3(0)² - 5(0)+2 = 2

Answer:

D. {(7, 11), (0,5), (11, 7), (7,13)}

Step-by-step explanation:

Because the relations (7, 11), (7,13) are false.

f(7) = 11

f(7) = 13

=> 11 = 13 (F)

Answer:

The correct answer option is: D. {(7, 11), (0,5), (11, 7), (7,13)}.

Step-by-step explanation:

We are given some paired values of inputs and outputs (x and y) for a function and we are to determine whether which of them is not a function.

For a relation to be function, no value of x should be repeated. It means that for each value of y, there should be a unique value of x (input).

In option D, two of the pairs have same value of x. Therefore, it is not a function.

{(7, 11), (0,5), (11, 7), (7,13)}

Answer:

(x+2) (x-1)

Step-by-step explanation:

x^2+x-2

What 2 numbers multiply together to give -2 and add together to give 1

2 * -1 = -2

2 + -1 = 1

(x+2) (x-1)

Answer:

(x+2) (x-1)

Step-by-step explanation:

Answer:

Option D.  sine theta equals plus or minus square root of thirty-five over six, tangent theta equals plus or minus square root of thirty five

Step-by-step explanation:

we have that

[tex]cos(\theta)=\frac{1}{6}[/tex]

If the cosine is positive, then the angle theta lie on the first or fourth Quadrant

therefore

The sine of angle theta could be positive (I Quadrant) or negative (IV Quadrant) and the tangent of angle theta could be positive (I Quadrant) or negative (IV Quadrant)

step 1

Find [tex]sin(\theta)[/tex]

Remember that

[tex]sin^{2} (\theta)+cos^{2} (\theta)=1[/tex]

we have

[tex]cos(\theta)=\frac{1}{6}[/tex]

substitute

[tex]sin^{2} (\theta)+(\frac{1}{6})^{2}=1[/tex]

[tex]sin^{2} (\theta)+\frac{1}{36}=1[/tex]

[tex]sin^{2} (\theta)=1-\frac{1}{36}[/tex]

[tex]sin^{2} (\theta)=\frac{35}{36}[/tex]

[tex]sin(\theta)=(+/-)\frac{\sqrt{35}}{6}[/tex]

so

sine theta equals plus or minus square root of thirty-five over six

step 2

Find [tex]tan(\theta)[/tex]

Remember that

[tex]tan(\theta)=\frac{sin(\theta)}{cos(\theta)}[/tex]

we have

[tex]sin(\theta)=(+/-)\frac{\sqrt{35}}{6}[/tex]

[tex]cos(\theta)=\frac{1}{6}[/tex]

substitute

[tex]tan(\theta)=(+/-)\sqrt{35}[/tex]

tangent theta equals plus or minus square root of thirty five

Answer:

Step-by-step explanation:

Here's the gameplan for this.  First of all we need a general formula, then we will define the variables for each.

The general formula for all of these is the same:

[tex]A(t)=P(1+\frac{r}{n})^{nt}[/tex]

where A(t) is the amount after the compounding, P is the initial investment, n is the number of compoundings per year, r is the interest rate in decimal form, and t is time in years.  

Then after we find the amount after the compounding, we will subtract the initial amount from that, because the amount at the end of the compounding is greater than the initial amount.  It's greater because it represents the initial amount PLUS the interest earned.  The difference between the initial amount and the amount at the end is the interest earned.

For A:

A(t) = ?

P = 450

n = 1

r = .06

t = 7

[tex]A(t)=450(1+\frac{.06}{1})^{(1)(7)}[/tex]

Simplifying gives us

[tex]A(t)=450(1.06)^7[/tex]

Raise 1.06 to the 7th power and then multiply in the 450 to get that

A(t) = 676.63 and

I = 676.63 - 450

I = 226.63

For B:

A(t) = ?

P = 450

n = 4 (there are 4 quarters in a year)

r = .06

t = 7

[tex]A(t)=450(1+\frac{.06}{4})^{(4)(7)}[/tex]

Simplifying inside the parenthesis and multiplying the exponents together gives us

[tex]A(t)=450(1.015)^{28}[/tex]

Raising 1.015 to the 28th power and then multiplying in the 450 gives us that

A(t) = 682.45

I = 682.45 - 450

I = 232.75

For C:

A(t) = ?

P = 450

n = 12 (there are 12 months in a year)

r = .06

t = 7

[tex]A(t)=450(1+\frac{.06}{12})^{(12)(7)}[/tex]

Simplifying the parenthesis and the exponents:

[tex]A(t)=450(1+.005)^{84}[/tex]

Adding inside the parenthesis and raising to the 84th power and multiplying in 450 gives you that

A(t) = 684.17

I = 684.17 - 450

I = 234.17

Answer:

792

Step-by-step explanation:

₁₂C₅ = (12!) / (5! (12-5)!)

₁₂C₅ = (12!) / (5! 7!)

₁₂C₅ = 792

There are 792 different selections are​ possible.

Each of the different groups or selections can be formed by taking some or all of a number of objects, irrespective of their arrangments is called a combination.

We are given that in a Power Ball​ lottery, 5 numbers between 1 and 12 inclusive are drawn. These are the winning numbers.

Therefore, the selections are​ possible as;

₁₂C₅ = (12!) / (5! (12-5)!)

₁₂C₅ = (12!) / (5! 7!)

₁₂C₅ = 792

Hence, there are 792 different selections are​ possible.

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Answer:

II. Angle C cannot be a right angle.

III. Angle C could be less than 45 degrees.

The given altitude of triangle ABC is h, is located inside the triangle and

extends from side AB to the vertex C.

The true statements are;

I. Triangle ABC could be a right triangle

II. Angle C cannot be a right angle

Reasons:

I. Triangle ABC could be a right triangle

The altitude drawn from the vertex C to the line AB = h

The length of h = AB

Where, triangle ABC is a right triangle, we have;

The legs of the right triangle are; h and AB

The triangle ABC formed is an isosceles right triangle

Therefore, triangle ABC could be an isosceles right triangle; True

II. Angle C cannot be a right angle: True

If angle ∠C is a right angle, we have;

AB = The hypotenuse (longest side) of ΔABC

Line h = AB is an altitude, therefore, one of the sides of ΔABC is hypotenuse to h, and therefore, longer than h and AB, which is false

Therefore, ∠C cannot be a right angle

III. Angle C could be less than 45 degrees; False

The minimum value of angle C is given by when triangle ABC is an isosceles right triangle. As the position of h shifts between AB, the lengths of one of the sides of ΔABC increases, and therefore, ∠C, increases

Therefore, ∠C cannot be less than 45°

The true statements are I and II

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Answer:

The length of arc AC is [tex]15\pi\ cm[/tex]

Step-by-step explanation:

step 1

Find the circumference of the circle

The circumference is equal to

[tex]C=\pi D[/tex]

we have

[tex]D=36\ cm[/tex]

substitute

[tex]C=\pi (36)[/tex]

[tex]C=36\pi\ cm[/tex]

step 2

Find the length of arc AC

Remember that the circumference subtends a central angle of 360 degrees

The measure of arc AC is equal to

arc AC+30°=180° ----> because the diameter divide the circle into two equal parts

arc AC=180°-30°=150°

using proportion

[tex]\frac{36\pi}{360}=\frac{x}{150}\\ \\x=36\pi*150/360\\ \\x=15\pi\ cm[/tex]

Answer:

The 1st & 2nd option

Step-by-step explanation:

Ans1: 16/3

Ans2: 30

Ans3: 11/7×surd6

Ans4: 2/3×surd11

Rational number is a number that can be expressed in ratio (quotient)

It can be expressed in the form of repeating or terminating decimal

Example:

16/3 is equal to

5.3333333333....(repeating decimal)

thus it is a rational number

1/4 is equal to

0.25 (terminating decimal)

thus it is a rational number

Answer:

P (Math or English) = 0.90

Step-by-step explanation:

* Lets study the meaning of or , and on probability

- The use of the word or means that you are calculating the probability

  that either event A or event B happened

- Both events do not have to happen

- The use of the word and, means that both event A and B have to

  happened

* The addition rules are:

# P(A or B) = P(A) + P(B) ⇒ mutually exclusive (events cannot happen

  at the same time)

# P(A or B) = P(A) + P(B) - P(A and B) ⇒ non-mutually exclusive (if they  

   have at least one outcome in common)

- The union is written as A ∪ B or “A or B”.  

- The Both is written as A ∩ B or “A and B”

* Lets solve the question

- The probability of taking Math class 71%

- The probability of taking English class 77%

- The probability of taking both classes is 58%

∵ P(Math) = 71% = 0.71

∵ P(English) = 77% = 0.77

∵ P(Math and English) = 58% = 0.58

- To find P(Math or English) use the rule of non-mutually exclusive

∵ P(A or B) = P(A) + P(B) - P(A and B)

∴ P(Math or English) = P(Math) + P(English) - P(Math and English)

- Lets substitute the values of P(Math) , P(English) , P(Math and English)

 in the rule

∵ P(Math or English) = 0.71 + 0.77 - 0.58 ⇒ simplify

∴ P(Math or English) =  0.90

* P(Math or English) = 0.90

To find the probability that a randomly selected student is taking a math class or an English class, use the principle of inclusion-exclusion. The probability is 90%.

To find the probability that a randomly selected student is taking a math class or an English class, we can use the principle of inclusion-exclusion. We know that 71% of students are taking a math class, 77% are taking an English class, and 58% are taking both.

To find the probability of taking either math or English, we add the probabilities of taking math and English, and then subtract the probability of taking both:

P(Math or English) = P(Math) + P(English) - P(Math and English)

= 71% + 77% - 58%

= 90%

Therefore, the probability that a randomly selected student is taking a math class or an English class is 90%.

Answer:

  a) growth will reach a peak and begin declining after about 42.6 days. 5000 people will be infected at that point

  b) the infected an uninfected populations will be the same after about 42.6 days

Step-by-step explanation:

We have assumed you intend the function to match the form of a logistic function:

[tex]P(t)=\dfrac{Kpe^{rt}}{K+p(e^{rt}-1}[/tex]

This function is symmetrical about its point of inflection, when half the population is infected. That is, up to that point, it is concave upward, increasing at an increasing rate. After that point, it is concave downward, decreasing at a decreasing rate.

a) The growth rate starts to decline at the point of inflection, when half the population is infected. That time is about 42.6 days after the start of the infection. 5000 people will be infected at that point

b) The infected and uninfected populations will be equal about 42.6 days after the start of the infection.

Suppose [tex]y_2(x)=y_1(x)v(x)[/tex] is another solution. Then

[tex]\begin{cases}y_2=vx^3\\{y_2}'=v'x^3+3vx^2//{y_2}''=v''x^3+6v'x^2+6vx\end{cases}[/tex]

Substituting these derivatives into the ODE gives

[tex]x^2(v''x^3+6v'x^2+6vx)-x(v'x^3+3vx^2)-3vx^3=0[/tex]

[tex]x^5v''+5x^4v'=0[/tex]

Let [tex]u(x)=v'(x)[/tex], so that

[tex]\begin{cases}u=v'\\u'=v''\end{cases}[/tex]

Then the ODE becomes

[tex]x^5u'+5x^4u=0[/tex]

and we can condense the left hand side as a derivative of a product,

[tex]\dfrac{\mathrm d}{\mathrm dx}[x^5u]=0[/tex]

Integrate both sides with respect to [tex]x[/tex]:

[tex]\displaystyle\int\frac{\mathrm d}{\mathrm dx}[x^5u]\,\mathrm dx=C[/tex]

[tex]x^5u=C\implies u=Cx^{-5}[/tex]

Solve for [tex]v[/tex]:

[tex]v'=Cx^{-5}\implies v=-\dfrac{C_1}4x^{-4}+C_2[/tex]

Solve for [tex]y_2[/tex]:

[tex]\dfrac{y_2}{x^3}=-\dfrac{C_1}4x^{-4}+C_2\implies y_2=C_2x^3-\dfrac{C_1}{4x}[/tex]

So another linearly independent solution is [tex]y_2=\dfrac1x[/tex].

The probability that a randomly selected dropout aged 16 to 17 is white, given the provided statistics, is 67.39%.

The student is asking a question related to conditional probability in the field of Mathematics. The question prompts us to find out the probability that a randomly selected high school dropout in the age range of 16 to 17 is white. To find the answer, we use the following formula:

P(A|B) = P(A ∩ B) / P(B)

Where:
P(A|B) is the probability of event A happening given that event B has occurred.
P(A ∩ B) is the probability of both event A and event B happening together.
P(B) is the probability of event B happening.

From the problem statement, we know that P(B), the percentage of dropouts who are 16-17 years old, is 9.2%. Also, P(A ∩ B), the percent of dropouts who are both white and 16-17 years old, is given as 6.2%. We are supposed to find P(A|B), the probability that a dropout is white given that they are 16-17 years old.

Therefore, by substituting these values into the formula, we get:

P(A|B) = 6.2% / 9.2% = 67.39%

Rounded to two decimal places, the answer is 67.39%. So, there is approximately a 67.39% chance that a random high school dropout aged 16-17 is white.

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Answer:

  (1/3(5-√34), 1/27(-205+32√34)) and (1/3(5+√34), 1/27(-205-32√34))

Step-by-step explanation:

The slope of the given line is the x-coefficient, 4. Then you're looking for points on the f(x) curve where f'(x) = 4.

  f'(x) = 3x^2 -10x +1 = 4

  3x^2 -10x -3 = 0

  x = (5 ±√34)/3 . . . . . x-coordinates of tangent points

Substituting these values into f(x), we can find the y-coordinates of the tangent points. The desired tangent points are ...

  (1/3(5-√34), 1/27(-205+32√34)) and (1/3(5+√34), 1/27(-205-32√34))

_____

The graph shows the tangent points and approximate tangent lines.

To find where the tangent line to the function f(x)=[tex]x^3-5x^2+x[/tex] is parallel to the line y=4x+23, we need to find where the derivative of the function is equal to the slope of the given line.

In the field of mathematics, the problem asks us to find at which points the tangent line to function f(x)=[tex]x^3-5x^2+x[/tex] is parallel to the line y=4x+23. A line is tangent to a function at a point where the derivative of that function equals the slope of the line. The first step in solving this problem is to calculate the derivative of function f(x).

The derivative of the function f(x) is f'(x) =[tex]3x^2 - 10x +1[/tex]. Next, we set this equal to the slope of the given line, which is 4. Solving this quadratic equation will give us the x-values where the tangent line is parallel to y=4x+23.

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Answer:

A. a false generalization

Step-by-step explanation:

All people who work do so in an office, at a computer. Bill works, so he works in an office, at a computer.

This is false generalization.

A false generalization is a mistake that we do when we generalize something without considering all the points or variables.

Like here, if people work, that does not generally means they always work in offices and work in front of computers.

There are many professions like various sports, civil engineering, police patrolling etc that cannot be done while sitting in an office and in front of a computer.

Hence, option A is the answer.

Answer: 61,750 miles

Step-by-step explanation:

Given : The p-value of the tires to outlast the warranty = 0.96

The probability that corresponds to 0.96 from a Normal distribution table is 1.75.

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

Standard deviation : [tex]\sigma=1000\text{ miles}[/tex]

The formula for z-score is given by  : -

[tex]z=\dfrac{x-\mu}{\sigma}\\\\\Rightarrow\ 1.75=\dfrac{x-60000}{1000}\\\\\Rightarrow\ x-60000=1750\\\\\Rightarrow\ x=61750[/tex]

Hence, the tread life of tire should be 61,750 miles if they want 96% of the tires to outlast the warranty.

The company looking to ensure 96% of the tires outlast the warranty should use a mileage warranty of 58,250. This ensures a failure rate of only 4%, as calculated using statistics and the concept of normal distribution.

The question deals with the concept of normal distribution in statistics, specifically, with an application to a real-life situation - to decide the warranty for a product (in this case, a type of tire). The mean and standard deviation given represent the average life and variation in life of the tires respectively. If the company wants 96% of the tires to outlast the warranty, then they are looking to find the lifespan beyond which only 4% of the tires would fail.

The Z-score corresponding to the 96th percentile in a standard normal distribution table is roughly 1.75. Since standard deviation is 1,000 miles, that implies that 1.75 standard deviations below the average is acceptable for the warranty. Therefore, we calculate the warranty as Mean - 1.75*Standard Deviation, which results in 60,000 - 1.75*1000 = 58,250 miles. Thus, the company should use a warranty of 58,250 miles.

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Answer:

whatever

Step-by-step explanation:

-789237

Answer:

Since every kilo of apples cost $3.12, it is directly proportional to the total weight (which is not exactly the same as mass).

Step-by-step explanation:

Answer:

:p

Step-by-step explanation:

Answer:

[tex]\large\boxed{x=\sqrt6-3}[/tex]

Step-by-step explanation:

[tex]Domain:\\2x+3\geq0\to x\geq-1.5[/tex]

[tex]f(x)=\sqrt{2x+3},\ g(x)=x^2\\\\f(g(x))-\text{substitute x = g(x) in}\ f(x):\\\\f(g(x))=f(x^2)=\sqrt{2x^2+3}\\\\g(f(x))-\text{substitute x = f(x) in}\ g(x):\\\\g(f(x))=g(\sqrt{2x+3})=(\sqrt{2x+3})^2=2x+3\\\\f(g(x))=g(f(x))\iff\sqrt{2x^2+3}=2x+3\qquad\text{square of both sides}\\\\(\sqrt{2x^2+3})^2=(2x+3)^2}\qquad\text{use}\ (\sqrt{a})^2=a\ \text{and}\ (a+b)^2=a^2+2ab+b^2\\\\2x^2+3=(2x)^2+2(2x)(3)+3^2\\\\2x^2+3=4x^2+12x+9\qquad\text{subtract}\ 2x^2\ \text{and 3 from both sides}[/tex]

[tex]0=2x^2+12x+6\qquad\text{divide both sides by 2}\\\\x^2+6x+3=0\qquad\text{add 6 to both sides}\\\\x^2+6x+9=6\\\\x^2+2(x)(3)+3^2=6\qquad\text{use}\ (a+b)^2=a^2+2ab+b^2\\\\(x+3)^2=6\iff x+3=\pm\sqrt6\qquad\text{subtract 3 from both sides}\\\\x=-3-\sqrt6\notin D\ \vee\ x=-3+\sqrt6\in D[/tex]

The integral is path-independent if there is a scalar function [tex]f[/tex] whose gradient is

[tex]\nabla f=(2e^x\cos2y,-\sin2y)[/tex]

(at least, that's what it looks like the given integrand is)

Then

[tex]\dfrac{\partial f}{\partial x}=2e^x\cos 2y\implies f(x,y)=2e^x\cos2y+g(y)[/tex]

Differentiating both sides with respect to [tex]y[/tex] gives

[tex]\dfrac{\partial f}{\partial y}=-4e^x\sin 2y\neq-\sin2y[/tex]

so the line integral *is* dependent on the path. (again, assuming what I've written above actually reflects what the question is asking)

The question asks about path independence in vector calculus, which indicates a property of a vector field where the value of the line integral is the same regardless of the path taken, as long as the vector field is conservative.

The student's question is focused on the concept of path independence in the context of line integrals in vector calculus. The subject matter implies they are dealing with a conservative vector field, where the integral of a function along any path depends only on the endpoints of that path, not the specific route taken. The goal is to check if a given vector field is path independent and, if so, to perform the integration from a starting point (0, 0, 0) to an endpoint (a, b, c). To establish path independence, one common method is to verify if the curl of the vector field is zero throughout the domain of interest. If it is, the field is conservative, and the path independence principle applies.

A vector field is path independent if the line integral between two points is the same regardless of the path taken between those points. Path independence typically occurs in conservative fields, where there exists a potential function such that the original vector field is its gradient.

If a field is conservative and path independent, the integral of the field over any path from point P1 to point P2 will yield the same result as the integral over any other path from P1 to P2 in the field's domain.