Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = t2 + 35 , y = ln(t2 + 35), z = t; (6, ln(36), 1)

To calculate the work done to lift the leaking water, consider the average weight of the water over each part of the journey and calculate force x distance. After doing this, it is found that the total work done is 160 ft-lb.

This problem is a example of work done against gravity. Gravity pulls the water downward, whereas the rope lifts it upward. Remember the formula for work is Work = force x distance. In this case, the force is the weight of the water being lifted (decreasing as the water leaks out) and the distance is the height the bucket is raised.

Let's start by assuming that the rate of the water leaking out is linear. This means that if the bucket is lifted halfway up the rope when it's half empty, then its average weight over the first 10 feet is 0.75 * 16 lb (12 lb), and its average weight over the next 10 feet is 0.25 * 16 lb (4 lb).

So the work done is calculated as follows:

Therefore, the total work done in lifting the water alone is Work1 + Work2 = 160 ft-lb.

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To solve this problem, you can use calculus to determine the work done lifting the water as it leaks out of the bucket. To find the amount of work done to lift the water to the top, you'll need to integrate the varying weight of the water over the distance it's lifted. Since the water leaks out at a constant rate, it will linearly decrease in weight from 16 pounds to 0 pounds over the course of the 20-foot ascent.
You're given that the bucket starts with 16 pounds of water, which is equal to 2 gallons. That's because 1 gallon of water weighs approximately 8 pounds. The water weight decreases to 0 pounds as the height reaches 20 feet.
The weight of the water as a function of height, \( w(x) \), can be modeled as a linear function that starts from 16 lb at the ground (when \( x = 0 \)) and goes to 0 lb at 20 ft (when \( x = 20 \)). Thus, the weight function is:
\[ w(x) = 16 - \frac{16}{20}x \]
This simplifies to:
\[ w(x) = 16 - 0.8x \]
The work done lifting the water from height \( x \) to \( x + dx \) is \( w(x) \cdot dx \).
Work, \( W \), is the integral of this force over the distance it's applied:
\[ W = \int_{0}^{20} w(x) \, dx \]
Substitute \( w(x) \) into the equation:
\[ W = \int_{0}^{20} (16 - 0.8x) \, dx \]
Evaluating this integral involves finding the antiderivative:
\[ W = \left[ 16x - 0.4x^2 \right]_{0}^{20} \]
Apply the bounds of the integration (from 0 to 20):
\[ W = \left( 16(20) - 0.4(20)^2 \right) - \left( 16(0) - 0.4(0)^2 \right) \]
\[ W = (320 - 0.4(400)) - (0 - 0) \]
\[ W = 320 - 160 \]
Therefore, the total work done lifting the water alone is:
\[ W = 160 \text{ foot-pounds} \]

Answer: The range is 13.  The mean is 10. The standard deviation is 4.62 .

Step-by-step explanation:

The given data : 3, 6, 9, 11, 15, 16

Total number of data values : n = 6

The mean of data is given by :-

[tex]\overline{x}=\dfrac{\sum^6_{i=1}x_i}{n}\\\\\Rightarrow\overline{x}=\dfrac{60}{6}=10[/tex]

The standard deviation is given by :-

[tex]\sqrt{\dfrac{1}{n}(\sum^6_{i=1}(x_i-\overline{x})^2)}\\\\=\sqrt{\dfrac{1}{6}(\sum^6_{i=1}(x_i-10)^2)}\\\\=\sqrt{\dfrac{1}{6}\times(49+16+1+1+25+36)}=4.61880215352\approx4.62[/tex]

The range of the data : Maximum value -Minimum value

[tex]=16-3=13[/tex]

Answer:

company B

Step-by-step explanation:

average salary of Company A(μ) = $51,500

standard deviation of Company A (σ)=  $2,175.

average salary of Company B(μ) = $46,820

standard deviation of Company B(σ) =$5,920

desired salary(x) = $55,000

z-score for company A = [tex]\dfrac{x-\mu}{\sigma}[/tex]

                                    = [tex]\dfrac{55000-51500}{ 2175} = 1.61[/tex]

z-score for company A = [tex]\dfrac{x-\mu}{\sigma}[/tex]

                                    =  [tex]\dfrac{55000-46820}{ 5920} = 1.38[/tex]

higher the value of z less chances of getting the desired salary hence company B has value of z is less so, the chances of getting desired salary is more in company B.

Jason has to calculate z-score to compare his desired salary with the average salaries at two different companies. The z-score for Company A is 1.61 and for Company B is 1.38. Hence, the desired wage of $55,000 is 1.61 and 1.38 standard deviations away from the mean salaries at Company A and Company B, respectively.

The z-score is a measure of how many standard deviations an observation or datum is from the mean. To calculate Jason's z-score at each company, we would subtract the mean salary at that company from $55,000 and then divide by the standard deviation for that company.

For Company A: [tex]Z_A = ($55,000 - $51,500) / $2,175 = 1.61.[/tex]

For Company B: [tex]Z_B = ($55,000 - $46,820) / $5,920 = 1.38.[/tex]

Therefore, Jason's desired salary of $55,000 is 1.61 standard deviations away from the mean at Company A and 1.38 standard deviations away from the mean at Company B.

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Answer:

1) So the students do not make the same mistakes

2) So the students can see the importance of their jobs, to save lives

Answer:

[tex][\frac{\pi}{2},\frac{3\pi}{2}][/tex]

Step-by-step explanation:

Let me first state that I am assuming your function is

[tex]f(x)=7cos^2(x)-14sin(x)[/tex]

If this is incorrect, then disregard this whole answer/explanation.

In order to find where the function is increasing or decreasing, we need to first find the first derivative, set it equal to 0, and then factor to find the values that cause the derivative to equal 0.  This is where you expect to find a max or a min value in the function itself.  But this function is not going to be easily solved for 0 once we find the derivative unless we make it in terms of either sin or cos right now, before taking the first derivative.  

Let [tex]cos^2(x)=1-sin^2(x)[/tex]

This is a Pythagorean trig identity, and I'm assuming that if you're in calculus solving for the intervals of increasing and decreasing values that you have, at one time, used trig identities.

Rewriting:

[tex]f(x)=7(1-sin^2(x))-14sin(x)[/tex] which simplifies to

[tex]f(x)=7-7sin^2(x)-14sin(x)[/tex] and in order of descending values of x:

[tex]f(x)=-7sin^2(x)-14sin(x)+7[/tex]

Now we can find the derivative.  For the first term, let u = sin(x), therefore,

[tex]f(u)=u^2[/tex], u' = cos(x), and f'(u) = 2u.  The derivative is found by multiplying f'(u) by u', which comes out to 2sin(x)cos(x)

The derivative for the next 2 terms are simple, so the derivative of the function is

[tex]f'(x)=-7[2sin(x)cos(x)]-14cos(x)[/tex] which simplifies down to

[tex]f'(x)=-14sin(x)cos(x)-14cos(x)[/tex]

We will set that equal to zero and solve for the values that cause that derivative to equal 0.  But first we can simplify it a bit.  You can factor out a -14cos(x):

[tex]f'(x)=-14cos(x)(sin(x)+1)[/tex]

By the Zero Product Property, either

-14cos(x) = 0 or sin(x) + 1 = 0

Solving the first one for cos(x):

cos(x) = 0

Solving the second one for sin(x):

sin(x) = -1

We now look to the unit circle to see where, exactly the cos(x) = 0.  Those values are

[tex]\frac{\pi}{2},\frac{3\pi}{2}[/tex]

The value where the sin is -1 is found at

[tex]\frac{3\pi}{2}[/tex]

We set up a table (at least that's what I advise my students to do!), separating the intervals in ascending order, starting at 0 and ending at 2pi.

Those intervals are

0 < x < [tex]\frac{\pi}{2}[/tex], [tex]\frac{\pi}{2}<x<\frac{3\pi}{2}[/tex], and [tex]\frac{3\pi}{2}<x<2\pi[/tex]

Now pick a value that falls within each interval and evaluate the derivative at that value and determine the sign (+ or -) that results.  You don't care what the value is, only the sign that it carries.  For the first interval I chose

[tex]f'(\frac{\pi}{4})=-[/tex] so the function is decreasing here (not what you wanted, so let's move on to the next interval).

For the next interval I chose:

[tex]f'(\pi)=+[/tex] so the function is increasing here.

For the last interval I chose:

[tex]f'(\frac{7\pi}{4})=-[/tex]

It appears that the only place this function is increasing is on the interval

[tex][\frac{\pi}{2},\frac{3\pi}{2}][/tex]

Final answer:

The interval on which the function f(x) = 7cos^2(x) - 14sin(x) is increasing is DNE.

Explanation:

To find the interval on which the function f(x) = 7cos^2(x) - 14sin(x) is increasing, we need to determine where the derivative of the function is positive. The derivative of f(x) can be found using the chain rule, which gives us f'(x) = -14cos(x) - 28sin(x)cos(x). To find where f'(x) > 0, we need to solve the inequality -14cos(x) - 28sin(x)cos(x) > 0.

We can simplify this inequality to cos(x)(-14 - 28sin(x)) > 0. Since cos(x) is positive on the interval 0 ≤ x ≤ 2π and -14 - 28sin(x) is negative on the interval 0 ≤ x ≤ 2π, the product of these two terms will be negative. Therefore, there are no values of x on the interval 0 ≤ x ≤ 2π where f(x) is increasing.

Answer:

[tex]H_0:\mu =873\\\\H_a: \mu<873[/tex]

Step-by-step explanation:

Given : In a study investigating a link between walking and improved​ health, researchers reported that adults walked an average of 873 minutes in the past month for the purpose of health or recreation.

Claim : The true average number of minutes in the past month that adults walked for the purpose of health or recreation is lower than 873 minutes.

i.e. [tex]\mu<873[/tex]

We know that the null hypothesis contains equal sign , then the set of hypothesis for the given situation will be :-

[tex]H_0:\mu =873\\\\H_a: \mu<873[/tex]

The null hypothesis assumes no difference, so it reflects an average walking time of 873 minutes (H0: μ = 873). The alternative hypothesis reflects the research query, suggesting the true average is less than 873 minutes (Ha: μ < 873).

In statistics, the null hypothesis and the alternative hypothesis are often used to test claims or assumptions about a population. In this case, the research is about whether the true average number of minutes in the past month that adults walked for the purpose of health or recreation is lower than 873 minutes.

The null hypothesis (H0) is often a statement of 'no effect' or 'no difference'. Here, it would be: H0: μ = 873. This means that the population mean (μ) of walking time is equal to 873 minutes.

The alternative hypothesis (Ha) is what you might believe to be true or hope to prove true. In this study, it would be: Ha: μ < 873. This means that the population mean of walking time is less than 873 minutes.

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Answer:

The slope is -6 and the y intercept is -1

Step-by-step explanation:

Lets put the  equation in slope intercept form (y=mx+b) where m is the slope and b is the y intercept

-6x-y =1

Add y to each side

-6x-y+y = 1+y

-6x = 1+y

Subtract 1 from each side

-6x-1 = y+1-1

-6x-1 =y

y = -6x-1

The slope is -6 and the y intercept is -1

Answer:

Probability: 0.1825 or 18.25%  .......  Unlikely

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Probability problems.

To start of we need to calculate the total amount of respondents that took the survey. We do this by adding all the answers together.

169 + 12,527 + 2,834 = 15,530 total people

Now that we know the total amount of people we can calculate the probability of each response by dividing the amount of people that had that response by the total amount of people that took the survey.

Never Used: [tex]\frac{169}{15,530} = 0.0109 = 1.09%[/tex]

Sometimes Used: [tex]\frac{12,527}{15,530} = 0.8066 = 80.66%[/tex]

Frequently used: [tex]\frac{2834}{15,530} = 0.1825 = 18.25%[/tex]

So we can see that the probability of a randomly selected person using a credit card frequently is 0.1825 or 18.25%

Unlikely

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

Final answer:

The probability that a randomly selected person uses a credit card frequently is calculated as 2834 divided by the total of 15530, resulting in 0.1825 (rounded to four decimal places). Since this is less than 20%, it is considered 'Unlikely' for someone to frequently use a credit card.

Explanation:

To calculate the probability that a randomly selected person uses a credit card frequently, we need to use the basic probability formula, which is the number of favorable outcomes divided by the total number of outcomes. In this case, the number of people who use a credit card frequently is the favorable outcome, and the total number of respondents is the sum of all categories of credit card usage.

Number of people who use a credit card frequently: 2834

Total number of respondents: 169 (never) + 12527 (sometimes) + 2834 (frequently) = 15530

Probability of frequent use: 2834 / 15530 = 0.1825 (rounded to 4 decimal places)

Now, let's interpret the result. A probability of 0.1825, when rounded, is about 18.25%. This number is less than 20%, which is generally considered the benchmark for something to be considered "likely". Therefore, it is Unlikely for someone to use a credit card frequently.

Answer: The store sold 288 tapes last month.

Step-by-step explanation:

Let the number of CDs be x , then the number of tapes is given by the expression : 4x

Also, the total quantity of these two items sold was 360.

Now, we have the following equation :-

[tex]x+4x=360\\\\\Rightarrow\ 5x=360\\\\\Rightarrow\ x=\dfrac{360}{5}\\\\\Rightarrow\ x=72[/tex]

The number of CDs sold in last month = 72

The number of tapes sold in last month =[tex]4\times72=288[/tex]

Hence, the store sold 288 tapes last month.

Answer:

0.0726mL

Step-by-step explanation:

Let's find the answer by using the following formula:

(subcutaneous dose)=(milliliters of the injection)*(drug concentration)/(child weight)

Using the given data we have:

(0.01mg/kg)=(milliliters of the injection)*(1mg/mL)/(16lbs)

milliliters of the injection=(0.01mg/kg)*(16lbs)/(1mg/mL)

Notice that the data has different units so:

1kg=2.20462lbs then:

16lbs*(1kg/2.20462lbs)=7.25748kg

Using the above relation we have:

milliliters of the injection=(0.01mg/kg)*(7.25748kg)/(1mg/mL)

milliliters of the injection=0.0726mL

Check the picture below.

so the hyperbola looks more or less like so, with a = 6, and its center at the origin.

[tex]\bf \textit{hyperbolas, vertical traverse axis } \\\\ \cfrac{(y- k)^2}{ a^2}-\cfrac{(x- h)^2}{ b^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h, k\pm a)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{ a ^2 + b ^2}\\ asymptotes\quad y= k\pm \cfrac{a}{b}(x- h) \end{cases} \\\\[-0.35em] \rule{34em}{0.25pt}[/tex]

[tex]\bf \begin{cases} a=6\\ h=0\\ k=0\\ \stackrel{asymptotes}{y=\pm\frac{3}{4}x} \end{cases}\implies \stackrel{\textit{using the positive asymptote}}{0+\cfrac{6}{b}(x-0)=\cfrac{3}{4}x}\implies \cfrac{6x}{b}=\cfrac{3x}{4}\implies 24x=3xb \\\\\\ \cfrac{24x}{3x}=b\implies 8=b \\\\[-0.35em] ~\dotfill\\\\ \cfrac{(y- 0)^2}{ 6^2}-\cfrac{(x- 0)^2}{ 8^2}=1\implies \cfrac{y^2}{36}-\cfrac{x^2}{64}=1[/tex]

Answer:

Accumulated amount will be $2504.90.

Step-by-step explanation:

Formula that represents the accumulated amount after t years is

A = [tex]A_{0}(1+\frac{r}{n})^{nt}[/tex]

Where A = Accumulated amount

[tex]A_{0}[/tex] = Initial amount

r = rate of interest

n = number of times initial amount compounded in a year

t = duration of investment in years

Now the values given in this question are

[tex]A_{0}[/tex] = $1000

n = 12

r = 4.6% = 0.046

t = 20 years

By putting values in the formula

A = [tex]1000(1+\frac{0.046}{12})^{240}[/tex]

  = [tex]1000(1+0.003833)^{240}[/tex]

  = [tex]1000(1.003833)^{240}[/tex]

  = 1000×2.50488

  = 2504.88 ≈ $2504.90

Therefore, accumulated amount will be $2504.90.

Answer:

2/7

Step-by-step explanation:

For the president position, there are 4 females from a total of 7 people.

For the vice president position, there are 3 males from 6 people left over.

So the probability is 4/7 × 3/6 = 2/7.

Answer: [tex]\dfrac{2}{7}[/tex]

Step-by-step explanation:

Given : Number of males = 3

Number of females = 4

The number of ways to select a female is elected president and a male is elected vice president :-

[tex]^3P_1\times ^4C_1=\dfrac{3!}{(3-1)!}\times\dfrac{4!}{(4-1)!}=3\times4=12[/tex]

The total number of ways to select 2 people from 7 :_

[tex]^7P_2=\dfrac{7!}{(7-2)!}=42[/tex]

Now, the probability that a female is elected president and a male is elected vice president will be :-

[tex]\dfrac{12}{42}=\dfrac{2}{7}[/tex]

Answer:

Step-by-step explanation:

See if the attachment below helps you with this.

Answer:

B. 78 and 82.

Step-by-step explanation:

We have been given that the average test score of the class was an 80 and the standard deviation was 2. We are asked to find two values between which 68% of class will score.

We know that in a normal distribution approximately 68% of the data falls within one standard deviation of the mean.

So 68% scores will lie within one standard deviation below and above mean that is:

[tex](\mu-\sigma,\mu+\sigma)[/tex]

Upon substituting our given values, we will get:

[tex](80-2,80+2)[/tex]

[tex](78,82)[/tex]

Therefore, about 68% of the class would score between 78 and 81 and option B is the correct choice.

Answer:

2.5 ml for a dosage of 20 mg.

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Ratio problems.

Since this is basically a ratio problem we can use the simple Rule of Three property to solve this problem. The Rule of Three property can be seen in the photo below. Now we just plug in the values and solve for x.

 8 mg.   ⇒   1 ml.

20 mg.   ⇒   x

[tex]\frac{20mg*1ml}{8mg} = 2.5ml[/tex]

Now we can see that we should administer a 2.5 ml for a dosage of 20 mg.

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a. This recurrence is of order 2.

b. We're looking for a function [tex]A(x)[/tex] such that

[tex]A(x)=\displaystyle\sum_{n=0}^\infty a_nx^n[/tex]

Take the recurrence,

[tex]\begin{cases}a_0=a_0\\a_1=a_1\\a_n-5a_{n-1}+6a_{n-2}=0&\text{for }n\ge2\end{cases}[/tex]

Multiply both sides by [tex]x^{n-2}[/tex] and sum over all integers [tex]n\ge2[/tex]:

[tex]\displaystyle\sum_{n=2}^\infty a_nx^{n-2}-5\sum_{n=2}^\infty a_{n-1}x^{n-2}+6\sum_{n=2}^\infty a_{n-2}x^{n-2}=0[/tex]

Pull out powers of [tex]x[/tex] so that each summand takes the form [tex]a_kx^k[/tex]:

[tex]\displaystyle\frac1{x^2}\sum_{n=2}^\infty a_nx^n-\frac5x\sum_{n=2}^\infty a_{n-1}x^{n-1}+6\sum_{n=2}^\infty a_{n-2}x^{n-2}=0[/tex]

Now shift the indices and add/subtract terms as needed to get everything in terms of [tex]A(x)[/tex]:

[tex]\displaystyle\frac1{x^2}\left(\sum_{n=0}^\infty a_nx^n-a_0-a_1x\right)-\frac5x\left(\sum_{n=0}^\infty a_nx^n-a_0\right)+6\sum_{n=0}^\infty a_nx^n=0[/tex]

[tex]\displaystyle\frac{A(x)-a_0-a_1x}{x^2}-\frac{5(A(x)-a_0)}x+6A(x)=0[/tex]

Solve for [tex]A(x)[/tex]:

[tex]A(x)=\dfrac{a_0+(a_1-5a_0)x}{1-5x+6x^2}\implies\boxed{A(x)=\dfrac{a_0+(a_1-5a_0)x}{(1-3x)(1-2x)}}[/tex]

c. Splitting [tex]A(x)[/tex] into partial fractions gives

[tex]A(x)=\dfrac{2a_0-a_1}{1-3x}+\dfrac{3a_0-a_1}{1-2x}[/tex]

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

[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]

so that for [tex]|3x|<1[/tex] and [tex]|2x|<1[/tex], or simply [tex]|x|<\dfrac13[/tex], we have

[tex]A(x)=\displaystyle\sum_{n=0}^\infty\bigg((2a_0-a_1)3^n+(3a_0-a_1)2^n\bigg)x^n[/tex]

which means the solution to the recurrence is

[tex]\boxed{a_n=(2a_0-a_1)3^n+(3a_0-a_1)2^n}[/tex]

d. I guess you mean [tex]a_0=2[/tex] and [tex]a_1=5[/tex], in which case

[tex]\boxed{\begin{cases}a_0=2\\a_1=5\\a_2=13\\a_3=35\\a_4=97\\a_5=275\end{cases}}[/tex]

e. We already know the general solution in terms of [tex]a_0[/tex] and [tex]a_1[/tex], so just plug them in:

[tex]\boxed{a_n=2^n+3^n}[/tex]

Answer:

  (a) increasing: (-ln(2)/3, ∞); decreasing: (-∞, -ln(2)/3)

  (b) minimum: (-ln(2)/3, (9/8)∛2) ≈ (-0.21305, 1.41741); maximum: DNE

  (c) inflection point: DNE; concave up: (-∞, ∞); concave down: DNE

Step-by-step explanation:

The first derivative of f(x) = e^(8x) +e^(-x) is ...

  f'(x) = 8e^(8x) -e^(-x)

This is zero at the function minimum, where ...

  8e^(8x) -e^(-x) = 0

  8e^(9x) -1 = 0 . . . . . . multiply by e^x

  e^(9x) = 1/8 . . . . . . .  add 1, divide by 8

  9x = ln(2^-3) . . . . . . take the natural log

  x.min = (-3/9)ln(2) = -ln(2)/3 . . . divide by the coefficient of x, simplify

This value of x is the location of the minimum.

__

The function value there is ...

  f(-ln(2)/3) = e^(8(-ln(2)/3)) + e^(-(-ln(2)/3))

  = 2^(-8/3) +2^(1/3) = 2^(1/3)(2^-3 +1)

  f(x.min) = (9/8)2^(1/3) . . . . . minimum value of the function

__

A graph shows the first derivative to have positive slope everywhere, so the curve is always concave upward. There is no point of inflection. The minimum point found above is the place where the function transitions from decreasing to increasing.

To find the intervals on which f(x) = [tex]e^8^x + e^(^-^x^)[/tex] is increasing and decreasing, analyze the first derivative. The function is increasing on (-1/8, ∞) and decreasing on (-∞, -1/8). The local minimum is at x = -1/8, and the function is concave up on (-∞, -1/8) and concave up on (-1/8, ∞).

To find the intervals on which the function f(x) = [tex]e^8^x + e^(^-^x^)[/tex] is increasing and decreasing, we need to analyze the first derivative of the function. The first derivative is f'(x) = [tex]8e^8^x - e^(^-^x^)[/tex]. We set this derivative equal to zero and solve for x to find the critical points. There is one critical point at x = -1/8. We can test intervals to the left and right of this critical point to determine the behavior of the function. The function is decreasing on (-∞, -1/8) and increasing on (-1/8, ∞). Therefore, the function is increasing on the interval (-1/8, ∞) and decreasing on the interval (-∞, -1/8).

To find the local minimum and maximum values of f, we analyze the second derivative of the function. The second derivative is f''(x) =[tex]64e^8^x + e^(^-^x^)[/tex]. We evaluate this second derivative at the critical point x = -1/8. The second derivative at x = -1/8 is positive, so the function has a local minimum at x = -1/8.

The inflection point of the function can be found by analyzing the points where the concavity changes. The second derivative changes sign at x = -1/8. Therefore, the inflection point of the function is (-1/8, f(-1/8)). To find the intervals on which the function is concave up and concave down, we analyze the sign of the second derivative. The second derivative is positive on (-∞, -1/8) and positive on (-1/8, ∞), meaning the function is concave up on (-∞, -1/8) and concave up on (-1/8, ∞).

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Answer:

Area of the rectangle is increasing with the rate of 84 cm/s.

Step-by-step explanation:

Let l represents the length, w represents width, t represents time ( in seconds ) and A represents the area of the triangle,

Given,

[tex]\frac{dl}{dt}=6\text{ cm per second}[/tex]

[tex]\frac{dw}{dt}=5\text{ cm per second}[/tex]

Also, l = 12 cm and w = 4 cm,

We know that,

A = l × w,

Differentiating with respect to t,

[tex]\frac{dA}{dt}=\frac{d}{dt}(l\times w)[/tex]

[tex]=l\times \frac{dw}{dt}+w\times \frac{dl}{dt}[/tex]

By substituting the values,

[tex]\frac{dA}{dt}=12\times 5+4\times 6[/tex]

[tex]=60+24[/tex]

[tex]=84[/tex]

Hence, the area of the rectangle is increasing with the rate of 84 cm/s.

Answer:

The solution of the given equation is [tex]\sqrt[3]{4}-\sqrt[3]{2}[/tex].

Step-by-step explanation:

According to the cardano's method, the solution of the equation is x=u-v. If the equation is

[tex]x^3+px=q[/tex]

Where [tex]u^3-v^3=q[/tex]

[tex]3uv=p[/tex]

The given equation is

[tex]x^3+6x=2[/tex]

Here p=6 and q=2.

[tex]u^3-v^3=2[/tex]                 .... (1)

[tex]3uv=6[/tex]

[tex]uv=2[/tex]

Taking cube both the sides.

[tex]u^3v^3=8[/tex]

Multiply both sides by 4.

[tex]4u^3v^3=32[/tex]             .... (2)

Taking square both the sides of equation (1).

[tex](u^3-v^3)^2=2^2[/tex]

[tex](u^3)^2-2u^3v^3+(v^3)^2=4[/tex]       .... (3)

Add equation (2) and (3).

[tex](u^3)^2-2u^3v^3+(v^3)^2+4u^3v^3=4+32[/tex]

[tex](u^3+v^3)^2=36[/tex]

Taking square root both the sides.

[tex]u^3+v^3=6[/tex]             .... (4)

On adding equation (1) and (4), we get

[tex]2u^3=8[/tex]

[tex]u^3=4[/tex]

[tex]u=\sqrt[3]{4}[/tex]

On subtracting equation (1) and (4), we get

[tex]-2v^3=-4[/tex]

[tex]v^3=2[/tex]

[tex]v=\sqrt[3]{2}[/tex]

The solution of the equation is

[tex]x=u-v=\sqrt[3]{4}-\sqrt[3]{2}[/tex]

Therefore the solution of the given equation is [tex]\sqrt[3]{4}-\sqrt[3]{2}[/tex].

Final answer:

Solving x³+6x=2 using Cardano's method involves rewriting the equation to match the standard form of a depressed cubic equation, calculating the required constants, and finally, applying these constants to find the roots.

First, let's rewrite the equation x³+6x-2 = 0 as x³+6x = 2 to match the standard form of a depressed cubic equation which is x³ +px = q. Here, p = 6 and q = 2.

Next, we calculate the value t = sqrt[(q/2)² + (p/3)³]. So, t = sqrt[(1)² + (2)³] = sqrt[1 + 8] = 3.

Using these values, we can now calculate the roots. We know the roots are given by the formulaes u-v where u = cubicroot(q/2 + t) and v = cubicroot(q/2 - t). So, u = cubicroot(1 + 3) = 2, and v = cubicroot(1 - 3) = - root(2).

Therefore, the roots of the given polynomial equation are x = u - v = 2 - (- root(2)) = 2 + root(2).

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Answer:

y=2/5x+3/5

Step-by-step explanation:

Use the slope formula to get the slope:

m=4/10

m=2/5

The y intercept is 3/5

The equation is y=2/5x+3/5

Answer:

y = (2/5)x + 3/5

Step-by-step explanation:

Points to remember

Equation of the line passing through the poits (x1, y1) and (x2, y2)  and slope m is given by

(y - y1)/(x - x1) = m    where slope m = (y2 - y1)/(x2 - x1)

To find the slope of line

Here (x1, y1) =  (-4, -1) and  (x2, y2) = (6, 3)

Slope = (y2 - y1)/(x2 - x1)

 = (3 - -1)/(6 - -4)

 = 4/10 = 2/5

To find the equation

(y - y1)/(x - x1) = m  

(y - -1)/(x - -4) = 2/5

(y + 1)/(x + 4) = 2/5

5(y + 1) = 2(x + 4)

5y + 5 = 2x + 8

5y = 2x + 3

y = (2/5)x + 3/5

Answer: Option (4) is correct.

Step-by-step explanation:

Given that,

B = {John, Paul, George, Ringo, Pete, Stuart}

Now, we have select the Set that is equal to the Set B.

From all the options given in the question, option (4) is correct.

It contains all the elements of Set B but only the arrangement or sequence of the Set is different.

Correct Set 4 = {Paul, Ringo, Pete, John, George, Stuart} = Set B

The set matching B = {John, Paul, George, Ringo, Pete, Stuart} from the options provided is {Paul, Ringo, Pete, John, George, Stuart}, as it contains all the same members regardless of order and no other elements.

The question asks to select all answers that are equal to the set B = {John, Paul, George, Ringo, Pete, Stuart}. A set, in this context, is defined as a collection of distinct objects, considered as an object in its own right. In a set, the order of elements does not matter, but duplication of elements is not allowed. From the provided options, the only answer that matches set B exactly is {Paul, Ringo, Pete, John, George, Stuart}, since it contains all the same elements as set B, regardless of order, and does not include any additional elements.

Answer: [tex]R^{2}[/tex] = 0.89

Step-by-step explanation:

Coefficient of determination is represented by [tex]R^{2}[/tex]. This tells us that how much of the variation in the dependent variable is explained by the independent variable.

It is the ratio of explained variation by the independent variables to the total variation in the dependent variable.

Hence,

Coefficient of determination = [tex]\frac{Explained\ Variation}{Total\ Variation}[/tex]

= [tex]\frac{18.592}{20.711}[/tex]

[tex]R^{2}[/tex]= 0.89

∴ 89% of the variation in the dependent variable is explained by the independent variables.

The coefficient of determination is calculated by dividing the explained variation by the total variation. For the student's data, the coefficient of determination is approximately 0.8978, which translates to about 89.78% of the variation in the dependent variable being explained by the regression line.

The student is asking about the coefficient of determination, which is a statistical measure in a regression analysis. To find the coefficient of determination, we use the explained variation and the total variation from the regression equation. It is calculated by dividing the explained variation by the total variation and then squaring the result if needed to find r squared.

In this case, the explained variation is 18.592 and the total variation is 20.711. The formula to find the coefficient of determination (r²) is:

r² = Explained Variation / Total Variation

Plugging in the values we have:

r² = 18.592 / 20.711

r² ≈ 0.8978

Expressed as a percentage, the coefficient of determination is approximately 89.78%, which means that about 89.78% of the variation in the dependent variable can be explained by the independent variable using the regression line.

Step-by-step explanation:

hi I have answered ur question

Answer:

Step-by-step explanation:

Sugar and Salt even thought they both dissolve in water they both dissolve in different ways. When salt dissolves in water, its individual types of ions are torn apart from each other, while Sugar molecules stay together when dissolved in water, and therefore the molecules remain the same when dissolved in water.

This being said in science using your senses can be just as valuable as using calculations. In this case both Sugar and Salt taste differently. Sugar is sweet while Salt is salty. Therefore tasting the substance can be the easiest and most accurate way of determining the substance.

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

Final answer:

To determine if an unknown sample contains sugar, salt, or both, examine their solubility properties and chemical reactivity. Comparing the density of the sample against reference values of pure substances can provide preliminary identification, while testing for chloride ions with silver nitrate confirms the presence of salt.

Explanation:

Identifying Sugar and Salt in an Unknown Sample

To determine whether an unknown sample contains sugar, salt, or both, we must identify the physical and chemical properties that distinguish these substances. Salt (sodium chloride) and sugar (sucrose) have distinct solubility properties and chemical reactivity, which we can use to identify them when dissolved in water.

Solubility and Density Test

Both sugar and salt are highly soluble in water, but we can compare their densities to make a preliminary identification. A known volume of each substance is weighed and their densities calculated. Salt generally has a greater density than sugar. If the unknown sample has a certain mass, comparing it with the reference densities may provide an initial indication.

Chemical Reactivity Test

To confirm the identity of the substances, a chemical reactant such as silver nitrate can be introduced to the water solution of the unknown sample. If a white precipitate forms, it indicates the presence of chloride ions, which suggests the presence of salt. Since sugar does not produce a precipitate with silver nitrate, its absence would indirectly indicate the presence of sugar.

Performing these tests will allow us to determine if the unknown sample is sugar, salt, or a mixture of both. The greater the discrepancy between the calculated density and the known densities of pure sugar or pure salt, the more likely it is that the sample is a mixture.

By the chain rule,

[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}[/tex]

Then for all [tex]t[/tex] the first derivative has a value of 7.

By the product rule,

[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}\right]=\dfrac{\mathrm d\left(\frac{\mathrm dy}{\mathrm dt}\right)}{\mathrm dx}\dfrac{\mathrm dt}{\mathrm dx}+\dfrac{\mathrm dy}{\mathrm dt}\dfrac{\mathrm d^2t}{\mathrm dx^2}[/tex]

but [tex]t=x\implies\dfrac{\mathrm dt}{\mathrm dx}=1\implies\dfrac{\mathrm d^2t}{\mathrm dx^2}=0[/tex], so we're left with

[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d\left(\frac{\mathrm dy}{\mathrm dt}\right)}{\mathrm dx}[/tex]

By the chain rule,

[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d\left(\frac{\mathrm dy}{\mathrm dt}\right)}{\mathrm dx}=\dfrac{\mathrm d\left(\frac{\mathrm dy}{\mathrm dt}\right)}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}=\dfrac{\mathrm d^2y}{\mathrm dt^2}[/tex]

but [tex]y=7t-2\implies\dfrac{\mathrm dy}{\mathrm dt}=7\implies\dfrac{\mathrm d^2y}{\mathrm dt^2}=0[/tex] so the second derivative is 0 for all [tex]t[/tex].

The derivative dy/dx for the given parametric equations is 7, and the second derivative d2y/dx2 is zero. The slope at t = 9 is 7, and the curve does not exhibit concavity since it is linear.

To find dy/dx for the parametric equations x = t and y = 7t - 2, we need to compute the derivatives of both x and y with respect to t and then use the chain rule to find dy/dx as dy/dt divided by dx/dt. Since the derivative of x with respect to t is 1, and the derivative of y with respect to t is 7 (as the derivatives of the constants -2 and 1 are zero), dy/dx equals 7/1, which is 7. To find the second derivative d2y/dx2, we note that since dx/dt is constant (equals 1), the second derivative is zero. Therefore, the concavity of the curve does not change and is neither concave up nor down.

At the given value of the parameter t = 9, the slope of the tangent line is 7, as it is for all values of t. Since the second derivative is zero, the curve is linear and does not exhibit concavity at any point, including t = 9.

Answer:

The correct answer is last option

Step-by-step explanation:

From the figure we can see two right angled triangle.

Points to remember

If two right angled triangles are congruent then their hypotenuse and one leg are congruent

To find the correct options

From the figure we get all the angles of 2 triangles are congruent.

one angle is right angle. But there is no information about the hypotenuse and legs.

So the correct answer is last option

There is not enough information to determine congruency.

Answer: 2.68

Step-by-step explanation:

Claim : The proportion of peas with yellow pods is equal to 0.25​

i.e. p=0.25

Sample size : [tex]460[/tex]

Proportion of peas with yellow pods in sample  :

[tex]P=\dfrac{91}{460}=0.19782608695\approx0.20[/tex]

Now, the test statistic for the population proportion is given by :-

[tex]z=\dfrac{p-P}{\sqrt{\dfrac{P(1-P)}{n}}}[/tex]

[tex]\Rightarrow\ z=\dfrac{0.25-0.20}{\sqrt{\dfrac{0.20(1-0.20)}{460}}}\Rightarrow\ z=2.68095132369\approx2.68[/tex]

Hence, the value of the test statistic is 2.68

The test statistic for the given population proportion and sample data is approximately -2.57. This result was calculated using the Z test formula for testing population proportions, with a sample proportion of 0.1978, an expected proportion of 0.25, and a sample size of 460.

To calculate the test statistic, we'll use the formula for Z: Z = (p' - p0) / sqrt[(p0(1 - p0)) / n]

Plugging these values into the formula, we get: Z = (0.1978 - 0.25) / sqrt[(0.25 * 0.75) / 460] ≈ -2.57

The estimated proportion p' in this formula represents the proportion of peas with yellow pods within our sample. We use this, along with the hypothesized proportion p0 and the sample size n, to calculate the test statistic, which gives us an idea of how far our observed data is from the expected hypothesis.

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Answer:Given below

Step-by-step explanation:

Using mathematical induction

For n=1

[tex]2!=2^1[/tex]

True for n=1

Assume it is true for n=k

[tex]2!\cdot 4!\cdot 6!\cdot 8!.......2k!\geq \left ( k+1\right )^{k}[/tex]

For n=k+1

[tex]2!\cdot 4!\cdot 6!\cdot 8!.......2k!2\left ( k+1\right )!\geq \left ( k+1\right )^{k}\dot \left ( 2k+2\right )![/tex]

because value of [tex]2!\cdot 4!\cdot 6!\cdot 8!.......2k!=\left ( k+1\right )^{k}[/tex]

[tex]\geq \left ( k+1\right )^{k}\dot \left ( 2k+2\right )![/tex]

[tex]\geq \left ( k+1\right )^{k}\left [ 2\left ( k+1\right )\right ]![/tex]

[tex]\geq \left ( k+1\right )^{k}\left ( 2k+\right )!\left ( 2k+2\right )[/tex]

[tex]\geq \left ( k+1\right )^{k+1}\left ( 2k+\right )![/tex]

Therefore [tex]2!\cdot 4!\cdot 6!\cdot 8!.......2k!2\left ( k+1\right )! must be greater than \left ( k+1\right )^{k+1}[/tex]

Hence it is true for n=k+1

[tex]2!\cdot 4!\cdot 6!\cdot 8!.......2k!2\left ( k+1\right )!\geq \left ( k+1\right )^{k+1}[/tex]

Hence it is true for n=k

Answer:

-1

Step-by-step explanation:

Move all terms containing  x

to the left side of the equation.

Tap for fewer steps...

Subtract  4x from both sides of the equation.

2 x−10−4x= −8 y=4x−8

Subtract  4 x  from  2 x − 2 x− 10= − 8 y = 4 x − 8

Move all terms not containing  x  to the right side of the equation.

Tap for more steps...

− 2 x = 2 y = 4 x − 8

Divide each term by  − 2  and simplify.

Tap for fewer steps...

Divide each term in  − 2 x= 2  by  − 2 .

− 2 x − 2 = 2 − 2 y = 4 x − 8

Simplify the left side of the equation by cancelling the common factors.

Tap for fewer steps...

Reduce the expression by cancelling the common factors.

Tap for more steps...

− ( − 1 ⋅ x ) = 2 2 y = 4 x − 8

Rewrite  

− 1 ⋅ x  as  - x . x = 2 − 2 y = 4 x − 8

Divide  2  by  − 2 .

x = − 1 y = 4 x − 8

Replace all occurrences of  x  with the solution found by solving the last equation for  x . In this case, the value substituted is  − 1 . x= − 1 y = 4 ( − 1 ) − 8

Simplify  4 ( − 1 ) − 8 .

Tap for fewer steps...

Multiply  4 by  − 1 .

x = − 1 y = − 4 − 8

Subtract  8  from  - 4 .

x = − 1 y = − 12

The solution to the system of equations can be represented as a point.

( − 1 , − 12 )

The result can be shown in multiple forms.

Point Form:  ( − 1 , − 12 )

Equation Form:  x  =− 1  ,y = − 12