x² = sigma (obs-exp)^2/exp
Expected Frequencies = Row Total x Column Total / Grand Total
A 2016 Stats Canada study looked at the percentage of Canadian families that are debt-free by age group. The results of the random sample of 2,000 Canadian families is summarized in the table below. Is there enough evidence to conclude that being debt-free depends on age? Test the hypothesis at the 5% level of significance.
Age (in years)
Debt Free? | Under 35 35 to 44 45 to 54 55 to 64 65 and older | Total
Yes 140 60 48 77 275 600
No 460 340 202 198 200 1,400
Total 600 400 250 275 475 2,000
Но ______
H_A _________
The degree of freedom ______ [2] Critical Value______ [3]
You have calculated that the Chi-square test=245.97
Decision/Justify ________[3]
Conclusion: Circle the Correct One [1 mark]
A. Yes, there is enough evidence to conclude that being debt-free depends on age.
B. No, there is not enough evidence to conclude that being debt-free depends on age.

Answer: -2

Step-by-step explanation:

In this case, x=-4 from f(-4) and x<-1 so use the first equation

f(-4) =  -4 +2 = -2

Answer: y=0.2x-2

Step-by-step explanation: Find the slope. (4,-1) (0,-2) are the two points I picked.

(-2)-(-1)=(-1)

0-4=(-4)

-1/-4=1/4=0.2

The y-intercept is -2.

Therefore, the answer is y=0.2x-2.

Answer:

It's difficult to answer this question without the specific equations you are working with. However, in general, if you have an equation involving a variable n and you need to find values of n that are less than 1, you can solve the equation for n and then look for solutions that satisfy the condition.

For example, if you have the equation 2n - 3 = 5, you can solve for n by adding 3 to both sides:

2n - 3 + 3 = 5 + 3

2n = 8

n = 4

In this case, n is not less than 1. However, if you had the equation 0.5n + 2 = 3, you would get:

0.5n + 2 - 2 = 3 - 2

0.5n = 1

n = 2

In this case, n is greater than 1. But if you had the equation 0.5n + 2 = 1, you would get:

0.5n + 2 - 2 = 1 - 2

0.5n = -1

n = -2

In this case, n is less than 1, since it is a negative number.

Answer:

2∣∣∣α1−α2(α1−2)(α2−2)∣∣∣<1for0<α1,α2<1

Step-by-step explanation:

785 in² is the volume of the cylinder shown .

What is a cylinder, simply defined?

A cylinder is a three-dimensional solid in mathematics that holds two parallel bases spaced at a constant distance apart from one another and connected by a curving surface.

                These bases often have a circular form (like a circle), and a line segment connecting the centres of the two bases is known as the axis.

the volume of the cylinder = πr²h

               r = 10

             h = 25 in

           the volume of the cylinder  = 3.14 * 10 * 25

                                                         = 785 in²

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The constant 'a' can be found by solving the integral equation: 2∫₀² 6(2x² + x - a)dx = 24. Simplifying the expression and solving for 'a' gives the value of 5/9.

To find the value of the constant 'a', we need to solve the integral equation:

2∫₀² 6(2x² + x - a)dx = 24

First, we'll integrate the function with respect to x:

2[∫(12x² + 6x - 6a)dx] = 24

Now, we'll find the antiderivative:

2[(4x³/3 + 3x²/2 - 6ax) |₀²] = 24

Next, we'll evaluate the antiderivative at the limits of integration:

2[(4(2³)/3 + 3(2²)/2 - 6a(2)) - (0)] = 24

Simplify the expression:

2[(32/3 + 12 - 12a)] = 24

Divide both sides by 2:

(32/3 + 12 - 12a) = 12

Now, we'll solve for 'a':

-12a = 12 - 12 - 32/3

-12a = -20/3

a = (-20/3) / -12

a = 5/9

So, the value of the constant 'a' is 5/9.

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The function f(x) = log₂(2x² + 2) has an absolute maximum value of log₂(2) ≈ 1 and an absolute minimum value of log₂(2) ≈ 1 on the interval [-1,∞).

To find the absolute maximum and minimum values of f(x) = log₂(2x² + 2) on the interval [-1,∞), we can use the following steps:

Take the derivative of f(x) with respect to x:

f'(x) = 4x / (2x² + 2) ln(2)

Find critical points by setting f'(x) equal to zero and solving for x:

f'(x) = 0 => 4x / (2x² + 2) ln(2) = 0 => x = 0

Check the value of f(x) at the critical point and at the endpoints of the interval:

f(-1) = log₂(2) ≈ 1

f(0) = log₂(2) ≈ 1

As x approaches infinity, f(x) approaches infinity.

Determine the absolute maximum and minimum values of f(x):

The absolute maximum value of f(x) on the interval [-1,∞) is log₂(2) ≈ 1, which occurs at x = -1 and x = 0. The absolute minimum value of f(x) on the interval [-1,∞) is also log₂(2) ≈ 1, which occurs at x = -1 and x = 0.

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The speed of the particle at time t=0 is approximately sqrt(25 * pi^2 + 16) units per time unit. The speed of the particle at time t=0 when given parametric equations x(t)=5sin(pit) and y(t)=(2t-1)^2, follow these steps:

Step:1. Differentiate the x(t) and y(t) equations with respect to time (t) to find the velocity components in the x and y directions:
  dx/dt = d(5sin(pit))/dt
  dy/dt = d((2t-1)^2)/dt
Step:2. Apply the chain rule and differentiation rules to compute the derivatives:
  dx/dt = 5 * pi * cos(pit)
  dy/dt = 2 * (2t-1) * 2
Step:3. Substitute t=0 into the expressions for dx/dt and dy/dt to get the velocity components at time t=0:
  dx/dt(0) = 5 * pi * cos(0) = 5 * pi
  dy/dt(0) = 2 * (2(0)-1) * 2 = -4
Step:4. Use the Pythagorean theorem to find the magnitude of the velocity, which represents the speed of the particle at time t=0:
  Speed = sqrt((dx/dt(0))^2 + (dy/dt(0))^2)
  Speed = sqrt((5 * pi)^2 + (-4)^2)
  Speed = sqrt(25 * pi^2 + 16)

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The rate of change of the area of the square when the side length is 8 inches and the side length is increasing at a rate of 14 inches per minute is 224 square inches per minute.

To find the rate of change of the area of the square, we need to use the formula for the area of a square:

A = s^2

where A is the area of the square and s is the length of the side of the square.

To find the rate of change of the area, we need to take the derivative of this formula with respect to time:

dA/dt = 2s(ds/dt)

where dA/dt is the rate of change of the area, ds/dt is the rate of change of the side length, and s is the side length of the square.

Since the side length is increasing at a rate of 14 inches per minute, we can substitute ds/dt = 14 into the above equation, and we are given that the side length is 8 inches, so we can substitute s = 8.

dA/dt = 2s(ds/dt)

dA/dt = 2(8)(14)

dA/dt = 224

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The probability that a sample of size n=186 is randomly selected with a mean less than 9.1 is approximately 0.0655 (rounded to 4 decimal places).

To find the probability that a single random value is less than 9.1, we can use the standard normal distribution and calculate the z-score:
z = (9.1 - 15.8) / 60.6 = -0.110
Using a standard normal distribution table or calculator, we can find that the probability of a value being less than -0.110 is 0.4564. Therefore, the probability that a single random value is less than 9.1 is approximately 0.4564 (rounded to 4 decimal places).
To find the probability that a sample of size n=186 is randomly selected with a mean less than 9.1, we need to use the sampling distribution of the mean. The sampling distribution of the mean has a mean equal to the population mean (15.8) and a standard deviation equal to the population standard deviation divided by the square root of the sample size:
standard deviation = 60.6 / sqrt(186) = 4.436
We can then calculate the z-score for this sampling distribution:

z = (9.1 - 15.8) / 4.436 = -1.508

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Answer:

  (B)  123.2 yd²

Step-by-step explanation:

You want the area of the decagon shown with side length 4 yd and apothem 6.16 yd.

The area is given by the formula ...

  A = 1/2Pa

where P is the perimeter and 'a' is the apothem, the distance from the center to the middle of one side of the regular polygon.

The perimeter is the sum of the side lengths. Each of the 10 sides has a length of 4 yd, so that sum is 10·4 yd = 40 yd.

Using the known values in the formula, we find the area to be ...

  A = 1/2(40 yd)(6.16 yd)

  A = 123.2 yd²

__

Additional comment

Effectively, you are computing 10 times the area of the triangle that is the area of one sector. That triangle has a base of 4 yd and a height of 6.16 yd. Its area is ...

  A = 1/2bh = 1/2(4 yd)(6.16 yd) = 12.32 yd²

The 10 sectors of the decagon will have an area of ...

  A = 10 × 12.32 yd² = 123.2 yd²

The cοst οf 5 packages οf grape drink will cοst is $13.75.

Wοrd prοblem must be sοlved step-by-step. Generally we gο by the fοllοwing way:

Tο master wοrd prοblems there is nο way οther than practicing mοre and mοre οf the kind. If yοu accept my advice, I'll say yοu nοt οnly practice frοm the bοοk given in yοur curriculum, but alsο try sοlving them frοm as much bοοks yοu can.

The tοtal cοst οf 4 packages = $11

Cοst οf 1 package = 11/4

= 2.7

Hence, the cοst οf 5 packages οf grape drink will cοst = 5 * 11/4

= 55/4

= 13.75

Hence, The cοst οf 5 packages οf grape drink will cοst is $13.75.

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Complete question:

A sample of size n = 8 from a Normal(p, o) population results in a sample standard deviation of s=5.4. A 95% lower bound for the true population standard deviation is [tex]\sigma >3.809[/tex]

The Chi-Square distribution and the provided information:

sample size (n = 8), sample standard deviation (s = 5.4), and a 95% confidence level.

Our goal is to find the lower bound for the true population standard deviation ([tex]\sigma[/tex]).
Identify the degrees of freedom (df)
[tex]df = n - 1 = 8 - 1 = 7[/tex]
Find the Chi-Square value corresponding to the given confidence level and degrees of freedom.
For a 95% confidence level and 7 degrees of freedom, the Chi-Square value [tex](X^2)[/tex]is 14.067.
Calculate the lower bound for the population standard deviation (σ)
Using the formula for the lower bound of the standard deviation:
[tex]\sigma > \sqrt {[(n - 1) \times s^2 / X^2]}[/tex]
Plug in the given values:
[tex]\sigma > \sqrt {[(7) \times (5.4)^2 / 14.067]}[/tex]
[tex]\sigma > \sqrt {[(7) \times (29.16) / 14.067]}[/tex]
[tex]sigma > \sqrt {[(203.12) / 14.067]}[/tex]
[tex]\sigma > \sqrt (14.431)[/tex]
[tex]\sigma > 3.80[/tex]
Based on the calculations, the correct answer is:
E: [tex]\sigma > 3.80[/tex]

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The least common multiple (LCM) of the expressions is 9000x⁴y⁵

From the question, we have the following parameters that can be used in our computation:

450x²y⁵

3000x⁴y

Factor each expression

So, we have

450x²y⁵ = 2 * 3 * 3 * 5 * 5x²y⁵

3000x⁴y = 2 * 2 * 2 * 3 * 5 * 5 * 5x⁴y

Multiply all factors

So, we have

LCM = 2 * 2 * 2 * 3 * 3 * 5 * 5 * 5x⁴y⁵

Evaluate

LCM = 9000x⁴y⁵

Hence, the LCM is 9000x⁴y⁵

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a)

The wind chill temperature is 35.74 + 0.6215T - 35.75(V^0.16) + 0.4275T(V^0.16).

b)

W'(40) = -0.0458T + 12.24

The derivative also tells us that as the wind speed increases, the rate of decrease of wind chill temperature slows down.

We have,

To find the equation of the line tangent to f(x) at x = 1, we need to find the slope of the tangent at that point and the point of tangency.

First, we find the derivative of f(x):

f'(x) = -6x + 6

At x = 1, the slope of the tangent is:

f'(1) = -6(1) + 6 = 0

So the equation of the tangent at x = 1 is simply:

y = f(1) = -3(1)^2 + 6(1) - 7 = -4

Therefore,

The equation of the line tangent to f(x) = -3x² + 6x - 7 at x = 1 is y = -4.

a)

The wind chill temperature is a function of the air temperature and the wind speed.

The formula to calculate wind chill temperature in degrees Fahrenheit is:

WCT = 35.74 + 0.6215T - 35.75(V^0.16) + 0.4275T(V^0.16)

where T is the air temperature in degrees Fahrenheit and V is the wind speed in miles per hour.

Since the wind is blowing at 40 mph, we need to know the air temperature to calculate the wind chill temperature.

Without that information, we cannot find the wind chill temperature.

b)

To find W'(40), we need to take the derivative of the wind chill temperature formula with respect to V and evaluate it at V = 40:

W'(V) = -5.6075V^(-0.84)T + 18.856(V^(-0.84)) - 1.5V^(-0.84)

W'(40) = -5.6075(40)^(-0.84)T + 18.856(40)^(-0.84) - 1.5(40)^(-0.84)

W'(40) = -0.0458T + 12.24

This means that for a given air temperature T, if the wind speed increases by 1 mph, the wind chill temperature decreases by 0.0458 degrees Fahrenheit, approximately.

The derivative also tells us that as the wind speed increases, the rate of decrease of wind chill temperature slows down.

Thus,
a)

The wind chill temperature is 35.74 + 0.6215T - 35.75(V^0.16) + 0.4275T(V^0.16).

b)

W'(40) = -0.0458T + 12.24

The derivative also tells us that as the wind speed increases, the rate of decrease of wind chill temperature slows down.

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The process of a baby picking up a crumb or Cheerio involves several different types of assessment data, including visual, perceptual, fine motor, and proprioceptive skills.

Firstly, the baby uses their visual and perceptual skills to locate the crumb or Cheerio. They may scan the surrounding environment or look directly at the object. This can be assessed through observation of the baby's eye movements and head orientation.

Next, the baby uses their fine motor skills to reach for the crumb or Cheerio. They may use their fingers or their whole hand to grasp the object. This can be assessed through observation of the baby's hand movements and coordination.

Finally, the baby uses their proprioceptive skills to adjust their grip and bring the crumb or Cheerio to their mouth. This can be assessed through observation of the baby's mouth movements and coordination.

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Correlation is a statistical measure that describes the strength and direction of a relationship between two variables. It indicates how much one variable tends to change in response to changes in the other variable.

The sample correlation coefficient between X and Y can be calculated using the following formula: r =[tex][nΣxy - (Σx)(Σy)] / [√(nΣx^2 - (Σx)^2) √(nΣy^2 - (Σy)^2)][/tex]

where n is the sample size, Σxy is the sum of the products of the corresponding x and y values, Σx and Σy are the sums of the x and y values, Σx^2 and Σy^2 are the sums of the squared x and y values, respectively.

Using the given information, we can calculate the necessary values as follows:

n = 45

Σx = 153.7

Σy = 231.2

Σxy = 712.5

Σx^2 = 718

Σy^2 = 1775.2

Substituting these values into the formula, we get:

r = [nΣxy - (Σx)(Σy)] / [√(nΣx^2 - (Σx)^2) √(nΣy^2 - (Σy)^2)]

r = [45(712.5) - (153.7)(231.2)] / [√(45(718) - (153.7)^2) √(45(1775.2) - (231.2)^2)]

r = 0.804

Therefore, the sample correlation coefficient between X and Y is 0.804. This indicates a strong positive linear relationship between the two variables.

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A. A post-lunch nap decreases the amount of time it takes males to sprint 20 meters.

To test if there is enough evidence to support the researcher's claim, we can perform a paired t-test. The null hypothesis is that there is no difference in the mean sprint time between without nap and with nap conditions. The alternative hypothesis is that the mean sprint time is shorter with a post-lunch nap.

(a) Calculate the differences between sprint times with and without nap for each male:
Male      Difference
1               0.01
2               0.01
3               0.02
4               0.01
5               0.017
6               0.06
7               0.01
8               0.03
9               0.01
10             0.03

(b) Calculate the mean difference:
mean difference = 0.022
(c) Calculate the standard deviation of the differences:
s = 0.026
(d) Calculate the t-statistic:
t = (mean difference - 0) / (s / sqrt(n)) = (0.022 - 0) / (0.026 / sqrt(10)) = 2.95
(e) Calculate the degrees of freedom:
df = n - 1 = 9
(f) Determine the critical value for a two-tailed test with alpha = 0.10 and df = 9:
t_critical = +/- 1.833
(g) Compare the absolute value of the t-statistic to the critical value:
|t| = 2.95 > 1.833
(h) The t-statistic falls in the rejection region, so we reject the null hypothesis.
(i) There is enough evidence to support the alternative hypothesis that the mean sprint time is shorter with a post-lunch nap.
(j) The p-value for this test is less than 0.10.
(k) We can conclude with 90% confidence that the mean difference in sprint times with and without nap is between 0.005 and 0.039.
(l) We can conclude with 95% confidence that the mean difference in sprint times with and without nap is between -0.002 and 0.046.
(m) We can conclude with 99% confidence that the mean difference in sprint times with and without nap is between -0.008 and 0.052.
(n) The assumptions for the test are that the samples are random and dependent, and the population is normally distributed.
(o) Based on the results of this test, we can support the researcher's claim that a post-lunch nap decreases the amount of time it takes males to sprint 20 meters after a night with only 4 hours of sleep.

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The solution are,

The solution is, the measure of the, inscribed angle: 30°, and,

central angle: 60°.

here, we have,

from the given figure, we get,

The central angle is double the inscribed angle for the same intercepted arc.

Since doubling the angle adds 30° to it,

the original inscribed angle must be 30°.

so, we get,

Then the central angle is 30°+30° = 2·30° = 60°.

The solution are,

the measure of angle C is 52°

the measure of angle B is 52°

the measure of angle BSD is 71°

the measure of angle CSE is 71°

the measure of angle E is 57°

the measure of arc BC 57°.

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The resultant component of the vector addition is (-9, 0).

The resultant component of the vector is calculated as follows;

a = (-9, 6)

b = (3, 1)

The result of 1/3a = ¹/₃ (-9), ¹/₃(6) = (-3, 2)

The result of 2b = 2(3, 1) = (6, 2)

The result of the vector addition is calculated as follows;

1/3a - 2b

= (-3, 2) - (6, 2)

= (-3 -6, 2 -2)

= (-9, 0)

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The statement "The use of the Poisson distribution requires a value n which indicates a definite number of independent trials" is false.

The Poisson distribution is a probability distribution that is used to model the occurrence of rare events in a given time or space interval. It does not require a definite number of independent trials, as it is a continuous probability distribution. Instead, it assumes that the events occur randomly and independently over time or space, with a constant mean rate.

The Poisson distribution is characterized by a single parameter, λ (lambda), which represents the average rate of occurrence of the event. Therefore, the Poisson distribution does not require a value n to indicate a definite number of independent trials.

Therefore, the statement "The use of the Poisson distribution requires a value n which indicates a definite number of independent trials" is false.

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To find the mean weight of 31 fruits at random, we can use the Central Limit Theorem. According to the theorem, the sample means of large sample size (n>=30) from any population will be normally distributed with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

The mean weight of 31 fruits at random will be normally distributed with a mean of 451 grams and a standard deviation of 9/sqrt(31) grams.

To find the weight that the mean will be greater than 20% of the time, we need to find the z-score corresponding to the 20th percentile of the normal distribution. Using a standard normal distribution table, we find that the z-score is -0.84.

Now we can use the formula z = (x - mu) / (sigma / sqrt(n)) to find the weight (x) that corresponds to the z-score. Plugging in the values, we get -0.84 = (x - 451) / (9 / sqrt(31)). Solving for x, we get x = 448.4 grams. Therefore, the mean weight of 31 fruits at random will be greater than 448.4 grams 20% of the time.

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To make 6/1 and 12/2 equivalent, we can simplify 12/2 to 6/1 by dividing both the numerator and denominator by 2. This results in two equivalent fractions, 6/1 and 6/1 and show them on number line also.

To make 6/1 and 12/2 equivalent, we can use a number line to represent both fractions and then compare them.

First, we can represent 6/1 on a number line by putting a point at 6 on the line, like

Next, we can represent 12/2 on the same number line by putting a point at 12, which is twice the value of 6, like

Now we can see that both points are on the same line, which means that 6/1 and 12/2 are equivalent fractions.

we can simplify both fractions to a common denominator and compare the resulting numerators. In this case, the common denominator is 2, so we can write

6/1 = 12/2 = 12/2

The numerators of both fractions are equal to 12, which means that 6/1 and 12/2 are equivalent fractions.

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The numerical value of a + b + c is 10(option a).

To find the prime factorization of 180, we start by dividing it by the smallest prime number, which is 2. We get:

180 ÷ 2 = 90

So 2 is a prime factor of 180. We continue dividing 90 by 2 until we can no longer divide by 2:

90 ÷ 2 = 45

45 ÷ 2 = 22.5 (not a whole number)

So we move on to the next smallest prime number, which is 3. We divide 45 by 3:

45 ÷ 3 = 15

Now we divide 15 by 3:

15 ÷ 3 = 5

Since 5 is a prime number, we can stop dividing. We have found the prime factorization of 180:

180 = 2 × 2 × 3 × 3 × 5

To express this in the form a²b²c, we group the prime factors in pairs of 2s and 3s:

180 = (2²) × (3²) × 5

Now we can see that a = 2, b = 3, and c = 5. We add them up to get:

a + b + c = 2 + 3 + 5 = 10

Hence the answer is A.

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The Prime factors of 36 are (2²) * (3²) or 2 * 2 * 3 * 3. Thus, option B is the correct answer.

Prime numbers are numbers that are not divisible by numbers other than 1 or the number itself. Composite numbers are numbers that have more than 2 factors other than 1 and the number itself.

Factors are numbers that when divided by another number leave no remainder. Prime factors are the prime numbers that when multiplied the product we get equal to the original number.

To calculate the prime factor, we use the division method.

In this method, firstly we divide the number by the smallest prime number it is when divided by is completely divisible. In this case, we divide 36 by 2 and get 18 as the quotient.

Again, divide the quotient of the previous division by the smallest prime number it is divisible. So, 18 is again divided by 2 and we get 9.

Repetition of the previous step takes place until we get 1. And 9 ÷ 3 = 3. Then  3 ÷ 3 = 1. Since we get 1, this is the final answer.

Finally, Prime factorization of 36 = 2 × 2 × 3 × 3 or we can write it as (2²) * (3²)

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By minimizing the total energy, the pigeon should fly at the point where the total energy is minimized. This point is 18.75 miles away from point A.

Energy is the ability to do work. It is the capacity to cause change, move objects, and affect the environment. It is a fundamental part of nature and exists in various forms, such as kinetic energy, potential energy, thermal energy, light energy, chemical energy, and electrical energy.

Point S should be located 18.750 miles away from point A. This is the point where the total energy required to reach point A is minimized. The total energy required is given by:

Total energy = (Energy rate over water) • (Distance over water) + (Energy rate over land) • (Distance over land)

Substituting in the given values, we get:

Total energy = (1.25 * 12) + (1 * 18.75)

Total energy = 24 + 18.75

Total energy = 42.75

By minimizing the total energy, the pigeon should fly at the point where the total energy is minimized. This point is 18.75 miles away from point A.

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The minimum distance between the curve and the point (3/2, 0) is [tex]\sqrt (5/4) = \sqrt (5/2)[/tex], which occurs at x = 1.

The curve comes to the point (3/2,0), we need to minimize the square of the distance between the point and the curve.

Let (x, y) be a point on the curve [tex]y = x^3 - 3x + 2[/tex]. Then, the square of the distance between (x, y) and (3/2, 0) is:

[tex]d^2 = (x - 3/2)^2 + y^2[/tex]

Substituting [tex]y = x^3 - 3x + 2[/tex], we get:

[tex]d^2 = (x - (3/2))^2 + (x^3 - 3x + 2)^2[/tex]

To minimize[tex]d^2[/tex], we take the derivative of [tex]d^2[/tex] with respect to x and set it equal to 0:

[tex]d^2/dx = 2(x - (3/2)) + 2(x^3 - 3x + 2)(3x^2 - 3) = 0[/tex]

Simplifying and factoring, we get:

[tex]2(x - (3/2)) + 6(x - 1)(x + 1)(x^2 - x - 1) = 0[/tex]

One solution to this equation is x = 1, which is a local minimum.

Since the curve is symmetric about the y-axis, there is another local minimum at x = -1.

We can check that these are the only two local minima by observing that the second derivative of d^2 is positive at these points.

The curve comes closest to the point (3/2, 0) at x = 1 and x = -1. To find the minimum distance, we substitute these values into the equation for [tex]d^2:[/tex]

[tex]d^2(1) = (1/2)^2 + (1)^2 = 5/4[/tex]

[tex]d^2(-1) = (5/2)^2 + (-3)^2 = 49/4[/tex]

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The maximum value of f(x,y) = x+y³ for x, y > 0 on the unit circle is (√37 + 2)/9.

We need to find the maximum value of the function f(x,y) = x+y³  on the unit circle, which is the set of all (x,y) points with radius 1 centered at the origin.

Since the domain of the function is restricted to x,y>0, we can use Lagrange multipliers to find the maximum value on the unit circle.

First, we set up the system of equations:

∇f = λ∇g

g(x,y) = x² + y² - 1

Where ∇f and ∇g are the gradient vectors of f and g, respectively, and λ is the Lagrange multiplier.

∇f = <1, 3y²>

∇g = <2x, 2y>

Setting ∇f = λ∇g, we get:

1/2x = λ

3y²/2y = λ

Simplifying, we get:

x = 3y²

Plugging this into the equation of the unit circle, we get:

9y⁴ + y² - 1 = 0

Using the quadratic formula, we get:

y² = (-1 ± √(1 + 36))/18

y² = (-1 ± √37)/18

Since y>0, we take the positive root:

y² = (√37 - 1)/18

Plugging this into x = 3y², we get:

x = 3(√37 - 1)/18

Therefore, the maximum value of f(x,y) = x+y³  on the unit circle is:

fmax = x+y³  = 3(√37 - 1)/18 + (√37 - 1)/54

fmax = (√37 + 2)/9

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Solving the equation we get, x= -2.

What is equation?

In  algebra, the definition of an equation is a mathematical statement which shows that two mathematical expressions are equal. For example, 3x - 7= 14 is an equation, in which 3x - 7 and 14 are two expressions separated by an 'equal( '=')' sign. Solving the equation we will get the value of the unknown x=7.

Given equation is

                  3/2+2x/5=7/10

    Taking the constants to the right hand side of the equation we get,

                    2x/5= 7/10 - 3/2

         The lowest common denominator of 7/10 and 3/2 is 10

Multiplying by 10 to the both sides of equation we get,

                     (2x/5)×10 = (7/10-3/2)×10

                   ⇒ 4x = (7/10)×10 - (3/2)×10

                   ⇒ 4x = 7- 15

                  ⇒ 4x = -8

         Dividing both sides by 4 we get,

                      x = -2

Hence, solving the equation we get, x= -2.

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The rate of change of the surface area of the sphere is -48 π square cm/ sec.

We're given that the radius of a sphere is dwindling at a rate of 2 cm/ sec. Let's denote the sphere's radius by r and the rate of change of the compass by dr/ dt. In this case, dr/ dt = -2( negative because the radius is dwindling).

We're asked to find the rate of change, in square cm/ sec, of the face area of the sphere at the moment when the compass is 3 cm. The face area S of a sphere with radius r is given by the formula S = 4π[tex]r^{2}[/tex].

We can use the chain rule of differentiation to find the rate of change of S with respect to time.

dS/ dt = dS/ dr * dr/ dt

We can find dS dr by secerning the formula for S with respect to r

dS/ dr = 8πr

Now we can substitute r = 3 and dr/ dt = -2 into the expression for dS/ dt

dS/ dt = dS/ dr * dr/ dt

dS/ dt = 8πr *(- 2)( substituting r = 3 and dr/ dt = -2)

dS/ dt = 8π( 3)(- 2)

dS/ dt = -48 π

thus, when the compass of the sphere is 3 cm, the rate of change of the face area of the sphere is -48 π square cm/ sec. The negative sign indicates that the face area is decreasing.

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