01:53:18
Which inequality represents the sentence below?

Two more than a number is less than 14.
2 + n less-than-or-equal-to 14
2 + n less-than 14
2 + n greater-than-or-equal-to 14
2 + n greater-than 14

The moment of inertia about the z-axis of a thin spherical shell with equation x² + y² + z² = 2a and constant density 8 is 8.

The moment of inertia of a solid object measures its resistance to rotational motion around a specific axis. For a thin spherical shell, the moment of inertia about the z-axis can be calculated using the formula:

I = ∫(r²) dm

where r is the perpendicular distance from the axis of rotation (z-axis) to an infinitesimally small mass element dm.

In this case, the spherical shell has constant density, so the mass per unit volume is constant. Therefore, dm = ρ dV, where ρ is the density and dV is the volume element.

Since the equation of the spherical shell is x² + y² + z² = 2a, we can rewrite it as r² + z² = 2a, where r is the distance from the z-axis to a point on the shell. The moment of inertia can be calculated by integrating over the volume of the shell:

I = ∫∫∫ (r²) ρ dV

Since the density is constant, ρ can be taken out of the integral:

I = ρ ∫∫∫ (r²) dV

The integral represents the volume of the spherical shell, which is 4πa². Therefore, we have:

I = ρ (4πa²)

Substituting the given density ρ = 8, we get:

I = 8 (4πa²) = 32πa²

So, the moment of inertia about the z-axis of the thin spherical shell is 32πa².

Learn more about moment of inertia here:

https://brainly.com/question/30051108

#SPJ11

To change the order of integration for the given triple integral, we can integrate with respect to one variable at a time.

The original order of integration is: ∫₀¹₆ ∫₀² ∫₃ʸ⁶ ₅cos(x²) ₄z dx dy dz

Let's change the order of integration. We start by integrating with respect to z first:

∫₀¹₆ ∫₀² ∫₃ʸ⁶ ₅cos(x²) ₄z dx dy dz

= ∫₀¹₆ ∫₀² [2z₃ʸ⁶ cos(x²)] dx dy

= ∫₀¹₆ [2z₃ʸ⁶ cos(x²)] x=₀² dy dz

Next, we integrate with respect to x:

∫₀¹₆ [2z₃ʸ⁶ cos(x²)] x=₀² dy dz

= ∫₀¹₆ [2z₃ʸ⁶ (sin(x²))|₀²] dy dz

= ∫₀¹₆ [2z₃ʸ⁶ (sin(4) - sin(0))] dy dz

= ∫₀¹₆ [2z₃ʸ⁶ sin(4)] dy dz

Finally, we integrate with respect to y:

∫₀¹₆ [2z₃ʸ⁶ sin(4)] dy dz

= [z₃ʸ⁷ sin(4)/7] ₀¹₆ dz

= ∫₀¹₆ z₃ sin(4)/7 dz

Now we can integrate with respect to z:

∫₀¹₆ z₃ sin(4)/7 dz

= [(z² sin(4))/14] ₀¹₆

= (16² sin(4))/14 - (0² sin(4))/14

= (256 sin(4))/14

= (128 sin(4))/7

Therefore, by changing the order of integration, the given triple integral becomes (128 sin(4))/7.

Learn more about triple integral here: brainly.com/question/32386151

#SPJ11

Step-by-step explanation:

The chi-square test for homogeneity is used if he wants to determine if the distributions are the same.

What is the chi-square test?

Chi-square is a statistical test that looks at how categorical variables from a random sample differ from one another to see if the expected and actual findings match together well.  It is a contrast of two sets of statistical data. Karl Pearson developed this test in 1900 for the analysis and distribution of categorical data.

Here,

We have to determine for which situations should the chi-square test for homogeneity be used.

We concluded from the given option that:

A researcher is trying to determine if salaries for men and women in the tech industry have the same distribution.

He surveys a random sample of men and women in the industry and records the distribution of salaries for each gender.

He wants to determine if the distributions are the same.

Hence, the chi-square test for homogeneity is used if he wants to determine if the distributions are the same.

To learn more about the chi-square test from the given link

https://brainly.com/question/31519775

#SPJ4

Responsible financial plan will take into consideration both short term and long term goals.

Given,

A financial plan is to be made.

A financial plan protects you from life's surprises. A Personal financial plan reduces doubt or uncertainty about your decisions and make adjustments to help overcome obstacles that could alter your lifestyle.

Now,

To make a better financial plan one should consider his/her short term as well long term goals.

Short term goals:

Short term goals include the goals that are needed to be achieved in the time frame 2-4 years.

For example,

One has to buy a car in the coming 3 years than this type of goals are considered short term and financial plan is to be made according to the price of car that is to be paid after 3 years while buying a car.

Long term goals:

Long term goals include the goals that are needed to be achieved in the time frame 10-12 years.

For example,

Retirement can be considered as long term plan for which one has to save a big amount of corpus so that after retirement his/her expenses will be well taken care off.

Learn more about Finances,

https://brainly.com/question/29763313

#SPJ1

The probability of flipping a coin and having it land on heads is always 50%, regardless of the previous outcomes. Each coin flip is an independent event, so the past outcomes do not affect the probability of future outcomes.

The experimental probability that Luke's next flip will be heads is 3/5.

Experimental probability (EP), also called empirical probability or relative frequency, is probability based on data collected from repeated trials.

Let n represent the total number of trials or the number of times an experiment is done. Let p represent the number of times an event occurred while performing this experiment n times.

[tex]\sf Experimental \ probability \ of \ an \ event = \dfrac{p}{n}[/tex]

Since heads was the result 3 times. There were 5 trials. So the probability is 3/5.

Thus, The experimental probability that Luke's next flip will be heads is 3/5.

Learn more about the experimental probability with the given link provided below.

https://brainly.com/question/30694293

S is transitive. Since S is reflexive, symmetric, and transitive, it is an equivalence relation.

To prove that S is an equivalence relation, we need to show that it satisfies three conditions: reflexivity, symmetry, and transitivity.

Reflexivity: For any ordered pair (a, b), we have a + b = b + a. So, (a, b) S (a, b), and S is reflexive.

Symmetry: If (a, b) S (c, d), then a + d = b + c. Rearranging this equation gives us d + a = c + b, which implies that (c, d) S (a, b). Therefore, S is symmetric.

Transitivity: If (a, b) S (c, d) and (c, d) S (e, f), then we have a + d = b + c and c + f = d + e. Adding these two equations gives us a + 2d + f = b + 2c + e. Rearranging this equation, we get (a, b) S (e, f). Hence, S is transitive.

Since S is reflexive, symmetric, and transitive, it is an equivalence relation.

Learn more about transitive here

https://brainly.com/question/24606230

#SPJ11

Similarly, if the consumer's income increases, they may choose to consume more of both function x1 and x2, or they may choose to consume more of one good and less of the other, depending on the relative prices and the marginal utility of each good.

The utility function represents the satisfaction or happiness a consumer derives from consuming two goods, x1 and x2. In this case, the utility function is u(x1, x2) = x1x2^2. This means that the consumer values x1 and x2 positively and that the value the consumer derives from x2 increases at a faster rate than x1 as they consume more of it.

The budget constraint, on the other hand, represents the limited resources or income of the consumer. It is given by p1x1 + p2x2 = w, where p1 and p2 are the prices of x1 and x2, respectively, and w is the consumer's income.

To find the optimal consumption bundle, we need to maximize the utility function subject to the budget constraint. This can be done using the method of Lagrange multipliers.

The Lagrangian function is given by:

L(x1, x2, λ) = x1x2^2 + λ(w - p1x1 - p2x2)

Taking partial derivatives with respect to x1, x2, and λ and setting them equal to zero, we get the following first-order conditions:

∂L/∂x1 = x2^2 - λp1 = 0

∂L/∂x2 = 2x1x2 - λp2 = 0

∂L/∂λ = w - p1x1 - p2x2 = 0

Solving these equations simultaneously, we can find the optimal values of x1 and x2 that maximize the utility function subject to the budget constraint. Once we have the optimal consumption bundle, we can use it to make predictions about how changes in prices or income will affect the consumer's consumption of x1 and x2. For example, if the price of x1 increases, the consumer will consume less of it and more of x2, assuming that the utility-maximizing bundle is still affordable.

To know more about function,

https://brainly.com/question/28193994

#SPJ11

The distance between the points (-7, -3) and (-3, -5) in simplest radical form is 2√5.

The distance formula used in finding the distance between two points is expressed as;

[tex]d = \sqrt{( x_2 - x_1 )^2 + ( y_2 - y_1)^2 }[/tex]

Given the points in the question:

Point 1 (-7,-3)

Point 2 (-3,-5)

Plug the given values into the distance formula and simplify.

[tex]d = \sqrt{( x_2 - x_1 )^2 + ( y_2 - y_1)^2 }\\\\d = \sqrt{( -3 - (-7) )^2 + ( -5 - (-3))^2 }\\\\d = \sqrt{( -3 + 7 )^2 + ( -5 + 3)^2 }\\\\d = \sqrt{( 4 )^2 + ( -2)^2 }\\\\d = \sqrt{16 + 4 }\\\\d = \sqrt{20 }\\\\d = 2\sqrt{5}[/tex]

Therefore, the distance between the points is 2√5 .

Learn more about the distance formula here: brainly.com/question/24509115

#SPJ1

Therefore, the amount of energy (in kJ) needed to ionize 7.5 mol of sodium is 3720 kJ. This is the long answer that contains 250 words

To determine the amount of energy (in kJ) needed to ionize 7.5 mol of sodium (Na(g) + 496 kJ → Na+(g) + e–), we can use dimensional analysis. The balanced chemical equation for the ionization of sodium is:Na(g) + 496 kJ → Na+(g) + e–The energy required to ionize one mole of sodium is 496 kJ/mol.

Therefore, the energy required to ionize 7.5 mol of sodium can be calculated as:7.5 mol × 496 kJ/mol = 3720 kJ Therefore, the amount of energy (in kJ) needed to ionize 7.5 mol of sodium is 3720 kJ. This is the long answer that contains 250 words.

To know more about energy, click here

https://brainly.com/question/1932868

#SPJ11

If m and n are both positive integers, the print statements will be executed m x n times.

The number of times the print statements are executed depends on the values of m and n.

Assuming that both m and n are positive integers, the print statements inside the nested for loops will be executed m x n times.

This is because the outer loop runs m times and the inner loop runs n times for each iteration of the outer loop.

Therefore, the total number of executions of the print statements will be the product of m and n.

This can be represented as:
Number of executions = m x n

In summary, if m and n are both positive integers, the print statements will be executed m x n times.

Know more about integers here:

https://brainly.com/question/17695139

#SPJ11

The number 1.088 x 10¹ is 13.6 times larger than 8 x 10⁻¹

To find how many times larger is (1.088 x 10¹)  than (8 x 10⁻¹), we just need to take the quotient between these two numbers. To do so remember that when we take the quotient between two powerswith the same base, we just need to subtract the exponents.

Then here we will get:

[tex]\frac{1.088*10^1}{8*10^{-1}} = \frac{1.088}{8} *10^{1 - (-1)} = 0.136*10^2[/tex]

We can rewrite that as:

1.36*10 = 13.6

Then the first number is 13.6 times larger than the second one.

Learn more about quotients at:

https://brainly.com/question/629998

#SPJ1

Approximately 10.8% of the original amount of Sr-90 will remain in the artifact after 68.8 years.

The decay of a radioactive substance is modeled by the equation:

N(t) = N₀ * (1/2)^(t / T)

where N(t) is the amount of the substance at time t, N₀ is the initial amount, T is the half-life, and t is the time elapsed since the initial measurement.

In this case, we are given that the half-life of Sr-90 is T = 28.1 years, and we want to find the percentage of Sr-90 remaining after 68.8 years, which is t = 68.8 years.

The percentage of Sr-90 remaining at time t can be found by dividing the amount of Sr-90 at time t by the initial amount N₀, and multiplying by 100:

% remaining = (N(t) / N₀) * 100

Substituting the values given, we get:

% remaining = (N₀ * (1/2)^(t/T) / N₀) * 100

= (1/2)^(68.8/28.1) * 100

≈ 10.8%

Therefore, approximately 10.8% of the original amount of Sr-90 will remain in the artifact after 68.8 years.

Learn more about artifact  here:

https://brainly.com/question/11408267

#SPJ11

The table should be completed to show different percentages of the questions Rita answered correctly on the pre-test as follows;

Number of questions correct           Percentage

7                                                       350% of the correct pre-test questions.

1                                                        50% of the correct pre-test questions.

2                                                       100% of the correct pre-test questions.

In Mathematics and Statistics, a percentage refers to any numerical value that is expressed as a fraction of hundred (100). This ultimately implies that, a percentage indicates the hundredth parts of any given numerical value.

Based on the information provided about this tape diagram that shows the number of questions Rita answered correctly on the pre-test, we can logically deduce that each of the box represents the number of questions and corresponds to a percentage of 50;

350%  ⇒ 350/50 = 7 questions.

50%  ⇒ 50/50 = 1 question.

100%  ⇒ 100/50 = 2 questions.

Read more on percentage here: brainly.com/question/28009735

#SPJ1

Missing information:

The question is incomplete and the complete question is shown in the attached picture.

Answer:

Solution is in attached photo.

Step-by-step explanation:

Do take note, when the square is split into 2 diagonal halves, we will see a isosceles triangle, from there, we can use sine rule (there is more than one way) to find the length of one side.

Answer:

Step-by-step explanation:

[tex]9y-3xy^2-4+x[/tex]

9y-3xy²-4+x

The statement "if the means of two distributions are equal, then the variance must also be equal" is false. While the mean and variance of a distribution are related, they are not always directly proportional to each other.

It is possible for two distributions to have the same mean but different variances. For example, imagine two distributions where one has all of its values clustered tightly around the mean, while the other has a wider range of values spread out more widely from the mean.

In this case, the first distribution would have a lower variance than the second, but both could still have the same mean. In summary, while there may be some cases where equal means correspond with equal variances, this is not always the case.

To know more about variance visit:

https://brainly.com/question/29727198

#SPJ11

The tuition at the local college in 2000, assuming it followed the same growth rate as after 2017, can be estimated to be approximately $4,018. The tuition in 2010, using the same growth rate, would be around $9,049.

To find the tuition in 2000, we need to calculate the value of f(x) when x represents the number of years since 2000. Since the given function models the cost of tuition x years since 2017, we need to determine how many years have passed between 2000 and 2017, which is 17 years. Plugging this value into the function, we get:

f(17) = 15(1.07)^17 ≈ $4,018

Therefore, the estimated tuition in 2000, assuming it followed the same growth rate as after 2017, would be approximately $4,018.

To determine the tuition in 2010, we need to calculate the value of f(x) when x represents the number of years since 2010. Since 2010 is 7 years before 2017, we have:

f(7) = 15(1.07)^7 ≈ $9,049

Hence, the estimated tuition in 2010, using the same growth rate, would be around $9,049. It is important to note that these calculations are based on the assumption that the tuition growth rate before 2017 was consistent with the growth rate after 2017 as provided by the function f(x).

Learn more about functions:

https://brainly.com/question/31062578

#SPJ11

The probability that a randomly chosen test-taker will score below 450 on the SAT is approximately 0.1587, and the probability of scoring above 600 is approximately 0.0228.

To find the probability that a randomly chosen test-taker will score below 450 on the SAT, we need to calculate the corresponding z-value and use a z-table or calculator to find the probability.

Step 1: Calculate the z-value using the formula z = (x - μ) / σ, where x is the score, μ is the mean, and σ is the standard deviation. In this case, x = 450, μ = 500, and σ = 100.

z = (450 - 500) / 100

z = -0.5

Step 2: Use a z-table or calculator to find the cumulative probability associated with the z-value. The cumulative probability represents the area under the standard normal distribution curve up to the given z-value. In this case, we want the area to the left of z = -0.5.

Using a z-table or calculator, the cumulative probability for z = -0.5 is approximately 0.3085.

Step 3: Subtract the cumulative probability from 0.5 to find the probability below 450. Since the standard normal distribution is symmetric, the probability below the z-value is equal to 0.5 minus the cumulative probability.

Probability below 450 = 0.5 - 0.3085

Probability below 450 ≈ 0.1915

Therefore, the probability that a randomly chosen test-taker will score below 450 on the SAT is approximately 0.1915, rounded to four decimal places.

For the second question, we need to find the probability that a randomly chosen test-taker will score above 600 on the SAT.

Step 1: Calculate the z-value using the formula z = (x - μ) / σ. In this case, x = 600, μ = 500, and σ = 100.

z = (600 - 500) / 100

z = 1

Step 2: Use a z-table or calculator to find the cumulative probability associated with the z-value. We want the area to the left of z = 1.

Using a z-table or calculator, the cumulative probability for z = 1 is approximately 0.8413.

Step 3: Subtract the cumulative probability from 1 to find the probability above 600. Since the standard normal distribution is symmetric, the probability above the z-value is equal to 1 minus the cumulative probability.

Probability above 600 = 1 - 0.8413

Probability above 600 ≈ 0.1587

Therefore, the probability that a randomly chosen test-taker will score above 600 on the SAT is approximately 0.1587, rounded to four decimal places.

Learn more about probability here:

https://brainly.com/question/32117953

#SPJ11

The given statement "For a square matrix A, vectors in ColA are orthogonal to vectors in NulA" is TRUE because they are indeed orthogonal to vectors in NulA (the null space of A).

This statement is a direct consequence of the fundamental theorem of linear algebra. When you multiply a matrix A by its corresponding null space vector x, you get the zero vector (Ax = 0).

The dot product of any vector in the column space of A and the null space vector x is also zero, which indicates that these vectors are orthogonal. In other words, the column space and null space are orthogonal subspaces, and their vectors are perpendicular to each other

Learn more about matrix at https://brainly.com/question/17215829

#SPJ11

The line integral along the boundary of the triangle C is 32/15.

To apply Green's , we need to find the curl of the vector field F:

∂F₂/∂x - ∂F₁/∂y = (2xy³) - (y)

The boundary of the triangle C, which consists of three-line segments:

C₁: From (0,0) to (1,0)

C₂: From (1,0) to (1,2)

C₃: From (1,2) to (0,0)

Using the parametric equations for each line segment, we can express the line integral as:

∫C F · dr = ∫∫R (∂F₂/∂x - ∂F₁/∂y) dA

R is the region enclosed by C.

Since R is a triangle with vertices (0,0), (1,0), and (1,2), we can use a double integral to compute the area of R:

∫∫R dA = [tex]\int_0^1 \int_0^{y_2} dx dy[/tex] = 1/2

Now we can apply Green's Theorem:

∫C F · dr = ∫∫R (∂F₂/∂x - ∂F₁/∂y) dA

= ∫∫R (2xy³ - y) dA

= [tex]\int_0^1 \int_0^{y_2} (2xy^3 - y) dx dy[/tex]

= [tex]\int_0^2 (4/5)y^5 - (1/2)y^2 dy[/tex]

= 32/15

For similar questions on triangle

https://brainly.com/question/1058720

#SPJ11

The mean absolute deviation (MAD) for the given set of data is 4.83 contacts.

The mean absolute deviation (MAD) for this set of data is 4.83 contacts. MAD is a measure of how much the data values deviate from the mean on average. It provides information about the variability or dispersion of the data set. In this case, the mean of the data set is calculated by summing up all the values and dividing by the number of values. The absolute deviation for each value is obtained by subtracting the mean from each individual value and taking the absolute value to eliminate any negative signs. These absolute deviations are then averaged to find the MAD.

MAD is a measure of how spread out the data values are from the mean. To calculate the MAD, we first find the mean of the data set, which is the sum of all the values divided by the number of values (68 + 72 + 77 + 64 + 69 + 76) / 6 = 426 / 6 = 71. Next, we find the absolute deviation for each value by subtracting the mean from each individual value and taking the absolute value. The absolute deviations for each value are: 68 - 71 = 3, 72 - 71 = 1, 77 - 71 = 6, 64 - 71 = 7, 69 - 71 = 2, and 76 - 71 = 5. Then, we calculate the mean of these absolute deviations, which is (3 + 1 + 6 + 7 + 2 + 5) / 6 = 24 / 6 = 4. Finally, the MAD is 4.83, rounded to two decimal places.

In simpler terms, the MAD of 4.83 means that, on average, each person's number of contacts deviates from the mean by approximately 4.83 contacts. This indicates that the number of contacts stored in the cell phones of these six individuals is relatively close together, with relatively small variations from the mean value.

Learn more about deviation here:

https://brainly.com/question/31835352

#SPJ11

The cube root of an irrational number is rational must be incorrect. Thus, we can conclude that the cube root of an irrational number is irrational.

To prove using contradiction that the cube root of an irrational number is irrational, we will assume the opposite: the cube root of an irrational number is rational.

Let x be an irrational number, and let y be the cube root of x (i.e., y = ∛x). According to our assumption, y is a rational number. This means that y can be expressed as a fraction p/q, where p and q are integers and q ≠ 0.

Now, we will find the cube of y (y^3) and show that this leads to a contradiction:

y^3 = (p/q)^3 = p^3/q^3

Since y = ∛x, then y^3 = x, which means:

x = p^3/q^3

This implies that x can be expressed as a fraction, which means x is a rational number. However, we initially defined x as an irrational number, so we have a contradiction.

Learn more about irrational number

brainly.com/question/17450097

#SPJ11

We reject the null hypothesis at a significance level of 0.05 and conclude that there is a significant main effect. The obtained value of 5.25 is the value of the test statistic calculated from the data.

In the main effect f(1,12) = 5.25, p < 0.05, the symbol p stands for the probability level or the significance level of the statistical test.

The probability level or significance level is the maximum probability of observing the test statistic or a more extreme value, assuming that the null hypothesis is true. In other words, it represents the probability of making a type I error, that is, rejecting the null hypothesis when it is actually true.

In this case, the value of p is less than 0.05, which means that the probability of obtaining a test statistic as extreme as or more extreme than the observed value, assuming  null hypothesis is true, it is less than 0.05.

Therefore, we reject the null hypothesis at a significance level of 0.05 and conclude that there is a significant main effect. The obtained value of 5.25 is the value of the test statistic calculated from the data.

To know more about null hypothesis refer here:

https://brainly.com/question/28920252

#SPJ11

If in the circle centered at "A", we have NA ≅ PA, MO⊥NA, and RO⊥PA, then the measure of the the segment PO is (d) 3.5 ft.

From the figure, we observe the triangles OAN and OAP are "right-triangles" where one "common-side" is OA and the two "congruent-sides" NA ≅ PA (given), it follows that they are congruent.

⇒ OP ≅ ON;

We know that, the perpendicular drawn from circle's center on chord divides it in two "congruent-segments",

So, We have;

PO ≅ RP, and NO ≅ MN;

​Which means that, PO = RO/2 and ON = MO/2 = 7/2;

Since, OP ≅ ON, we get:

⇒ PO = 7/2 = 3.5,

Therefore, the correct option is (d).

Learn more about Circle here

https://brainly.com/question/31109208

#SPJ1

The function f takes a binary string of length 2, and returns the first bit of that string, which is either 0 or 1.

Therefore, the range of f is {0, 1}.

To know more about  binary string refer here:

https://brainly.com/question/15766517

#SPJ11

The monomers arranged in decreasing order of reactivity towards cationic polymerization are ii > i > iii.

Cationic polymerization is a process where a cationic initiator initiates the polymerization of monomers. In this case, monomer ii is the most reactive towards cationic polymerization, followed by monomer i, and then monomer iii. Monomer ii exhibits the highest reactivity due to its chemical structure, which enables it to readily undergo cationic polymerization. Monomer i has slightly lower reactivity compared to ii, while monomer iii is the least reactive among the three monomers. The arrangement ii > i > iii implies that monomer ii will polymerize the fastest, followed by monomer i, and then monomer iii. This ordering of monomers is based on their relative abilities to undergo cationic polymerization

Learn more about Cationic polymerization here:

https://brainly.com/question/29726222

#SPJ11

Answer:

Step-by-step explanation:

To simplify the fraction (−4c^−1)^3 / (12c^−8), we can apply the rules of exponents.

First, let's simplify the numerator: (-4c^(-1))^3. To raise a power to a power, we multiply the exponents, so we have:

(-4c^(-1))^3 = (-4)^3 * (c^(-1))^3

= -64 * c^(-3)

Now, let's simplify the denominator: 12c^(-8).

Putting the simplified numerator and denominator together, the fraction becomes:

(-64 * c^(-3)) / (12c^(-8))

To simplify further, we can divide the coefficients and subtract the exponents of the variable:

(-64 / 12) * (c^(-3 - (-8)))

= (-64 / 12) * (c^5)

= -16/3 * c^5

So, the fraction (−4c^−1)^3 / (12c^−8) simplifies to (-16/3) * c^5.

The equation of the tangent plane is -x/√2 - y/2 + z = 1/2√2.

To parametrize the surface A, we can use spherical coordinates:

x = cosθ sinϕ

y = sinθ sinϕ / √2

z = cosϕ

where 0 ≤ θ ≤ 2π and 0 ≤ ϕ ≤ π.

Substituting these expressions into the equation of A, we get:

(cosθ sinϕ)^2 + 2(sinθ sinϕ / √2)^2 + cos^2ϕ = 1

Simplifying and rearranging, we get:

sin^2ϕ(cos^2θ + sin^2θ/2) + cos^2ϕ = 1

sin^2ϕ + cos^2ϕ = 1

So this parametrization satisfies the equation of A.

To find the tangent plane at the point (1/√2, 1/2, 0), we need the partial derivatives of x, y, and z with respect to θ and ϕ:

∂x/∂θ = -sinθ sinϕ

∂y/∂θ = cosθ sinϕ / √2

∂z/∂θ = 0

∂x/∂ϕ = cosθ cosϕ

∂y/∂ϕ = sinθ cosϕ / √2

∂z/∂ϕ = -sinϕ

Evaluating these partial derivatives at (1/√2, 1/2, 0), we get:

∂x/∂θ = -1/2

∂y/∂θ = 1/2√2

∂z/∂θ = 0

∂x/∂ϕ = 1/√2

∂y/∂ϕ = 1/2

∂z/∂ϕ = 0

So the normal vector to the tangent plane at (1/√2, 1/2, 0) is given by:

n = (-∂x/∂θ, -∂y/∂θ, ∂x/∂ϕ) × (∂x/∂ϕ, ∂y/∂ϕ, -∂z/∂ϕ)

 = (-1/2, 1/2√2, 0) × (1/√2, 1/2, 0)

 = (-1/2, -1/4√2, 1/2)

So the equation of the tangent plane is:

(-1/2)(x - 1/√2) + (-1/4√2)(y - 1/2) + (1/2)(z - 0) = 0

Simplifying, we get:

-x/√2 - y/2 + z = 1/2√2

Learn more about "tangent ":

https://brainly.com/question/4470346

#SPJ11

[tex]4x^2 + 9y^2 = 36[/tex] is the rectangular form of the curve using parametric equations.

A set of equations known as a parametric equation expresses point coordinates in terms of one or more parameters. In other words, it establishes a connection between one or more variables that specify a point's or an object's location in space. Curves, surfaces, and other geometric shapes are frequently described using parametric equations. Due to their greater versatility in forming complicated shapes than conventional equations, they are excellent for visualising complex shapes and producing computer-generated visuals. In physics, engineering, and mathematics, parametric equations are frequently utilised because they offer a potent tool for modelling and analysing complicated systems.

To eliminate the parameter, we need to solve for the parameter (in this case, theta) in terms of x and y and then substitute that expression into the other equation.

From the first equation, we have cos(theta) = x/3.

From the second equation, we have sin(theta) = y/6.

We can use the Pythagorean identity [tex]sin^2(theta) + cos^2(theta) = 1[/tex]to eliminate theta:

[tex]sin^2(theta) + cos^2(theta) = (y/6)^2 + (x/3)^2 = 1[/tex]

Multiplying both sides by 36:

[tex]4x^2 + 9y^2 = 36[/tex]

This is the rectangular form of the curve using parametric equations.

Learn more about parametric equations here:
https://brainly.com/question/28537985

#SPJ11