Let C be a cylindrical can (including top and bottom lids) with height h and radius r.

(a) Write a multivariable function S(h, r) for the surface area of the can.
(b) Calculate S(3,2), S (3,2), and S,(3,2).
(c) Give a linear approximation for S(2.75, 2.1).

The solution is,  the volume of the solid is (5/6)π.

To use polar coordinates, we need to first express the equations of the surfaces in polar coordinates.

Here, we have,

In polar coordinates, we have x = r cosθ and y = r sinθ. Therefore, the equation x^2 + y^2 = 1 becomes r^2 = 1.

To find the volume of the solid, we can integrate over the region in the xy-plane bounded by the circle r=1. For each point (r,θ) in this region, the corresponding point in 3D space has coordinates (r cosθ, r sinθ, r^2+3)

Thus, the volume of the solid can be expressed as the double integral:

V = ∬R (r^2+3) r dr dθ

where R is the region in the xy-plane bounded by the circle r=1.

We can evaluate this integral using the limits of integration 0 to 2π for θ, and 0 to 1 for r:

V = ∫₀^¹ ∫₀^(2π) (r^3 + 3r) dθ dr

= ∫₀^¹ [(r^3/3 + 3rθ)]₀^(2π) dr

= ∫₀^¹ (2πr^3/3 + 6πr) dr

= 2π[(1/12) + (1/2)]

= 2π(5/12)

= (5/6)π

Therefore, the volume of the solid is (5/6)π.

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complete question:

Use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations z=x2+y2+3,z=0,x2+y2=1

.

As per the given information, the value of the house is 165,116

The current value of the house is = 213,000

The percentage with which the value of the house has decreased = 29%

Let the price of the house when it was purchased be = x

The value of the house is decreased by 29% = 0.29

It is required to understand that an exponential function is involved whenever we discuss a rise or decline.

Thus,

According to the question,

x + 29% of x = 213000

Solving,

x + 0.29x = 213000

1.29x = 213000

x = 213000/1.29

x = 165,116

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The concentration of Br- in the seawater sample is approximately 66 ppm. To find the concentration of Br- in seawater in parts per million (ppm), we will first convert the given concentration from moles per liter (M) to grams per liter (g/L). Then, we will convert it to parts per million.

Given:
- Concentration of Br- in seawater = 8.3 * 10^(-4) M
- Mass of 1 liter of seawater = 1.0 kg (1000 g)

Step 1: Convert the concentration from M to g/L.
We will use the molar mass of Br-, which is approximately 79.9 g/mol.

(8.3 * 10^(-4) mol/L) * (79.9 g/mol) = 0.06637 g/L

Step 2: Convert the concentration from g/L to ppm.
1 ppm is equivalent to 1 mg of solute per kg of solution.

(0.06637 g/L) * (1000 mg/g) = 66.37 mg/L

Since the mass of 1 L of seawater is 1.0 kg, the concentration in ppm is:

66.37 mg/kg = 66.37 ppm

So, the concentration of Br- in the seawater sample is approximately 66 ppm.

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The area of the triangle with sides a = 7, b = 5, and c = 5 is approximately 12.015 square units, rounded to three decimal places.

To find the area of a triangle with given side lengths a = 7, b = 5, and c = 5, we can use Heron's formula. Heron's formula states that the area (A) of a triangle can be calculated using the semi-perimeter (s) and the side lengths:

1. Calculate the semi-perimeter: s = (a + b + c) / 2
  s = (7 + 5 + 5) / 2
  s = 17 / 2
  s = 8.5

2. Apply Heron's formula: A = √(s * (s - a) * (s - b) * (s - c))
  A = √(8.5 * (8.5 - 7) * (8.5 - 5) * (8.5 - 5))
  A = √(8.5 * 1.5 * 3.5 * 3.5)

3. Calculate the area:
  A ≈ √(144.375)
  A ≈ 12.015

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We found that (a) the partial derivative of f(x, y) with respect to x evaluated at x=5 is 90y^3, and (b) the partial derivative of f(x, y) with respect to y evaluated at y=1 is 27x^2.

We have the function f(x, y) = 9x^2y^3, and we need to find (a) the partial derivative with respect to x and then evaluate it at 5, and (b) the partial derivative with respect to y and then evaluate it at 1.

(a) To find the partial derivative of f(x, y) with respect to x, we treat y as a constant and differentiate f(x, y) with respect to x:

∂f(x, y)/∂x = d(9x^2y^3)/dx = 18xy^3

Now, we need to evaluate this derivative at x=5:

∂f(5, y)/∂x = 18(5)y^3 = 90y^3

(b) To find the partial derivative of f(x, y) with respect to y, we treat x as a constant and differentiate f(x, y) with respect to y:

∂f(x, y)/∂y = d(9x^2y^3)/dy = 27x^2y^2

Now, we need to evaluate this derivative at y=1:

∂f(x, 1)/∂y = 27x^2(1)^2 = 27x^2

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Answer: the answer would be 6a.

Step-by-step explanation:

since only 3 can be multiplied, we end up with 6a

Answer:

3a × 2 = 6a

Step-by-step explanation:

3a multiplied by 2 is equal to 6a

Answer: To fill the whole bucket, it will take 64 seconds so the remaining time is 40 seconds

Step-by-step explanation: As we are given 3/8 th part of the bucket is filled in 24 seconds. So by simply applying the unitary method we can say -

3/8 th part -----> 24 seconds

To fill the whole bucket multiply both sides by 8/3 in order to make the 1 unit of the bucket on the L.H.S, we get

1 bucket ----> 64 seconds.

The remaining times as it already passes 24 seconds and 3/8 th part of the bucket is filled, 64-24 seconds i.e 40 seconds is remaining in which bucket is full.

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a) The equivalent double integral with the order of integration reversed is ∫∫D f(x, y) dy dx.

b)  To reverse the order of integration, we need to sketch the region of integration D and rewrite the original double integral with the opposite order of integration.

Since the provided information is incomplete, it is not possible to determine the specific region of integration or the function f(x, y) involved.

However, in general, reversing the order of integration involves swapping the order of integration limits and rewriting the integrand accordingly. This allows for the evaluation of the integral in a different order, which can be useful in certain cases for simplifying calculations or applying different integration techniques.

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The author would have sold  365396 books over the first 20 months.

The author sold 22000 books in the first month, and the sales declined by 7% each month.

This means that the number of books sold in each subsequent month is 93% of the number sold in the previous month (since 100% - 7% = 93%). Therefore, the number of books sold in the second month is:

0.93×22000

= 20460

The number of books sold in the third month is:

0.93 × 20460 = 19007

To find the total number of books sold over the first 20 months, we can use the formula for the sum of a geometric series:

S = a(1 - rⁿ) / (1 - r)

Substituting the values, we get:

S = 22000(1 - 0.93²⁰) / (1 - 0.93)

S = 22000(0.766)/0.07

S=240743

Therefore, the author would have sold 365396 books over the first 20 months.

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The margin of error for the survey is 0.053.

To calculate the margin of error, we need to use the formula:

Margin of Error = Critical Value x Standard Error

We can start by finding the critical value using a z-table. For a 95% confidence interval, the critical value is 1.96.

Next, we need to calculate the standard error. The formula for the standard error of a proportion is:

Standard Error = [tex]\sqrt{(p*(1-p))/n}[/tex]

where p is the sample proportion (0.372) and n is the sample size (500).

Standard Error = [tex]\sqrt{(0.372*(1-0.372))/500}[/tex] = 0.027

Now that we have the critical value and the standard error, we can calculate the margin of error:

Margin of Error = 1.96 x 0.027 = 0.053

Therefore, the margin of error for the survey is 0.053 or approximately 5.3%. This means that we can be 95% confident that the true proportion of people who watched the Avalanche game in the entire state of Colorado is within 5.3% of the sample proportion of 37.2%.

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The surface area of the rectangular prism is 588 square yards. The total region or area covered by all the faces of a rectangular prism is defined as the surface area of a rectangular prism.

It is a three-dimensional shape. It has six faces, and all the faces are rectangular-shaped. Therefore, both the bases of a rectangular prism must also be rectangles.

- Face 1: 9 yards long and 6 yards high, so its area is 9 x 6 = 54 square yards.

- Face 2: 9 yards long and 6 yards high, so its area is 9 x 6 = 54 square yards.

- Face 3: 16 yards wide and 6 yards high, so its area is 16 x 6 = 96 square yards.

- Face 4: 16 yards wide and 6 yards high, so its area is 16 x 6 = 96 square yards.

- Face 5: 9 yards long and 16 yards wide, so its area is 9 x 16 = 144 square yards.

- Face 6: 9 yards long and 16 yards wide, so its area is 9 x 16 = 144 square yards.

The surface area =  54 + 54 + 96 + 96 + 144 + 144

= 588 square yards.

Surface area of the rectangular prism is 588 square yards.

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The cube that is a unit cube is the one that is 15 centimeters long, 15 centimeters wide, and 15 centimeters high because all its edges are of the same length and measure 1 unit.

A unit cube is a cube with edges that are all of equal length and measure 1 unit. Therefore, out of the given options, the cube that is a unit cube is the one that is 15 centimeters long, 15 centimeters wide, and 15 centimeters high. This is because all its edges are of the same length and measure 1 unit, which is 15 centimeters. The other cubes given in the options are not unit cubes because their edges are not of the same length or do not measure 1 unit. For example, the cube that is 7 feet long, 7 feet tall, and 7 feet high has edges that are 7 feet long, which is not the same length as its height and width. Similarly, the cube that is 5 inches long, 5 inches wide, and 5 inches high has edges that measure 5 inches, which is not 1 unit.

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The area of the regular nonagon is 1,582.5 ft².

The area of the nonagon is calculated by applying the following formula as shown below;

A = (9/4) a² (cos 20/sin 20)

where;

The given length of the nonagon = 16 ft.

The area of the nonagon is calculated as follows;

A = (9/4) a² (cos 20/sin 20)

A = (9/4) (16)² (cos 20/sin 20)

A = 1,582.5 ft²

Thus, the area of the regular nonagon is calculated by applying the formula given.

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We used the given measurements of mean and standard deviation to determine the z-score of the value we were interested in, which allowed us to look up the corresponding probability in the standard normal distribution table or use a calculator.

The first step is to standardize the value of 61.5 inches using the formula z = (x - mu) / sigma, where x is the value we want to find the probability for, mu is the mean, and sigma is the standard deviation.

z = (61.5 - 56.9) / 4 = 1.15

Next, we look up the probability corresponding to this z-value in the standard normal distribution table or use a calculator. The probability that a randomly chosen child has a height less than 61.5 inches is the same as the probability that a standard normal variable is less than 1.15.

Using a table or calculator, we find that this probability is approximately 0.8749.

Therefore, the probability that a randomly chosen child has a height of less than 61.5 inches is approximately 0.8749 or 87.49%.

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The side lengths are given as follows:

The side lengths for this problem are obtained considering the triangle midsegment theorem, which states that the midsegment of the triangle divided the laterals of the triangle into two segments of equal length.

The congruent segments(segments of equal length) are given as follows:

Hence the lengths are given as follows:

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4000 cm^3 is the total volume of this octahedron

The volume of a pyramid is given by the formula:

V = (1/3) * base area * height

Since the base is a square with sides of 20 cm, its area is:

base area = 20^2 = 400 cm^2

The height of each pyramid is half the overall height of the octahedron, which is 30 cm. So the height of each pyramid is:

height = 30/2 = 15 cm

Now we can find the volume of one pyramid:

V = (1/3) * base area * height

V = (1/3) * 400 cm^2 * 15 cm

V = 2000 cm^3

Therefore, the total volume of the octahedron is twice the volume of one pyramid:

V_total = 2 * V

V_total = 2 * 2000 cm^3

V_total = 4000 cm^3

So the volume of the octahedron is 4000 cubic centimeters.

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A. When an instrument is weak, then the TSLS estimator is biased (even in large samples), and TSLS t-statistics and confidence intervals are unreliable.

B. When an instrument is irrelevant, the TSLS estimator is consistent but the large-sample distribution of TSLS estimator is not that of a normal random variable, but rather the distribution of the product of two normal random variables.
C. When an instrument is irrelevant, the consistency of the TSLS estimator breaks down. The large-sample distribution of the TSLS estimator is not that of a normal random variable, but rather the distribution of a ratio of two normal random variables.

A. It is not possible to statistically test the exogeneity of the instruments when the coefficients are overidentified; i.e., when m>k.
B. It is possible to statistically test the exogeneity of instruments when the coefficients are exactly identified; i.e., when m=k.
C. It is possible to statistically test the exogeneity of the instruments when the coefficients are overidentified; i.e., when m>k.
D. It is not possible to statistically test the exogeneity of instruments when the coefficients are exactly identified; i.e., when m=k.

In large samples, if the instruments are not weak and the errors are homoskedastic, then, under the null hypothesis that the instruments are exogenous, the J-statistic follows a distribution with degrees of overidentification, which are also the degrees of freedom.
The consequences of estimating the two stage least square (TSLS) coefficients using either a weak or an irrelevant instrument include:

A. When an instrument is weak, the TSLS estimator is biased (even in large samples), and TSLS t-statistics and confidence intervals are unreliable.
C. When an instrument is irrelevant, the consistency of the TSLS estimator breaks down. The large-sample distribution of the TSLS estimator is not that of a normal random variable, but rather the distribution of a ratio of two normal random variables.

Regarding the possibility of statistically testing the exogeneity of instruments:

B. It is possible to statistically test the exogeneity of instruments when the coefficients are exactly identified; i.e., when m=k.
C. It is possible to statistically test the exogeneity of the instruments when the coefficients are overidentified; i.e., when m>k.

In large samples, if the instruments are not weak and the errors are homoskedastic, then, under the null hypothesis that the instruments are exogenous, the J-statistic follows a distribution with degrees of overidentification, which are also the degrees of freedom.

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Rewrite the statements in if-then form: If I catch the 8:05 bus, then I will be on time for work.

To write this statement in if-then form, we start with the "if" part of the statement, which is the condition that needs to be satisfied for the conclusion to follow. In this case, the condition is "catching the 8:05 bus". The "then" part of the statement is the conclusion that follows if the condition is satisfied, which is "being on time for work".

Therefore, the statement "Catching the 8:05 bus is a sufficient condition for my being on time for work" can be written as "If I catch the 8:05 bus, then I will be on time for work" in if-then form. This form of expressing the statement makes it clear what the condition and conclusion are and how they are related to each other.

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Answer: Missing the statments

Step-by-step explanation:

To determine if two pools are similar, the pool company needs to check if the corresponding sides are proportional and the corresponding angles are equal. If these conditions are met, then the two pools are considered similar in geometry.

In mathematics, specifically in geometry, similar figures are figures that have the same shape but may differ in size. To determine if pool ABCD is similar to pool EFGH, the pool company needs to check the proportionality of corresponding sides and the equality of corresponding angles.

For instance, if the length and width of pool ABCD is twice that of pool EFGH, and all the corresponding angles are equal, then the two pools are similar. It's crucial to note that all corresponding sides should be in proportion and all corresponding angles should be equal for the figures to be considered similar.

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The probability of Zack getting exactly 4 chocolate pastries is approximately 0.275, or 27.5%. To find the probability that Zack buys four chocolate pastries out of the eight he selects, we'll use the hypergeometric distribution. In this case, there are 20 pastries in total, with 12 being chocolate and 8 being non-chocolate.

Zack is buying 8 pastries, and we want to know the probability that exactly 4 of them are chocolate.

The hypergeometric probability formula is: P(X = k) = (C(K, k) * C(N - K, n - k)) / C(N, n)

Here, N = 20 (total pastries), K = 12 (chocolate pastries), n = 8 (pastries Zack buys), and k = 4 (desired number of chocolate pastries).

Plugging these values into the formula, we get:

P(X = 4) = (C(12, 4) * C(20 - 12, 8 - 4)) / C(20, 8)

P(X = 4) = (C(12, 4) * C(8, 4)) / C(20, 8)

Calculating the combinations, we find:

P(X = 4) = (495 * 70) / 125,970

P(X = 4) = 34,650 / 125,970

P(X = 4) ≈ 0.275

The probability of Zack getting exactly 4 chocolate pastries is approximately 0.275, or 27.5%.

However, this answer is not among the provided options (0.09, 0.50, 0.60). Double-check the question and options to ensure the values are correct.

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The given sequence {0,1,0,0,1,0,0,0,1} converges to 1.

A convergent sequence is a sequence of numbers where the sequence approaches on something, that is the the limit exists.

It is finite.

Divergent sequence are sequences which does not contain the limit. The limit will be ±∞.

Given sequence is {0,1,0,0,1,0,0,0,1}.

Here there are 9 numbers in the sequence.

This sequence isfinite and approaches to 1.

So the limit is 1 and the sequence converges.

If the sequence was {0,1,0,0,1,0,0,0,1, ......}, the sequence is continuing.

The terms are either 0 or 1.

But it does not end.

So the sequence diverges.

Hence the given sequence converges to 1.

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Answer:

see below, answers are underlined

Step-by-step explanation:

1. Vertical angles are 2 angles that are on opposite sides of each other and are the same value.  So, 2 angles that are considered vertical angles would be MPN (5y) and LPO (95)

2. Adjacent angles are 2 angles that are right next to each other on the same line and when added, they equal 180.  So, 2 angles that are adjacent angles are MPL (5x) and MPN (5y)

3. Using what we know about adjacent and vertical angles, we can solve to find x in 2 different ways:

--> 5x=85 (vertical angles), x=17

--> 5x+95=180 (adjacent angles), x=17

4. Using what we know about adjacent and vertical angles, we can solve to find y also in 2 different ways:

-->5y=95 (vertical angles), y=19

-->5y+85=180 (adjacent angles), y=19

Hope this helps! :)

The solution to the system of equations 2x + y = 10 and x - y = 4 is x = 14/3 and y = 2/3.

Given the system of equation in the question;

2x + y = 10

x - y = 4

We can use substitution to solve this system of equations.

From the second equation, we can write:

x = y + 4

Now we can substitute this value of x into the first equation:

2(y + 4) + y = 10

Simplifying and solving for y, we get:

3y + 8 = 10

3y = 2

y = 2/3

Now that we know the value of y, we can substitute it back into the equation x = y + 4 that we obtained earlier:

x - y = 4

Plug in y = 2/3

x - 2/3 = 4

x = 2/3 + 4

x = 14/3

Therefore, the solution is (2/3, 14/3).

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Question 1: State where the function f(z) = 3 + 2i - z^3 + z is analytic.

Answer: The function f(z) = 3 + 2i - z^3 + z is a polynomial function, and polynomial functions are analytic everywhere in the complex plane. Therefore, this function is analytic for all complex numbers z.

Question 2: State where the function f(z) = e^z is analytic.

Answer: The function f(z) = e^z is an exponential function, and exponential functions are also analytic everywhere in the complex plane. Therefore, this function is analytic for all complex numbers z.

Question 3: State where the function 2 + z f(z) = (2 + 1 - 2i)(2 + 1 + 2i) is analytic.

Answer: The function 2 + z f(z) is a rational polynomial, and it is analytic everywhere except at the poles. In this case, the poles are -1 + 2i and -1 - 2i.

Question 4: State where the function f(z) = Im(z) + |z| + Re(z) is analytic.

Answer: The function f(z) = Im(z) + |z| + Re(z) involves the modulus (absolute value) of z, which is not an analytic function. Therefore, this function is not analytic anywhere in the complex plane.

Question 5: State where the function f(z) = 1/(22 + 16) is analytic.

Answer: The function f(z) = 1/(22 + 16) is a constant function, and constant functions are analytic everywhere in the complex plane. Therefore, this function is analytic for all complex numbers z.

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Answer:

In a 30°-60°-90° right triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is √3 times the length of the shorter leg.

The length of the television is 15√3 inches, and the width of the television is 15 inches.

From Algebra, the earning points she have if she completes 4 levels is equals to the 112 points. The unit rate and equivalent ratio are 28 points per level.

Biannca plays a video game quest, where players earn the same amount of points for completing a level. The above figure represents the levels and earning amount of Biannca. Number of levels completed by Biannca = 2

Earning points of Biannca after completing two level of video game = 56 points

We have to determine the earing points by her after completing the 4 levels of game. So, the earing points of Biannca in each level of game = two levels total earning points divided by number of levels [tex]= \frac{56}{2} [/tex]

Multipling the numentor and denominator by 1/2, so that

= 28 points/level

which is an equivalent ratio and the unit rate of Biannca earing. So, using multiplcation operation for obtaining the earning points on completing 4 levels of game equals to number of levels × earing points for each level. The required earing points = 4 × 28 points = 112 points. Hence, required value is 112 points.

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Complete question:

The above figure complete the question. In the video game unicorn quest, players earn the same amount of points for completing a level Biannca completed 2 levels and earned 56 points how many points will she have if she completes 4 levels what is the equivalent and unit rate?

Answer:

23

Step-by-step explanation:

23.

Given fraction is 1218.⟹ 1218 = 6×26×3.= 23.

So the lowest form of 1218 = 23

2:3

both 12 and 18 is divided by 6

The correct answer is $4.50 option (c).

To calculate the cost of 2.6 lb of oatmeal at $1.73/lb, we simply multiply the weight of the oatmeal by the cost per pound.

2.6 lb × $1.73/lb = $4.498

Rounding to two decimal places, the cost of 2.6 lb of oatmeal is $4.50.

Therefore, the correct response is $4.50.

o find the cost of 2.6 lb of oatmeal, we can multiply the price per pound by the number of pounds. So:

Cost of oatmeal = price per pound x number of pounds

= $1.73/lb  × 2.6 lb

= $4.498

Rounding this to two decimal places gives us $4.50. Therefore, the correct answer is $4.50.

To calculate the cost of 2.6 lb of oatmeal at a price of $1.73/lb, we can use the formula:

Cost = Price per unit  × Quantity

In this case, the price per unit is $1.73/lb and the quantity is 2.6 lb. So the cost would be:

Cost = $1.73/lb  × 2.6 lb = $4.498

Rounding to the nearest cent, the cost of 2.6 lb of oatmeal would be $4.50. Therefore, the correct response is $4.50.

To calculate the cost of 2.6 lb of oatmeal at $1.73/lb, we need to multiply the weight (in pounds) by the price per pound.

So, the cost would be:

2.6 lb  × $1.73/lb = $4.498

Rounding this to two decimal places gives us $4.50, which is one of the options provided. Therefore, the correct answer is $4.50.

Complete Question:

oatmeal costs $1.73/lb. how much would 2.6 lb of oatmeal cost? responses

a. $1.50  

b. $4.48  

c. $4.50  

d. $4.58

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Answer:

Step-by-step explanation:

*if this is wrong its because they didnt line the square up properly*

But the answer os 28 because, what you will need to do is multiply side and top

The side has 4 squares in the box

and the top has 7

so multipy 7 x 4 or 4 x 7

7 x 4 = 28

Two power series solutions of the differential equation (x^2+1)y''-6y=0 about x=0 are y1=x^2-3x^4/10+O(x^6) and y2=1-7x^2/6+O(x^4).

The given differential equation can be written as:

y''-6(x^2+1)^(-1)y=0 ...(1)

Let us assume the power series solutions of (1) about x=0 as:

y=∑_(n=0)^∞▒〖a_n x^n 〗Differentiating y with respect to x, we get:

y'=∑_(n=1)^∞▒na_n x^(n-1)

y''=∑_(n=2)^∞▒n(n-1)a_n x^(n-2)

Substituting these in (1), we get:

∑_(n=2)^∞▒n(n-1)a_n x^(n-2) - 6∑_(n=0)^∞▒a_n (x^2+1)^(-1) x^n=0

Multiplying throughout by x^2, we get:

∑_(n=4)^∞▒n(n-1)a_n x^(n-2) - 6∑_(n=2)^∞▒a_n (x^2+1)^(-1) x^(n)=0

omparing coefficients of like powers of x, we get the following recurrence relations:

a_2=0, a_3=0, a_4=3a_0/5, a_5=0, a_6=-(21a_0+5a_4)/70, a_7=0, a_8=(429a_0+245a_4)/1575, ...

Thus, we get the power series solution y1:

y1=a_0 + 0.x + 0.x^2 + (3a_0/5).x^3 - 0.x^4 - ((21a_0+5(3a_0/5))/70).x^5 + ...

Simplifying the above expression, we get:

y1=x^2-3x^4/10+O(x^6)

Similarly, we can solve for the second power series solution by using a different initial condition. We assume the second solution in the form of:

y=∑_(n=0)^∞▒b_n x^n

and substitute it in (1). On solving the recurrence relations, we get the power series solution:

y2=1-7x^2/6+O(x^4)

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The two power series solutions of the given differential equation are

y1(x) = a_3 * x^3 + a_4 * x^4 + ...

y2(x) = x^3 + a_4 * x^4 + ...

To find two power series solutions of the given differential equation (x^2 + 1)y'' - 6y = 0 about the ordinary point x = 0, we can assume a power series solution of the form:

y(x) = Σ(a_n * x^n)

where a_n are coefficients to be determined and Σ represents the sum over the values of n.

Let's differentiate y(x) twice to find the values of y''(x):

y'(x) = Σ(n * a_n * x^(n-1))

y''(x) = Σ(n * (n-1) * a_n * x^(n-2))

Now, we substitute y(x), y'(x), and y''(x) into the differential equation:

(x^2 + 1) * Σ(n * (n-1) * a_n * x^(n-2)) - 6 * Σ(a_n * x^n) = 0

Expanding and rearranging the terms, we get:

Σ(n * (n-1) * a_n * x^n + a_n * x^(n+2)) - 6 * Σ(a_n * x^n) = 0

Grouping the terms by their powers of x, we have:

Σ((n * (n-1) * a_n - 6 * a_n) * x^n) + Σ(a_n * x^(n+2)) = 0

Now, we equate the coefficients of like powers of x to zero to obtain a recursion relation for the coefficients a_n.

For n = 0:

(n * (n-1) * a_n - 6 * a_n) = 0

(-6 * a_0) = 0

a_0 = 0

For n = 1:

(n * (n-1) * a_n - 6 * a_n) = 0

(1 * 0 * a_1 - 6 * a_1) = 0

-5 * a_1 = 0

a_1 = 0

For n = 2:

(n * (n-1) * a_n - 6 * a_n) = 0

(2 * 1 * a_2 - 6 * a_2) = 0

-4 * a_2 = 0

a_2 = 0

For n = 3:

(n * (n-1) * a_n - 6 * a_n) = 0

(3 * 2 * a_3 - 6 * a_3) = 0

0 * a_3 = 0

a_3 can be any value

From the recursion relation, we see that a_0 = a_1 = a_2 = 0, indicating that the terms of y(x) involving these coefficients will vanish.

Therefore, we can write the first power series solution y1(x) as:

y1(x) = a_3 * x^3 + a_4 * x^4 + ...

For the second power series solution, we can choose a different value for a_3 to obtain a linearly independent solution. Let's choose a_3 = 1:

y2(x) = x^3 + a_4 * x^4 + ...

These are the two power series solutions of the given differential equation about the ordinary point x = 0.

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