find the radius of convergence, r, of the series. [infinity] xn 3n − 1 n = 1

Answer:

\( - 4x + 9 = 21 \\ - 4x = 21 - 9 \\ = 12 \\ x = \frac{12}{ - 4} \\ x = - 3\)

The dimensions of the can that minimize the cost of the material are approximately:

Radius (r) ≈ 6.324 cm

Height (h) ≈ 3.984 cm

To determine the dimensions of the can that minimize the cost of the material, we need to find the dimensions that minimize the surface area of the can.

Let's assume the can has a height of h and a radius of r.

The volume of a cylinder is given by:

V = πr²h

Given that the can should hold 500 cubic centimeters of liquid, we have:

500 = πr²h

We want to minimize the cost, which depends on the surface area of the can.

The surface area of the can is the sum of the areas of the top, bottom, and side surfaces.

The cost of the top and bottom surfaces is twice as expensive per square centimeter as the sides.

Let's assume the cost per square centimeter of the sides is c, so the cost per square centimeter of the top and bottom surfaces is 2c.

The surface area of the sides of the cylinder is given by:

A_sides = 2πrh

The surface area of the top and bottom surfaces (each) is given by:

A_top_bottom = 2πr²

The total surface area (cost) is given by:

Cost = 2(2c)A_top_bottom + cA_sides

= 4cπr² + 2c(2πrh)

= 4cπr² + 4cπrh

To minimize the cost, we need to minimize the surface area. To do this, we can express the surface area in terms of a single variable, such as the radius (r) or the height (h).

From the volume equation, we have:

h = 500 / (πr²)

Substituting this value of h into the surface area equation, we get:

Cost = 4cπr² + 4cπr(500 / (πr²))

= 4cπr² + 2000c/r

Now, we can take the derivative of the cost function with respect to r, set it equal to zero, and solve for r to find the critical points:

dCost/dr = 8cπr - 2000c/r² = 0

8cπr = 2000c/r²

8πr³ = 2000

r³ = 250 / π

r ≈ 6.324

Now, we can substitute this value of r back into the equation for h:

h = 500 / (π(6.324)²)

≈ 3.984

So, the dimensions of the can that minimize the cost of the material are approximately:

Radius (r) ≈ 6.324 cm

Height (h) ≈ 3.984 cm

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The domain and the range for the relations are given as follows:

For items 13 and 16, the functions are continuous, hence it is over the real numbers, and both the domain and range are all real values.

For items 14 and 15, the functions are discrete, hence it is over the integer numbers, and the domain is composed by the values assumed by x and the range is composed by the values assumed by y.

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1) The wavelength of A is equal to 6.98 × \(10^{-8}\)meters

2) The wavelength of B is equal to 3.45 × \(10^{-11}\) meters

Since we know that the wavelength = speed of light / frequency

The speed of light is 3.00 × \(10^8\) meters per second.

For light sample A with a frequency of 4.30 × 10^15 Hz can be calculated as;

wavelength of A = (3.00 ×  \(10^8\)  m/s) / (4.30 × 10^15 Hz)

wavelength of A = 6.98 ×  \(10^{-8}\) meters

For light sample B with a frequency of 8.70 × \(10^18\) Hz can be calculated as;

wavelength of B = (3.00 ×  \(10^8\) m/s) / (8.70 ×\(10^18\) Hz)

wavelength of B = 3.45 ×  \(10^{-11}\)  meters

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Answer:

Answer in the attachment

Step-by-step explanation:

Cut off both semicircles at the top and bottom.

Answer:

0.00000797

Step-by-step explanation:

Answer:

Adult tickets sold = 240

Step-by-step explanation:

If student tickets = x,

Adult tickets = 3x + 6

Since the total amount of tickets sold was 318, the equation would be:

x + 3x + 6 = 318

Simplify this:

4x + 6 = 318

4x = 312

x = 78

Student tickets = 78

and so:

Adult tickets = (3 x 78) + 6 = 240

If t = [0 1 1 0] is a transformation matrix, then the expression that correctly applies t to v is t × v.

Explanation:

In linear algebra, a transformation is a function that maps a vector space to itself. A transformation matrix is a matrix that represents a linear transformation.

The transformation matrix t = [0 1 1 0] is a 2x2 matrix that represents a transformation that switches the coordinates of a vector.

It can also be written as:

t = [0 1]  [1 0]

To apply this transformation to a vector v = [x y], we can use matrix multiplication. The result is a new vector v = [x y]`:

v = t × v

v = [0 1 1 0] × [x y]

v = [y x]

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Answer:

n<2

Step-by-step explanation:

-2n+5>1

-2n>1-5

(-2n>-4)divide by -2

Answer:

n<2

Step-by-step explanation:

Solve the Inequality for n

Isolate the variable by dividing each side by factors that don't contain the variable.

-2n+5>1

-2n>1-5

(-2n>-4)divide by -2

Interval Notation: (−∞,2)

Feasible region - the set of all possible solutions that satisfy all constraints.Binding constraint - a constraint that is met exactly at the optimal solution.Sensitivity analysis - the process of analyzing how changes in the parameters of a linear programming problem affect the optimal solution

1. Feasible region: A set of values for decision variables that satisfy all constraints in an optimization problem.
2. Binding constraint: A constraint that holds as an equality at the optimal solution, affecting the optimal value of the objective function.
3. Sensitivity analysis: A technique used to determine how different values of an independent variable affect a dependent variable under given constraints.
4. Constraint: A condition or limitation imposed on decision variables in an optimization problem.
5. Decision variables: The variables that represent the choices available to the decision-maker in an optimization problem.
6. Objective function: The mathematical expression representing the goal of an optimization problem, which is to be minimized or maximized.
7. Shadow price: The change in the objective function value due to a one-unit increase in the availability of a limited resource, holding other factors constant.

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Answer:

parallelogram

Step-by-step explanation:

Answer:

parallelogram

Step-by-step explanation:

did the test

A dealer gains 7% by selling a sofa set for Rs.3852. If the dealer sells the sofa set for Rs. 4050, his gain percent will be 12.5%.

The selling price of the sofa set at which the dealer gains 7% is Rs. 3852.

Let's first calculate the cost price of the sofa set:

Gain% = (Profit/Cost Price) x 100

7% = (Profit/Cost Price) x 100

Profit = 0.07 x Cost Price

Selling Price = Cost Price + Profit

3852 = Cost Price + 0.07 x Cost Price

3852 = 1.07 x Cost Price

Cost Price = 3600

Now, we need to find the gain percent when the selling price is Rs. 4050:

Profit = Selling Price - Cost Price

Profit = 4050 - 3600

Profit = 450

Gain% = (Profit/Cost Price) x 100

Gain% = (450/3600) x 100

Gain% = 12.5%

Therefore, if the dealer sells the sofa set for Rs. 4050, his gain percent will be 12.5%.

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Answer:

y = 9

Step-by-step explanation:

it's perpendicular to x = 8 which is a vertical line, so our line will be horizontal line with slope 0 and it goes through (6,9) so at x = 6, y =9

but since the line is horizontal, (slope = 0) it must have the same y-value at all x- points so,

the equation is y = 9

The equation is :

Solution:

If two lines are perpendicular, their slopes are opposite reciprocals of each other.

Since we have a vertical line, its slope is undefined. Now, remember the definition of perpendicular lines. They intersect at a right angle.

And the line that is perpendicular to a vertical line is a horizontal line, which has a slope of 0.

Lines with 0 slope have the appearance of y = b where b is the y intercept.

Now there are two ways to find the y-intercept, which are :

Allow me to demonstrate the first one.

Method 1. Point slope

Point slope is \(\bf{y-y_1=m(x-x_1)}\).

Plugging in the numbers \(\bf{y-9=0(x-6)}\)

Simplifying \(\bf{y-9=0}\)

Simplifying further

\(\bf{y=9}\)

In order to know if a triangle is a right triangle on a coordinate plane, you can find the lengths of all three sides of the triangle using the distance formula and apply the Pythagorean theorem.

Find the lengths of the three sides of the triangle using the distance formula.

Once you have the lengths of the sides, check if any of the three sides satisfy the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In other words, if a² + b² = c², where c is the longest side, then the triangle is a right triangle.

If one of the sides satisfies the Pythagorean theorem, then the triangle is a right triangle.

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The  line with a slope of –5 that passes through the points (7,–2) and (8,q) has q as -7.

The equation of a line can be described as follows:

y = mx + b

where

Therefore, the line passes through  (7, –2) and (8, q) with a slope of - 5.

Hence, let's find q.

y = - 5x + b

Let's find y-intercept

-2 = -5(7) + b

b = -2  + 35

b = 33

Therefore,

y = -5x + 33

Hence, let's find q.

y = -5(8) + 33

q = -40 + 33

q = -7

Therefore, q = - 7

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Answer:

The expression is 1.07d.

Answer:

\(Solution\\diameter(d)=14cm\\Now, \\Circumference=\pi d\\ =3.14*14cm\\ =43.96cm\)

Step-by-step explanation:

Hope it will help you a lot.

Answer:

Circumference = 43.98

Step-by-step explanation:

Circumference = 2 x pi x r

r= diameter/2 = 14cm/2= 7

Circumference = 2 x 3.14 x 7

Circumference = 43.98cm

Answer:

63 cm²

Step-by-step explanation:

If you put two triangles together, you can form a parallelogram. Now, what is the formula to find the area of a parallelogram? Simple!

Length × width = area.

So, we can take half of this area to find the area of the triangle we originally started with.

Formula:

base × height ÷ 2 = area

21 × 6 = 126

126 ÷ 2 = 63

Therefore, 63 would be the correct answer.

As the marginal cost and average cost are both equal to $8 at x = 5, we can conclude that the marginal cost and average cost are equal at this production level.

To find the production level at which the average cost is a minimum, we need to first find the average cost function. The average cost function is given by:

\(AC(x) = C(x)/x\)

Substituting C(x) from the given equation, we get:

\(AC(x) = (x^3 - 10x^2 + 33x)/x\)

Simplifying this, we get:

\(AC(x) = x^2 - 10x + 33\)

To find the production level at which the average cost is a minimum, we need to find the value of x that minimizes the average cost function. We can do this by taking the derivative of the average cost function and setting it equal to zero:

\(d/dx (x^2 - 10x + 33) = 2x - 10 = 0\)

Solving for x, we get:

x = 5

Therefore, the production level at which the average cost is a minimum is x = 5.

To show that the marginal cost and average cost are equal at this production level, we need to first find the marginal cost function. The marginal cost function is given by the derivative of the cost function:

\(MC(x) = d/dx (x^3 - 10x^2 + 33x) = 3x^2 - 20x + 33\)

Substituting x = 5, we get:

\(MC(5) = 3(5)^2 - 20(5) + 33 = 8\)

Therefore, the marginal cost at x = 5 is $8.

To find the average cost at x = 5, we can substitute x = 5 into the average cost function:

\(AC(5) = 5^2 - 10(5) + 33 = 8\)

Therefore, the average cost at x = 5 is also $8.

Since the marginal cost and average cost are both equal to $8 at x = 5, we can conclude that the marginal cost and average cost are equal at this production level.

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Answer:

hundred thousands

Step-by-step explanation:

You would move it to the left one place, so it would be the hundred thousands place.

Answer: The answer is hundred thousands place.

Step-by-step explanation: The question is wanting you to name the space the 8 is in. So 10's, 100's, 1000's and so on. But in this case an 8 in the hundred thousand's place has 10 times more value then an 8 in the ten thousand's place.

Answer:

Step-by-step explanation:

The question doesn't seem complete, but the question is likely to ask for 2 things: the equation of the ellipse and it's foci. So I'll respond to that. Also, the ellipse will be at the center since the artist wants the largest.

Firstly, the equation of an ellipse at the center is given by:

(x²/a²) + (y²/b²) = 1 where a > b.

Since the artist wants to make the largest ellipse, then the length of the rectangle will be the length of the ellipse, same applies to the width.

Since the length is 60, then a = 60/2 = 30

Since the width is 40, then b = 40/2 = 20

Therefore, the equation of the ellipse is:

(x²/30²) + (y²/20²) = 1

To find the foci we need to know the center and this can be gotten by

c² = a² – b²

c² = 30² – 20² = (30 + 20)(30 – 20) = (50)(10) = 500

c = √(500) = 10√(5).

The foci, since it's at th center would be:

(–10√(5), 0) and (+10√(5), 0)

The largest ellipse that the artist can make on the considered rectangular wooden piece will have area as: \(\dfrac{x^2}{900} + \dfrac{y^2}{400} = 1\\\)(assuming its major and minor axes lie on x and y axis respectively)

Suppose that the major axis is of the length 2a units, and that minor axis is of 2b units, then if major axis is on x-axis and minor axis is on y-axis, then the equation of that ellipse would be:
\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} =1\)

For this case, as we have rectangular piece of board the maximum ellipse will have it's major axis of length 60 inches, and of minor axis as 40 inches.

Thus, we have:

\(a = 60/2 = 30\: \rm inches\\b = 40/2 = 20 \: inches\)

Thus, if the rectangle is placed such that the major axis is on x-axis, and minor axis is on y-axis, then the equation of that ellipse would be:

\(\dfrac{x^2}{30^2} + \dfrac{y^2}{20^2} = 1\\\\\dfrac{x^2}{900} + \dfrac{y^2}{400} = 1\\\)

Thus, the largest ellipse that the artist can make on the considered rectangular wooden piece will have area as: \(\dfrac{x^2}{900} + \dfrac{y^2}{400} = 1\\\)(assuming its major and minor axes lie on x and y axis respectively)

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Answer:

2

Step-by-step explanation:

\(\frac{1}{4}\) a cup detergent for \(\frac{1}{2}\) a load of clothes would mean for each full load, it would take \(\frac{1}{2}\) a cup of detergent.

Thus if you have 4 load of clothes, which is 8 half loads. You can take that 8 half loads and multiply \(\frac{1}{4}\) by 8. This gives you 2 cups of detergent.

The amount of the discount that was given is $300.

A discount reduces the price of an item. When a discount is given, the item becomes cheaper. VAT is value added tax. VAT increases the price of an item.

The equation that can be used to represent the total cost of the bicycle is:

Total cost after VAT and discount = (1 + VAT) x (marked price - discount)

1921 = (1 + 0.13) x (2,000 - d)

1921 = 1.13 x (2,000 - d)

In order to determine the value of discount, take the following steps:

Divide both sides of the above equation by 1.13

1921 / 1.13 = 2000 - d

1,700 = 2,000 - d

Combine similar terms: 2000 - 1700 = d

Add similar terms together: d = 300

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Answer: 1/3

Step-by-step explanation: if there are three students then the denominator is 3 (?/3)  and if each of them want an equal share then the numerator would 1 (1/3)

just to check:

1/3 + 1/3 + 1/3

= 3/3

therefore the answer is 1/3

Answer:

1/3 or 5 pounds each

Step-by-step explanation:

if there's 3 kids splitting up 15 pounds worth of apples, the 15 pounds are divided into thirds. so each child takes home 1/3 or one third of the 15 pounds. which is equal to 5 pound of apples per child

Answer:

23 years.

Step-by-step explanation:

It's is b cause it is and yeah

Answer:

a

Step-by-step explanation:

Answer:

(PE) (MD) (AS)

applies to all if you must multiply or divide at the same time do the left side first

Answer:

x = 24

Step-by-step explanation:

if a and b are parallel then

62 and 5x - 2 are same- side interior angles and sum to 180° , that is

5x - 2 + 62 = 180

5x + 60 = 180 ( subtract 60 from both sides )

5x = 120 ( divide both sides by 5 )

x = 24

thus for a to be parallel to b , then x = 24

The margin of error for a 95% confidence interval estimate for the proportion of all students with cell phones is approximately 0.0932.

To find the margin of error for a 95% confidence interval estimate for the proportion of all students with cell phones, we can use the formula:

Margin of Error = Z* * sqrt(p*(1-p)/n)

where:
Z* is the z-score corresponding to the desired level of confidence (in this case, 1.96 for 95% confidence)
p is the sample proportion (55/75 = 0.7333)
n is the sample size (75)

Plugging in the values, we get:

Margin of Error = 1.96 * sqrt(0.7333*(1-0.7333)/75)
Margin of Error ≈ 0.0932

Therefore, the margin of error for a 95% confidence interval estimate for the proportion of all students with cell phones is approximately 0.0932.

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