s varies directly as t, and = 8 when = 16. Find the constant of variation and write a direct variation equation relating s and t. Then find t when = 16. (Write answers in decimal form.)
constant of variation:
direct variation equation:
when s=16, t=

Answer:

125

Step-by-step explanation:

Calculate the area of all the triangles (base x height/2) and add to the area of the square.

The set of solutions for each of the inequalities are given by:

(b) - 4.8 < m < 6.4

(e) x [tex]\geq[/tex] 5

(g) - 11.6 < z < 14

(h) either k < -19/3 or k > 7

Solving the given absolute value inequalities we get,

(b) The given inequality,

|2 - 5m/2| < 14

- 14 < 2 - 5m/2 < 14

-14 - 2 < -5m/2 < 14 - 2

-16 < -5m/2 < 12

-12 < 5m/2 < 16

-12*2 < 5m < 16*2

-24/5 < m < 32/5

- 4.8 < m < 6.4

(e) The given inequality,

2> - |(x-8)/5 + 3/5|

|(x-8)/5 + 3/5| > -2

|(x-8)/5 + 3/5| [tex]\geq[/tex] 0 [Since absolute value is always positive or zero]

(x-8)/5 + 3/5 [tex]\geq[/tex] 0

(x-8+3)/5 [tex]\geq[/tex] 0

x - 5 [tex]\geq[/tex] 0

x [tex]\geq[/tex] 5

(g) The given inequality,

|(5z - 6)/8| < 8

-8 < (5z - 6)/8 < 8

-64 < 5z - 6 < 64

-64 + 6 < 5z < 64 + 6

-58 < 5z < 70

-58/5 < z < 70/5

-58/5 < z < 14

- 11.6 < z < 14

(h) The given inequality,

|(3k - 1)/4| > 5

either, (3k - 1)/4 < -5

3k - 1 < -20

3k < -19

k < -19/3

or, (3k - 1)/4 > 5

3k - 1 > 20

3k > 20 + 1 = 21

k > 21/3

k > 7

Hence the solution sets are:

(b) - 4.8 < m < 6.4

(e) x [tex]\geq[/tex] 5

(g) - 11.6 < z < 14

(h) either k < -19/3 or k > 7

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The  error interval for the depth of the lake would be:

72.5 m ≤ d < 73.5 m

Since the depth of the lake is given as 73 m, truncated to an integer, the actual depth could be anywhere between 72.5 m and 73.5 m. This is because if the depth is between 72.5 m and 73 m, it would be rounded down to 73 m, and if it is between 73 m and 73.5 m, it would be rounded up to 73 m.

This error interval takes into account the rounding that occurred when the depth was truncated to an integer and gives a range of possible depths that the lake could be within.

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The missing angle in the triangle is 70 degrees.

The missing angle in the quadrilateral is 120 degrees.

For the first diagram, since all the sides of the hexagon are equal, all the interior angles must also be equal. Therefore, each angle measures 120 degrees.
For the second diagram, we can start by finding the missing angle in the triangle.

Since the sum of the angles in a triangle is 180 degrees,

we can subtract the known angles from 180 to find the missing angle:
180 - 60 - 50 = 70
To find the missing angle in the quadrilateral, we can start by noticing that opposite angles in a parallelogram are equal.

Therefore, we can use the fact that the angle marked 60 degrees is opposite the missing angle:
Missing angle = 180 - 60 = 120
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the scale factor for the dilation is 1.0625. According to the given question

how to find scale factor?

To find the scale factor for the dilation that transforms quadrilateral QRST to Q'R'ST, we need to compare the corresponding side lengths of the two figures. Since the figures are similar, the corresponding sides are proportional to each other.

Let the scale factor be represented by k. Then, we have:

|QR| / |Q'R'| = |RS| / |R'S'| = |ST| / |S'T'| = k

We can use the given information about the coordinates of the vertices to calculate the lengths of the sides.

|QR| = √((4-(-2))² + (1-(-4))²) = √(85)

|RS| = √((2-4)²+ (7-1)²) = √(40)

|ST| = √(((-4)-2)² + (1-7)²) = √(72)

|Q'R'| = √(((-4)-(-8))²+ ((-1)-(-7))²) = √(80)

|R'S'| = √(((-8)-(-2))² + ((-7)-1)²) = √(170)

|S'T'| = √(((-2)-(-4))² + (1-7)²) = √(20)

Therefore, we have:

√(85) /√(80) = √(40) / √(170) =√(72) / √(20) = k

Simplifying, we get:

k = √(85/80) = √(17/16) = 1.0625

Thus, the scale factor for the dilation is 1.0625.

Note: It's important to keep track of the order of the vertices when calculating the lengths of the sides. In this case, we assumed that the vertices were listed in clockwise order around the quadrilateral.

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After 8 hours, the freezer is at a temperature of 8°C.The temperature sequence for the first 8 hours after the freezer was unplugged is as follows:

Time (hours) Temperature (°C)

0                          -14

1                           -11

2                             -8

3                             -5

4                             -2

5                               1

6                               0

7                               4

8                                8

Temperature is a physical quantity that expresses the degree of hotness or coldness of an object or a living being.

The direction in which heat energy will spontaneously flow from a hotter body (one at a higher temperature) to a colder body (one at a lower temperature) is indicated by temperature, and it is expressed in terms of any of several arbitrary scales

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Answer:

Step-by-step explanation:

We can use the Pythagorean Theorem to find the distances from B to C and from A to B.

The Pythagorean Theorem 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 lengths of the other two sides.

Let's first find the distance from B to C. We can see that the length from B to C is the hypotenuse of a right triangle with legs of length 10 feet and 20 feet. So, using the Pythagorean Theorem, we can write:

distance from B to C = sqrt(10^2 + 20^2)

distance from B to C = sqrt(500)

distance from B to C ≈ 22.36 feet (rounded to the nearest tenth)

Therefore, the distance from B to C is approximately 22.36 feet.

Now, let's find the distance from A to B. We can see that the width of the room, from A to B, is the hypotenuse of a right triangle with legs of length 10 feet and 48 feet. So, using the Pythagorean Theorem, we can write:

distance from A to B = sqrt(10^2 + 48^2)

distance from A to B = sqrt(2354)

distance from A to B ≈ 48.5 feet (rounded to the nearest tenth)

Therefore, the distance from A to B is approximately 48.5 feet.

The circle equation is [tex](x-5)^{2} +(y-2)^{2}[/tex][tex]=52[/tex]. Hence the appropriate answer is [tex](x-3)^{2} +(y+1)^{2} =52[/tex]

A centre is a point in geometry that connects to a polyhedron or other structure. (or centre). The centre of the figure or object may have distinctive characteristics that are significant when studying it.

For example, the centre of a circle is thought to be the spot that is evenly spaced from all other points on the circle. This term, which is commonly represented by the character "O," is defined in terms of the radius, diameter, and circumference of a circle.

Additionally, other geometric constructions like tangent lines and inscribed polygons use the middle of a circle as well. Other geometric shapes may define the middle differently.

Given

The circle's centre is found at the midpoint of the circumference, which can be established by averaging the [tex]x[/tex]- and[tex]y[/tex]-coordinates of the ends:

[tex]centre = (3+7)/2,(-1+5)/2(,5,2)[/tex]

Distance =[tex]\frac{1}{2}[/tex] of diameter= circle radius

[tex]r=\sqrt{(7-3)}[/tex]

[tex]\sqrt{52/2}[/tex][tex]= \sqrt{2+(5-(-1)2)/2}[/tex][tex]13[/tex]

Therefore the circle equation[tex](x-5)^{2} +(y-2)(y-2)^{2}= \sqrt[2]{13^{2} }[/tex]

[tex](x-5)^{2} +(y-2)^{2} =52[/tex]

Hence the appropriate answer is [tex](x-3)^{2} +(y+1)^{2} =52[/tex]

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Check the picture below, so that's the picture of a parabolic path with a certain initial velocity.

so anyhow, not to bore you to death, the maximum or peak point occurs at the vertex, as you see in the picture, and the x-coordinate is how long it took to get there.

[tex]\textit{vertex of a vertical parabola, using coefficients} \\\\ P(t)=\stackrel{\stackrel{a}{\downarrow }}{-1698}t^2\stackrel{\stackrel{b}{\downarrow }}{+85000}t\stackrel{\stackrel{c}{\downarrow }}{+10000} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right) \\\\\\ \left(-\cfrac{ 85000}{2(-1698)}~~~~ ,~~~~ 10000-\cfrac{ (85000)^2}{4(-1698)}\right) \implies \left( - \cfrac{ 85000 }{ -3396 }~~,~~10000 - \cfrac{ 7225000000 }{ -6792 } \right)[/tex]

[tex]\left( \cfrac{ -21250 }{ -849 } ~~~~ ,~~~~ 10000 +\cfrac{ 903125000 }{ 849 } \right) \\\\\\ \left( \cfrac{ 21250 }{ 849 } ~~~~ ,~~~~ \cfrac{ 911615000 }{ 849 } \right) ~~ \approx ~~ (\stackrel{ hrs }{25}~~,~~\stackrel{ population }{1,074,000})[/tex]

ANSWER:

Therefore, Stephanie cannot pass the exam even if she gets full marks in Paper 2.

Step-by-step explanation:

Pass mark = 65 + 75 = 140

Stephanie got 80% of the marks for Paper 1, which is:

0.8 x 65 = 52

To pass the exam, Stephanie must get a total of 140 marks, and she already has 52 from Paper 1. Therefore, she needs to get the remaining marks from Paper 2:

140 - 52 = 88

So, Stephanie must get at least 88 marks out of 75 in Paper 2 to pass the exam. However, this is not possible as the maximum marks for Paper 2 are 75.

Since there are four tellers, the capacity of the bank in customers per hour = 4 x 7.5 = 30.

Valley Bank has four tellers. It takes a teller 8 minutes to serve one customer.

The correct option is D. 30.

We are given that

Valley Bank has four tellers. It takes a teller 8 minutes to serve one customer.

To calculate the capacity of the bank in customers per hour, we need to find how many customers each teller can serve in an hour. To do this, we first need to convert the time taken to serve one customer from minutes to hours.

1 minute = 1/60 hoursSo, time taken to serve one customer

= 8 minutes

= 8/60 hours

= 2/15 hours

One teller can serve one customer in 2/15 hours.In one hour, the number of customers one teller can serve = 1/(2/15) = 15/2 = 7.5 (customers/hour)

Therefore, the capacity of one teller in customers per hour is 7.5.

Now, we need to find the capacity of the bank in customers per hour. Since there are four tellers, the capacity of the bank in customers per hour = 4 x 7.5 = 30.

So, the correct option is D. 30.

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The measure of angle BCD is 62 degrees.

When a transversal intersects two parallel lines, alternate interior angles are congruent.

This is given by the Alternate Interior Angles Theorem. Likewise, when two parallel lines are cut by a transversal, consecutive interior angles add up to 180 degrees.

This is given by the Consecutive Interior Angles Theorem.

In the following image, AB is parallel to DC, and BC is a transversal intersecting both parallel lines.

The measure of angle ABC is 118°, and the measure of DAB is 132.

Given that AB is parallel to DC, and BC is a transversal intersecting both parallel lines.

The measure of angle ABC is 118°, and the measure of DAB is 132.

We need to find the measure of angle BCD.

By the Consecutive Interior Angles Theorem, we know that angle BCD + angle ABC = 180 degrees

Therefore, angle BCD = 180 - angle ABC angle BCD = 180 - 118angle BCD = 62 degrees

Therefore, the measure of angle BCD is 62 degrees.

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The volume of sphere a is 32π/3 cubic cm, and the volume of sphere b is 256π/3 cubic cm.

What is volume of sphere?

The volume of a sphere is given by the formula:

V = (4/3)πr³

where r is the radius of the sphere, and π (pi) is a mathematical constant approximately equal to 3.14159.

The volume of a sphere can be calculated using the formula:

V = (4/3)π[tex]r^3[/tex]

where r is the radius of the sphere and π is the mathematical constant pi.

Using this formula, we can find the volumes of spheres a and b as follows:

Volume of sphere a:

V = (4/3)π([tex]2^3[/tex]) = (4/3)π(8) = 32π/3 cubic cm

Volume of sphere b:

V = (4/3)π([tex]4^3[/tex]) = (4/3)π(64) = 256π/3 cubic cm

Therefore, the volume of sphere a is 32π/3 cubic cm, and the volume of sphere b is 256π/3 cubic cm.

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The answers are:

a). The required monthly percentage rate of APR of 19% is 1.59%.

b). The Monthly percentage rate is 2.08%

c). Oscar pays $0.96 more than Felix.

a). Felix received a card with an APR of 19%; it is unknown what his monthly percentage rate was.

APR = 19%

Given that APR is increased monthly throughout the year,

Monthly rate = 19% / number of months in year

Monthly rate = 19% / 12

Monthly rate = 1.59%

Therefore, 1.59% is the minimum monthly percentage rate for an APR of 19%.

b) Oscar received a credit card with a 21% APR.

Monthly percentage rate = 21%/12 = 1.75%

c). We are informed that for a particular month, each of them had an average daily amount of $800.

Thus;

Felix's payments in full = 800 * 1.63%

Felix's payments in full  = $13.04

Amount Oscar issued = 800 * 1.75%

Amount Oscar issued = $14

Variation in Payments = $14 - $13.04 = $0.96

Thus, Oscar pays $0.96 more than Felix.

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The probability that teacher send 3 or 4 text every day is option B = 0.5

Probability is the measure of the likelihood of an event occurring. It is typically represented as a number between 0 and 1, where 0 indicates that the event is impossible, and 1 indicates that the event is certain.

To find the probability of an event, you need to identify the number of ways the event can occur and the total number of possible outcomes.

The formula for probability is:

Probability = Number of favorable outcomes / Total number of outcome

The probabilities of P(X=3) and P(X=4) must be added in order to determine the likelihood that the teacher sends 3 or 4 texts per day:

P(X=3 or X=4)=P(X=3) + P(X=4)=0.1 + 0.4 = 0.5

The likelihood that the teacher will send three or four texts every day is therefore 0.5.

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The weights of 1,000 men in a certain town follow a normal distribution with a mean of 150 pounds and a standard deviation of 15 pounds.

Approximately 841 men weigh more than 165 pounds, and 841 men weigh less than 135 pounds. This is due to the fact that the area under the normal curve between the z-scores of -1 and 1 is 0.8413, or 84.13%.

Standard deviation is a measure of how spread out a set of data is. It is calculated by taking the square root of the variance, which is the average of the squared differences from the mean. Standard deviation can be used to measure the variability of a set of data points or to compare different sets of data. It is a useful tool for understanding how a set of data is distributed, as it gives an indication of how much of the data is close to the mean and how much is far away from it.

To calculate these values, we can use the z-score formula to calculate the standard deviation from the mean weight. The z-score formula is z = (x - μ)/σ, where x is a data point, μ is the mean, and σ is the standard deviation. Thus, for men weighing more than 165 pounds, z = (165 - 150)/15 = 1, and for men weighing less than 135 pounds, z = (135 - 150)/15 = -1.

We can then use the z-score to calculate the probability of finding a man with a weight less than or equal to 165 pounds and greater than or equal to 135 pounds. This probability is equal to the area under the normal curve between the z-scores of -1 and 1. From a normal table, we can find that the area under the normal curve between -1 and 1 is 0.8413, which is equal to 84.13%.

Therefore, the number of men weighing more than 165 pounds is equal to 0.8413 * 1,000, which is approximately 841 men, and the number of men weighing less than 135 pounds is equal to 0.8413 * 1,000, which is also approximately 841 men.

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Complete questions as follows-

The weights of 1,000 men in a certain town follow a normal distribution with a mean of 150 pounds and a standard deviation of 15 pounds.

From the data, we can conclude that the number of men weighing more than 165 pounds is about , and the number of men weighing less than 135 pounds is about .

The factors of 6[tex]x^{2}[/tex]+ 37x - 60 are 2(x - 2) and 3(x + 5), which is answer c.

To factor the polynomial 6[tex]x^{2}[/tex] + 37x - 60, we need to find two numbers whose product is -360 (the product of the leading coefficient and the constant term) and whose sum is 37 (the coefficient of the linear term).

One way to do this is to list all the possible factor pairs of -360 and look for a pair that adds up to 37. Some of the factor pairs are:

1, -360

2, -180

3, -120

4, -90

5, -72

6, -60

8, -45

9, -40

10, -36

12, -30

15, -24

18, -20

We can see that 15 and -24 add up to 37, so we can use them as the coefficients of the linear term. To get the correct sign, we need to use -24 and 15 instead of 15 and -24.

So we have:

6[tex]x^{2}[/tex] + 37x - 60 = 6[tex]x^{2}[/tex] + 15x - 24x - 60

= 3x(2x + 5) - 12(2x + 5)

= (3x - 12)(2x + 5)

= 6(x - 2)(x + 5)

Therefore, the factors of 6[tex]x^{2}[/tex] + 37x - 60 are 2(x - 2) and 3(x + 5), which is answer c.

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a) The series 1/pi = 2sqrt(2)/9801 × summation n=0 to infinity of (4n)!(1103+26390n)/((n!)^4*396^4n) is convergent by the ratio test.

b) If we use just the first term of the series, we get an approximation of pi to one correct decimal place: pi ≈ 6533.008.

If we use two terms of the series, we get an approximation of pi to 14 correct decimal places: pi ≈ 3.14159265358979324.

a) To verify the convergence of the given series, we can use the ratio test.

Let's take the limit of the ratio of the (n+1)th term to the nth term as n approaches infinity:

limit as n approaches infinity of [(4(n+1))!(1103+26390(n+1))/((n+1)!^4396^4(n+1))] / [(4n)!(1103+26390n)/(n!)^4396^4n]

= [(4n+4)(4n+3)(4n+2)(4n+1)(1103+26390n+26390)/(n+1)^4*396^4]

= [(4n+1)^4(1103+26390n+26390)/(n+1)^4*396^4]

= (4n+1)^4(1103+26390n+26390)/(n+1)^4(396^4)

= (4n+1)^4(1103/n+26390+26390/n)/(396^4)

As n approaches infinity, the terms inside the parentheses approach constant values, and we can ignore the n-dependent terms in the numerator and denominator. Thus, the limit simplifies to

= (4^4 × 1103) / (396^4) = 1/(\pi)

Since the limit is less than 1, the series converges by the ratio test.

b) If we use just the first term of the series, we get

1/pi ≈ (2sqrt(2)/9801)×(4!/396^4) = 1.2337 x 10^-4

Taking the reciprocal of both sides, we get

pi ≈ 807104 / 1.2337 ≈ 6533.008

This approximation gives us only one correct decimal place of pi.

If we use two terms of the series, we get

1/pi ≈ (2sqrt(2)/9801)[(4!(1103)+(8!26390))/(396^4(1!^4))]

= 3.1415927300133055 x 10^-1

Taking the reciprocal of both sides, we get

pi ≈ 3.14159265358979324

This approximation gives us 14 correct decimal places of pi.

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The given question is incomplete, the complete question is:

Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula:

1/pi = 2sqrt(2)/9801 * summation n=0 to infinity of (4n)!(1103+26390n)/((n!)^4*396^4n)

William Gosper used this series in 1985 to compute the first 17 million digits of pi.

a) Verify that the series in convergent.

b) How many correct decimal places of pi do you get if you use just the first term of the series? What if you use two terms?

Answer:

Step-by-step explanation:

%218.18

first you add then subtract do the math work coordinate planes and such graphing math problems

As a result, Mr. Rodriguez can produce 10 sacks, each containing two blueberries and three grapes.

So, 10 bags must be the solution.

The largest number that can split evenly into two other numbers is known as the greatest common factor in mathematics. For instance, the number 6 is the most frequent factor between 12 and 30. The greatest common denominator is another name for the greatest common component.

We must find the GCF, or greatest common factor, between 20 and 30 to calculate how many bags Mr. Rodriguez can produce. This is necessary because each bag must contain the same quantity of blueberries and grapes, requiring that they both have a similar factor of 20 or 30.

1, 2, 4, 5, 10, and 20 are the elements that make up 20.

1, 2, 3, 5, 6, 10, 15, and 30 make up the number 30.

1, 2, 5, and 10 are the common variables between 20 and 30.

Since the quantity of blueberries and grapes in each bag must be the same, we can split both 20 and 30 by the GCF of 10, which is 10.

The correct number of bags needed is 10 .

This results in:

2 blueberries are in each container of 20/10.

3 grapes per container (30/10)

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The conclusion supported by the image of the lines is that when two parallel lines are rotated about the origin, they remain parallel to each other.

1. Observe the initial graph with two parallel lines.
2. Rotate the lines about the origin by the given angle.
3. Observe the image of the lines after the rotation.
4. Notice that the lines still maintain the same distance apart and do not intersect, meaning they are still parallel to each other.

If two lines do not intersect, they are said to be parallel.

The lines' slopes are identical. If m1=m2, then f(x) =m1x + b1 and

g(x)= m2x + b2 are parallel.

If m 1 = m 2, then f (x) = m 1 x + b 1 and g (x) = m 2 x + b 2 are parallel equations.

The coordinate axes are rotated about the origin (0,0) in counter clockwise direction through an angle of 60∘ .

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Answer:

48 cubic inches.

Step-by-step explanation:

The volume of a rectangular prism is given by the formula:

Volume = length x width x height

Since the rectangular prism has a volume of 6 cubic inches, we can write:

6 = length x width x height

When the prism is dilated using a scale factor of 2, each of its dimensions (length, width, and height) is multiplied by 2. Therefore, the new dimensions of the prism are:

2 x length, 2 x width, and 2 x height

The volume of the dilated prism can be calculated as:

Volume of dilated prism = (2 x length) x (2 x width) x (2 x height)

= 2³ x length x width x height

= 8 x (length x width x height)

Since the original rectangular prism had a volume of 6 cubic inches, we know that:

length x width x height = 6

Substituting this value into the equation for the volume of the dilated prism, we get:

Volume of dilated prism = 8 x 6 = 48 cubic inches

Therefore, the volume of the dilated rectangular prism is 48 cubic inches.

We can use the exponential function [tex]N(t) = N0 * 2^(t/2)[/tex] to determine the number of transistors in a car that doubles every two years.

An exponential function with the following form can be used to calculate the number of transistors in an automobile whose number doubles every two years:

[tex]N(t) = N0 * 2^(t/2)[/tex]

Where t is the amount of time in years after the initial measurement, N0 is the number of transistors at the start, and N(t) is the number of transistors at time t.

We can use this technique to determine the number of transistors in the car in any odd year since 1974, when there were 4100 in it.

For instance, we may insert in t = 15 to determine the quantity of transistors in 1989:

transistors N(15) = 4100 * 2(15/2) = 261,632

Similarly, we may enter t = 19 to calculate the number of transistors in 1993:

522,724 transistors are found in N(19) = 4100 * 2(19/2)

In 1997, we can enter t = 23:

1,045,449 transistors make up N(23) = 4100 * 2(23/2)

In summary, we may calculate the number of transistors in an automobile whose number doubles every two years using the exponential equation [tex]N(t) = N0 * 2(t/2)[/tex]. We can determine the number of transistors in a car in any odd year with an initial measurement of 4100 transistors in 1974.

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Answer:

Step-by-step explanation:

To find the frequency for which the particular solution to the differential equation has the largest amplitude, we first need to know the differential equation we are working with. However, since you didn't provide the specific differential equation, let's work with a general example: a forced harmonic oscillator. The equation for a forced harmonic oscillator can be written as:

m * d²x/dt² + c * dx/dt + k * x = F0 * cos(ωt)

where:

m is the mass of the oscillator

x is the displacement of the oscillator

c is the damping coefficient

k is the spring constant

F0 is the amplitude of the external force

ω is the angular frequency of the external force

For this type of equation, we can find the particular solution in the form:

x_p(t) = X * cos(ωt - δ)

where:

X is the amplitude of the particular solution

δ is the phase angle

We can rewrite the differential equation in the frequency domain by substituting x(t) = X * cos(ωt - δ) and its derivatives into the original equation, then applying the trigonometric identities. After simplifying, we can find the expression for X, the amplitude of the particular solution:

X = F0 / sqrt((k - mω²)² + (cω)²)

To find the frequency for which the particular solution has the largest amplitude, we need to maximize X with respect to ω. To do this, we can find the critical points by differentiating X with respect to ω and setting the result to zero:

dX/dω = 0

To simplify the problem, we can define the damping ratio ζ = c / (2 * sqrt(m * k)) and the undamped natural frequency ω_n = sqrt(k / m). The expression for X becomes:

X = F0 / sqrt((ω_n² - ω²)² + (2 * ζ * ω_n * ω)²)

Now, we differentiate X with respect to ω and set it to zero. Solving for ω, we get:

ω = ω_n * sqrt(1 - 2ζ²)

This is the frequency for which the particular solution to the differential equation has the largest amplitude, assuming a positive frequency and that the damping ratio ζ is less than 1 / sqrt(2). Otherwise, the system will be overdamped, and there will be no resonant frequency.

the frequency for which the particular solution to the differential equation has the largest amplitude is:

ω = √(γ/2 - β^2)

To find the frequency for which the particular solution to the differential equation has the largest amplitude, we can assume that the particular solution is of the form:

y(t) = A*cos(ωt + φ)

where A is the amplitude, ω is the frequency, and φ is the phase angle.

Substituting this form of y(t) into the differential equation gives:

-ω^2Acos(ωt + φ) - 2βωAsin(ωt + φ) + γA*cos(ωt + φ) = f(t)

Simplifying this equation gives:

(A/|D|)[γcos(ωt + φ) - ω^2cos(ωt + φ) - 2βω*sin(ωt + φ)] = f(t)

where |D| is the modulus of the denominator of A*cos(ωt + φ) and is given by:

|D| = √[ (γ - ω^2)^2 + (2βω)^2 ]

To find the frequency for which the amplitude of the particular solution is largest, we need to minimize the modulus of the denominator |D| over all frequencies ω. We can do this by finding the critical points of |D| with respect to ω and then checking which of these critical points correspond to a minimum.

Differentiating |D| with respect to ω gives:

d|D|/dω = [2ω(γ - ω^2) - 4β^2ω]/|D|

Setting this equal to zero and solving for ω gives:

ω = ±√(γ/2 - β^2)

We can see that there are two critical points for |D|, one positive and one negative. To check which of these corresponds to a minimum, we can use the second derivative test:

d^2|D|/dω^2 = (2γ - 6ω^2)/|D|^3

Substituting ω = ±√(γ/2 - β^2) into this expression gives:

d^2|D|/dω^2 = ±4√2β^3/γ^(3/2)

Since β and γ are both positive, the second derivative is negative for both critical points, which means that they both correspond to maxima of |D|. The positive critical point corresponds to the frequency for which the amplitude of the particular solution is largest.

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Answer: To find the lateral surface area of the rectangular prism formed by folding the net, we need to identify the faces that make up the sides of the prism.

In the net, we can see that the two rectangles on the top and bottom will form the bases of the prism, and the four rectangles around the sides will form the lateral faces.

The dimensions of the net are labeled in the diagram as follows:

a = 8 cm

b = 6 cm

c = 10 cm

To find the lateral surface area of the prism, we need to calculate the area of each of the four rectangles around the sides and then add them up.

The dimensions of the rectangles are:

8 cm by 10 cm (on the front and back)

6 cm by 10 cm (on the sides)

So the area of each of the four rectangles is:

8 cm x 10 cm = 80 cm²

6 cm x 10 cm = 60 cm²

Adding up the areas of the four rectangles, we get:

80 cm² + 80 cm² + 60 cm² + 60 cm² = 280 cm²

Therefore, the lateral surface area of the rectangular prism formed by folding the net is 280 square centimeters.

Step-by-step explanation:

The length of the third side is x= 8 (nearest rounded to the tenth).

The line segment in a triangle connecting the midway of two sides of the triangle is said to be parallel to its third side and is also half the length of the third side, according to the midpoint theorem.

By using the midpoint theorem, we get

The line is parallel to its third side x and is also half the length of the third side, therefore we can write

X= [tex]\frac{1}{2} * 15[/tex]

X = 7.5

Rounded the value nearest tenth

X = 8

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5

We know that there are 28 students in total, so:

G + H + J + K = 28

We also know the following probabilities:

P(G) = 7/28

P(G or H) = 5/14

The probability of selecting a student whose last name begins with G or H can be expressed as:

P(G or H) = P(G) + P(H) - P(G and H)

Since the events "selecting a student whose last name begins with G" and "selecting a student whose last name begins with H" are mutually exclusive (a student cannot have a last name that begins with both G and H), P(G and H) = 0. Therefore, we have:

5/14 = 7/28 + P(H)

Simplifying the equation, we get:

P(H) = 5/14 - 7/28 = 5/28

So the probability of selecting a student whose last name begins with H is 5/28. To find the number of students whose last name begins with H, we can multiply this probability by the total number of students:

H = P(H) x 28 = 5/28 x 28 = 5

Therefore, there are 5 students whose last name begins with H.

*IG: whis.sama_ent*

The social science majors at your university compare to national mac computer preference norms for college students statistical decision is  5.83.

The gathering, characterization, analysis, and drawing of inferences from quantitative data are all tasks that fall under the purview of statistics, a subfield of applied mathematics. Probability theory, linear algebra, and differential and integral calculus play major roles in the mathematical theories underlying statistics.

Finding valid inferences about big groups and general occurrences from the behaviour and other observable features of small samples is a major challenge for statisticians, or persons who study statistics. These tiny samples are representative of a small subset of a larger group or a small number of isolated occurrences of a widespread phenomena.

Proportion of students who preferred PC's to Mac (P) = 0.62

Sample size (n) = 200

Sample proportion of students who prefer Mac's to PCs = 0.82

To compare the proportion of students with Mac preference for college students a Z-test for single proportion will e conducted. The hypothesis is formulated as follows:

H o: There is no significant difference in the national proportion and college preference

H1 : There is a significant difference in the national proportion and college preference

The test statistic is computed below:

[tex]z=\frac{p-p_0}{\sqrt{\frac{p_0(1-p_0)}{n} } }[/tex]

[tex]z=\frac{0.82-0.62}{\sqrt{\frac{0.62(1-0.62)}{200} } }[/tex]

= 5.8272

Thus, the value of the test statistic is 5.83.

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