11. The velocity, V of a car moving with a constant acceleration is partly constant and partly
varies as the time taken, t. The velocity of the car after 8s and 12s are 9 m/s and 11
m/s respectively. Find

i)The relationship between the velocity and the time taken.

ii) The time taken when the velocity is 15 m/s.

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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Answer:

3/7

Step-by-step explanation:

The total is 7 blocks. There are three stars. Probability is PART/WHOLE.

3/7

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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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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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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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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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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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*

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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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.

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.

If a $1 million grant is to be divided among four charities, J, K, L, and M.  M will be awarded $200,000 of the grant.

Let the amount awarded to K be x. Then the amounts awarded to L and M will be x + 125,000 and y - 325,000, respectively.

Since the total grant is $1 million, we have:

x + (x + 125,000) + (y - 325,000) + y = 1,000,000

Simplifying this equation, we get:

2x + 2y - 200,000 = 1,000,000

2x + 2y = 1,200,000

x + y = 600,000

We also know that:

y - 125,000 = x + 325,000

y = x + 450,000

Substituting this into the equation x + y = 600,000, we get:

x + (x + 450,000) = 600,000

2x + 450,000 = 600,000

2x = 150,000

x = 75,000

Therefore, the amount awarded to M is:

y - 325,000 = x + 450,000 - 325,000 = $200,000

So, M will be awarded $200,000 of the grant.

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The probability of the spinner landing on keychain is 1/4 or 0.25, which means that for every 4 spins, the spinner is expected to land on keychain once on average.

The measure of the likelihood that an event will occur is known to be Probability.

This is a number between 0 and 1, where 0 represents an impossible event (has no chance of occurring) and 1 represents a definite event (100% chance of occurring).

An event with a probability of 0.5 is equally likely to occur and not to occur. 

To find the probability of the spinner landing on keychain, we need to first determine the total number of spins and the number of times the spinner landed on keychain.

From the given data, we can see that the spinner was spun 8 times, and it landed on keychain twice. Therefore, the probability of the spinner landing on keychain is:

Probability of keychain = Number of times the spinner landed on keychain / Total number of spins

Probability of keychain = 2/8 = 1/4

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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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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]

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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A triangle has two angles that are 73° and 24° in size. The third angle is 83 degrees in length.

The sum of all the angles in a triangle is always 180 degrees. So, we can utilize this fact to get the third angle's measurement:

Let's call the third angle "x". Then we have:

73° + 24° + x = 180°

Simplifying this equation, we get:

97° + x = 180°

Subtracting 97 degrees from both sides, we get:

x = 83°

Therefore, the measure of the third angle is 83 degrees.

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Answer:Tan 5pi/2 can also be expressed using the equivalent of the given angle (5pi/2) in degrees (450°).

Step-by-step explanation:

5π/2 radians is equivalent to 450 degrees.

We have,

To convert radians to degrees, we can multiply the value in radians by the conversion factor of 180 degrees/π radians.

To convert 5π/2 radians into degrees, we can use the conversion factor that states 180 degrees is equal to π radians.

Let's set up the conversion:

(5π/2 radians) × (180 degrees/π radians)

Here, the π radians in the numerator and denominator cancel out, leaving us with:

(5/2) × 180 degrees

Simplifying further, we have:

(5/2) × 180 = 900/2 = 450 degrees

Therefore,

5π/2 radians is equivalent to 450 degrees.

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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?

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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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 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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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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Answer:

Step-by-step explanation:

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.

The correct order of the set of numbers from least to greatest is 2 3 over 4, 2.71 repeating 71, 5 over 2, square root 5, the correct option is D..

To order the given set of numbers from least to greatest, we need to compare their values. We start by noticing that 2 and 3 over 4 are equivalent to 2.75. Next, we compare the values of the remaining numbers. Since the square root of 5 is approximately 2.236 and 5 over 2 is equivalent to 2.5, we have:

2 3 over 4 < 2.71 repeating 71 < 2.5 < square root 5

Therefore, the set of numbers in order from least to greatest is 2, 3 over 4, 2.71 repeating 71, 5 over 2, square root 5.

Ordering the set of numbers from least to greatest, we get:

2 3 over 4, 2.71 repeating 71, square root 5, 5 over 2.

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Answer:

The triangles are not similar because

12/24 is not equal to 3/5.

"Neither" is the correct answer.

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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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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