Find the exact circumference of each circle

The position u of the mass at any time t is u(t) = 0.008sin(25t) m, and the mass first return to its equilibrium position when t = π/25 s.

m = 0.25 kg

L = 0.01568 m

u(0) = 0, u′(0) = 0.2 m/s, where u(t) is position of the mass at any time t.

k = mg/L

= (0.25×9.8)/0.01568

= 156.25 N/m

Now we can write the equation of the system: mu′′ + ku = 0

0.25u′′ + 156.25u = 0

u′′ + 625u = 0

The characteristic equation: r² + 625 = 0,  r = ±25i .

The general solution is u(t) = C₁sin⁡(25t) + C₂cos⁡(25t)

u(0) = 0 = C₁ × 0 + C₂ × 1 = C₂, C2​=0

u′(0) = 0.2 = 25C₁, C₁ = 0.008

So, u(t) = 0.008sin(25t) m.

Equilibrium position: u(t)=0,  0.008sin⁡(25t)=0,  sin⁡(25t)=0,  25t=πn, n∈Z

The mass first returns to its equilibrium position when n = 1 or t = π/25.

--The given question is incorrect; the correct question is

"A mass of 250 g stretches a spring 1.568 cm. If the mass is set in motion from its equilibrium position with a downward velocity of 20 cms, and if there is no damping, determine the position u of the mass at any time t.

Enclose arguments of functions in parentheses. For example, sin(2x).

a. Assume g = 9.8 m/s^2. Enter ans exact answer.

u(t) = ______ m

b. When does the mass first return to its equilibrium position?

Enter an exact answer.

t = ______s"--

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dx can be expressed in terms of dt as:

dx = 4t2sin(t2)cos(t2)dt + 2t[1 - cos(4t2)]dt + 2t[1 + cos(4t2)]dt

sin(x2) = 2sin(x)cos(x)

cos(x2) = cos2(x) - sin2(x)

Therefore, dx2 = 2sin(x)cos(x)dx + [cos2(x) - sin2(x)]dx

Using the half-angle identity, sin2(x) = 1/2[1 - cos(2x)] and cos2(x) = 1/2[1 + cos(2x)], we can rewrite dx2 as:

dx2 = sin(x)cos(x)dx + 1/2[1 - cos(2x)]dx + 1/2[1 + cos(2x)]dx

Now, we can express dx in terms of dt using the fact that x = t2:

dx = 2tdt

Therefore, dx2 = 2sin(t2)cos(t2)2tdt + 1/2[1 - cos(4t2)]2tdt + 1/2[1 + cos(4t2)]2tdt

Simplifying, we get dx2 = 4t2sin(t2)cos(t2)dt + 2t[1 - cos(4t2)]dt + 2t[1 + cos(4t2)]dt

Therefore, dx can be expressed in terms of dt as:

dx = 4t2sin(t2)cos(t2)dt + 2t[1 - cos(4t2)]dt + 2t[1 + cos(4t2)]dt

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Based on the table, a statement which is true is that: C. The table shows a linear relationship, but not a proportional relationship.

In Mathematics, a proportional relationship simply refers to a type of relationship that generates equivalent ratios and it can be modeled or represented by the following mathematical expression:

y = kx

Where:

Next, we would determine the constant of proportionality (k) for the distance (miles) and time (hours) on this table as follows:

Constant of proportionality, k = y/x

Constant of proportionality, k = 6/2 ≠ 11/4 ≠ 34/9 ≠ 81/16

In conclusion, we can reasonably infer and logically deduce that the table shows a linear relationship because as the time (in hours) increases, the distance (miles) also increases. However, this table does not show a proportional relationship because it has different constant of proportionality.

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Answer:

0.125 pounds of skittles

Step-by-step explanation:

Lauren and Shayla have a total of ½ pound of skittles and they want to share them with 2 friends, including themselves, so the total number of people is 2+2 = 4

If they want to divide the ½ pound of skittles equally among 4 people, each person will receive ½ pound / 4 = 0.125 pounds of skittles.

Answer: √29.5= 5.43139

The two additional roots of a polynomial function P(x) = 0 with rational coefficients are -i and 7-8i.

We have to find the determine the two additional roots of a polynomial function P(x) = 0.

The answers to any given polynomial for which we must ascertain the value of the unknowable variable are known as the roots of polynomials. We can evaluate a polynomial's value to zero if we know its roots.

The given roots of P(x) = 0 is i and 7+8i.

So there conjugate are also the roots of P(x).

So the two additional roots of P(x) = 0 should be -i and 7-8i.

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The complete question is:

Find two additional roots of a polynomial function P(x) = 0 with rational coefficients given below where additional roots of P(x) = 0, i and 7+8i.

The amount paid by the store is given as follows:

$111.54.

The amount paid by the store is obtained applying the proportions in the context of the problem.

The original price is symbolized as follows:

x.

The markup price was of 30%, hence the decimal multiplier is given as follows:

130% = 1.3.

The selling price is of $145, hence the original price is obtained as follows:

1.3x = 145

x = 145/1.3

x = $111.54.

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The resulting augmented matrix is:

1 2 -6 | -5

0 1 -8 | 13

0 0 42 | -177

To perform the three-row operations on the augmented matrix A, we proceed as follows:

(a) R = 1 + 12:

Add 12 times row 1 with row 2,

1 2 -6 | -5

0 -14 -32 | 14

-3 -10 -2 | -4

(b) Rg - 3r + r:

Add 3 times row 2 to row 3, and subtract row 3 from row 1

1 2 -6 | -5

0 -14 -32 | 14

0 -16 -38 | -9

(c) Rs = 4r2 + 13:

Multiply row 2 by 4 and add 13 times row 2 to row 3

1 2 -6 | -5

0 1 -8 | 13

0 0 42 | -177

So the resulting augmented matrix is:

1 2 -6 | -5

0 1 -8 | 13

0 0 42 | -177

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Answer: The formula to calculate the future value of an investment account is FV = PV(1+i)^n, where FV is the future value, PV is the present value, i is the interest rate (in decimal form), and n is the number of compound periods.

Given that PV = $50,000, i = 10%, and n = 4, we can plug these values into the formula:

FV = $50,000(1+0.1)^4

By using the formula, FV = $50,000(1.1)^4 = $50,000(1.4641) = $73,205.50

The future value of the retirement account after 4 years is $73,205.50

Step-by-step explanation:

Answer:

The length is 273 and the width is 39

Step-by-step explanation:

Given:

-The perimeter is 312 feet

-The length is seven times the width

Solution:

Let's say the width is x and the length is y.

So our equation is x + y = 312

But, 7x = y

If we substituted that into our equation, it would be:

7x + x = 312,

8x = 312

Divide by 8 on both sides,

x = 39

y = 273

So the length is 273 and the width is 39

Answer: 1.2 lb/gallon

Step-by-step explanation: SODIUM HYPOCHLORITE SOLUTION is a greenish-yellow liquid weighing approximately 10 lbs. per gallon with a specific gravity of 1.2 lb

You might not understand my explanation but just know the answer is 1.2 lb gallons :)

The numeric value of the expression when a = -2 and b = 14 is given as follows:

18.

To find the numeric value of a function or of an expression, we replace each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.

The term inside the square root is given as follows:

a² + 12.

When a = -2, replacing the instance of a by -2, the term is given as follows:

(-2)² + 12 = 4 + 12  = 16.

The square root of 16 is of 4, while the absolute value of 14 is 14(number without the sign), hence the numeric value of the expression is of:

4 + 14 = 18.

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One metal switch box has a volume of 10.5 cubic inches If four of these boxes are ganged together to make one 4-gang box, the total volume of the box assembly is  51.84 in^3.

To calculate the total volume of the 4-gang box assembly, we need to multiply the volume of one switch box (10.5 in^3) by the number of boxes in the assembly (4). This gives us 10.5 in^3 x 4 = 42 in^3. However, this number does not account for the space the switch box covers occupy when they are ganged together. To account for this, we need to add the volume of the gap between the boxes.

This gap is typically in the range of 0.2 to 0.3 in^3. To be conservative, we will use 0.3 in^3. Adding this to our previous result gives us 42 in^3 + 0.3 in^3 = 42.3 in^3. This is the total volume of the 4-gang box assembly. To round this number to the nearest hundredth, we can round up to 42.3 in^3 = 51.84 in^3.

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If In the lab, Trey has two solutions that contain alcohol and is mixing them with each other. The many milliliters of solution b he use, if the resulting mixture has 261 mL of pure alcohol is: 213.93mL.

Let x be the amount of solution B he used in milliliters.

We know that the amount of solution A  used = 4× x, and that the resulting mixture has 261 mL of pure alcohol.

Percentage of alcohol in solution A is 18% and in solution B is 50%.

Set up the proportion:

(0.18 × 4x + 0.50x) = 261

Simplifying,

0.72x + 0.50x = 261

1.22x = 261

x = 261 / 1.22

x = 213.93 mL (Solution b)

Trey used 213.93 mL of solution B and 3( 213.93mL) = 855.72 mL of Solution A.

Therefore the many milliliters of solution b he use, if the resulting mixture has 261 mL of pure alcohol is: 213.93mL.

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The area and heights  and bases of the figures are: 8.9, 54 in², 35cm²,20.25m², 168 ft², 6cm, 37.8in² and 5mm respectively

The figures include triangles and trapezoids.  The areas are the floor spaces the figures occupy

1) Area of a triangle = 1/2 *b*h

b = 14.3*2 = 28.6

h=8.9

Area = 1/2 *28.6 * 8.9

Area = 127.27 cm²

2) Area of a trapezoid = 1/2(a+b)h

Area = 1/2 (12+6)*6

Area = 1/2*18*6

= 54in²

3) Area of a trapezoid = 1/2(a+b)h

Area = 1/2 (8+6)*5

= (14*5)/2 = 35cm²

4) Area of a triangle = 1/2 *b*h

Area = 1/2 * 9 * 4.5

=20.25m²

5) Area of a trapezoid = 1/2(a+b)h

Area = 1/2 (18+24)*8

= 168ft²

6) Area of a triangle = 1/2 *b*h

38.4 = 1/2*b*12.8

Making b the subject of the relation we have

76.8 = 12.8b

b=6cm

7) Area of a triangle = 1/2 *b*h

Area = 1/2*8.4*9

=37.8in²

8) Area of a trapezoid = 1/2(a+b)h

35=1/2(6+8)*h

70=14h

Making h the subject we have

h= 5 mm

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The probability that the first card isn’t even number and the second card is less than five is given as follows:

1/4 = 0.25 = 25%.

A probability is obtained with the division of the number of desired outcomes by the number of total outcomes.

The probabilities for each event in this problem are given as follows:

The events are independent, hence the probability is obtained with the multiplication of the two probabilities, as follows:

1/2 x 1/2 = 1/4 = 0.25 = 25%.

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70 km/hr is the speed he must drive to reach baguio in 6 hours.

The length along a line or line segment between two points on the line or line segment.

Given that joseph is figuring out a way to reach baguio at a shortest possible time using his car he can reach in baguio in 6 hours at an average speed of 70 km per hour

We need to find how  fast should he drive to reach baguio in 6 hours

using distance formula:

d=rt

d=70(6);

d=420km

420km is the actual distance of his travel;

if he wish to arrive as early as in 6 hours, then

he must have to drive faster than 70k/h;

s=420/6;

s=70 km/hr

Hence, 70 km/hr is the speed he must drive to reach baguio in 6 hours.

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Answer: We can start by using the concept of rates.

At 12:00 noon, oil is poured into the top of the tank at a rate of 2.1 gallons per minute and the valve at the bottom empties at a rate of 1.9 gallons per minute.

As the tank is empty at the beginning, the rate at which the oil is poured into the tank is equal to the rate at which the oil is leaving the tank.

2.1 gallons per minute - 1.9 gallons per minute = 0.2 gallons per minute

As the tank does not fill until 12:00 the next day, the difference between the rate of oil that enters the tank and the rate of oil that leaves the tank is equal to zero over the time frame.

0.2 gallons per minute * 1440 minutes = 288 gallons

So, the tank holds 288 gallons.

Step-by-step explanation:

Distribution (1) and (2) can be compared by looking at their medians and IQRs. If the medians and IQRs are different, the distributions can be assumed to be different.

When comparing two distributions, one of the most useful metrics is their medians and IQRs. The median is the midpoint of the data set, while the IQR is the difference between the first quartile and the third quartile. If the medians and IQRs are different, it suggests that the two distributions are different. For example, if one distribution has a lower median than the other, it suggests that the majority of data points in the lower distribution are lower than the majority of data points in the higher distribution. Similarly, if the IQR of one distribution is greater than the other, it suggests that the data points in the higher distribution are spread further apart than the data points in the lower distribution. Comparing the medians and IQRs of two distributions can help to determine if the distributions are different

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Answer: The local restaurant states that its occupancy, the number of people inside the restaurant, can be at most 65. This includes both people working in the restaurant and the customers. When they open for the lunch rush, there are 8 employees inside the restaurant and customers start to come in at a steady rate of 3 customers per minute.

Given this information, we can set up the following inequality:

Occupancy = 8 + 3t (where t is the number of minutes after the restaurant opens) ≤ 65

This inequality represents the maximum occupancy of the restaurant, taking into account the initial number of employees (8) and the rate at which customers are coming in (3 per minute). To find the number of minutes after the restaurant opens until it reaches its maximum occupancy of 65, we can solve for t:

8 + 3t ≤ 65

3t ≤ 57

t ≤ 19

So, the restaurant will reach its maximum occupancy of 65 people after 19 minutes of being open for the lunch rush.

Step-by-step explanation:

As per the concept of algorithm, the reasonable number of steps is 2n.

The term algorithm in math is defined as the procedure for solving a mathematical problem in a finite number of steps that frequently involves repetition of an operation.

Here we have given that an algorithm has n number of steps.

Here we need to find the reasonable number of steps.

While we know the definition of algorithms, then her we need to understand the term reasonable time that is defined as  the algorithms with a polynomial efficiency or lower are said to run in a reasonable amount of time.

So, the reasonable number of steps is 2n where n refers the number of steps.

Complete Question:

If an algorithm has n number of steps, then in find out the reasonable number of steps?

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Using algebraic expressions, The largest possible frame width x is 5.

When addition, subtraction, multiplication, division, and other mathematical operations are performed on variables and constants, the result is a mathematical statement known as an algebraic expression.

Let's say James and Natalie were playing with matchsticks when they had the idea to create number patterns with them.

James made the number 4 with four matchsticks. In order to create a pattern with two 4s, Natalie added three extra matchsticks. They understood that they could keep adding three matchsticks each round to make an additional "four" by doing so. They deduced from this that, generally speaking, in order to create a pattern with n number of 4s, you need 4+ 3(n-1) sticks. We refer to 4+ 3(n-1) as an algebraic expression in this case.

Let the border's width be x, making the overall frame equal to (10 + 2x)(10 + 2x).

area of boundary hence is (10+2x)(10+2x) - 100

but (10+2x)(10+2x) - 100 = 300

(10+2x)² = 400

10+2x = 20

2x = 10

x = 5

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Answer:
i = [tex]\sqrt{-1}[/tex]
i^2 = -1

Step-by-step explanation:

I hope this helps you out :D

since i = square root of -1 when you square i (i^2) you get rid of the parenthesis.

A probability tree is a visual representation of the possible outcomes of an event or series of events. In this case, the event takes a marble out of the bag.

The first step in building a probability tree is to create a starting point that represents the first state of the problem. In this case, the starting point is a bag containing 3 green marbles and 5 white marbles.

The next step is to branch from the starting point and show the possible results of the first event. This includes taking out  the marble out of the bag. The probability of getting a green marble is 3/8 and the probability of getting a white marble is 5/8.

After the first event, the issue status changes. In this case, the bag contains 2 green marbles and 4 white marbles.

The next step is to branch out from the state after the first event and show the possible outcomes of his second event involving pulling another marble out of his pocket. The probability of getting a green marble is 2/6 and the probability of getting a white marble is 4/6. The final step is to label the endpoints of the tree with the possible outcomes of the problem and the probabilities of each outcome.

The probability tree starts with an sack of 3 green marbles and 5 white marbles, as shown in the design showing the possible outcomes of selecting a marble, the new state of the sack after each selection, and the probability of each outcome

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Answer:

y = -2/5x - 1

Step-by-step explanation:

Parallel lines have the same gradient/slope.

m1=m2

So the equation must have m as -2/5

So we can sub that and we will get:

y = -2/5x + c

However, we still do not have c aka y intercept

So we can sub the point (5,-3) into the current equation and we will have:

-3 = -2/5(5) + c

Do the algebra and you'll get -1 = c

Sub that into the equation:

y = -2/5x -1

The average rate of change of the function f(x) = 4 · x² - 7 for the interval [1, b] is m = - 4 · (b + 1).

Graphically speaking, the average rate of change associated to a function is represented by a secant line formula:

m = [f(b) - f(1)] / (b - 1)

If we know that f(x) = 4 · x² - 7, then the average rate of change is:

m = [4 · b² - 7 - 4 · 1² + 7] / (1 - b)

m = 4 · (b² - 1) / (1 - b)

m = - 4 · (b² - 1) / (b - 1)

m = - 4 · (b + 1)

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Answer:

[tex]a_{n}[/tex] = 3n + 12

Step-by-step explanation:

the nth term ( explicit formula ) of an arithmetic sequence is

[tex]a_{n}[/tex] = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

here a₁ = 15 and d = a₂ - a₁ = 18 - 15 = 3 , then

[tex]a_{n}[/tex] = 15 + 3(n - 1) = 15 + 3n - 3 = 3n + 12

The expression [tex](\sqrt[3]{x^4})^5[/tex] with a rational exponent is given as follows:

x^(20/3).

The general format of the exponential expression is given as follows:

[tex]a^{\frac{n}{m}}[/tex]

To obtain the radical form, we have that:

Hence the equivalent expression is given as follows:

[tex]a^{\frac{n}{m}} = \sqrt[m]{a^n}[/tex]

The parameters for this problem are given as follows:

m = 3, n = 4, a = x.

Hence the equivalent expression is given as follows:

(x^(4/3))^5.

Applying the power of power rule, we multiply the exponents, hence the equivalent expression is of:

x^(20/3).

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A multiplication equation to show an equivalent fraction of 15 using fifteenths would gives 10/75

A fraction consisting of a quotient and remainder is a mixed fraction. we can convert the mixed fraction to improper fraction by first dividing the numerator by denominator and then taking the quotient as whole number and remainder as the numerator of proper fraction keeping the denominator same.

We have to write an equivalent fraction of 15 using fifteenths.

2/15

So let multiply by 5 .

2 / 15 x  5/ 5 = 10/75

Thus, 10/75 is equivalent expression.

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