7. (x³ +7x² + 2)÷(x-1)

Answer:

y = 90°

x = 63°

Step-by-step explanation:

The unknown y angle is a right angle, meaning it is a 90° angle.

We know a triangle is 180°. We know 2 angles one is 90°, one is 27°, so to find the other missing angle

We Take

180 - (27 + 90) = 63°

So,  x = 63°      y = 90°

Answer:

18+m=24, 6

Step-by-step explanation:

You will get the first part by understanding that 24 is the whole and 18 is the part. Part + the other part, m, is the whole. You will then solve this by isolating the variable m, and subtracting 18 on both sides of the equation. Since 24-18=6, that is the final answer.

The total amount repaid by Melvin was $1,440. Which is option C. $1,440.

Melvin borrowed $1,200 for furniture

His monthly payments were $60 for 24 months

To find the total amount repaid, we can multiply the monthly payment with the number of months. We can write this in mathematical terms as:

The total amount repaid = Monthly payment × Number of months

Using the above formula, let's calculate the total amount repaid by Melvin.

Total amount repaid = $60 × 24= $1,440

Therefore, the total amount repaid by Melvin was $1,440. Therefore, the correct option is C. $1,440.

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Answer:

Let's use algebra to solve this problem.

Let's start by defining some variables to represent the number of cows and chickens the farmer sold. Let c be the number of cows and h be the number of chickens. We know that the farmer sold a total of 11 animals, so:

c + h = 11 (Equation 1)

We also know that the total amount of money the farmer received was $2475. The amount he received from selling cows was $350 times the number of cows, or 350c. The amount he received from selling chickens was $75 times the number of chickens, or 75h. So:

350c + 75h = 2475 (Equation 2)

Now we have two equations with two unknowns. We can solve for one variable in terms of the other in the first equation:

c + h = 11

c = 11 - h

We can substitute this expression for c into the second equation:

350c + 75h = 2475

350(11 - h) + 75h = 2475

Simplifying and solving for h:

3850 - 350h + 75h = 2475

-275h = -1375

h = 5

So the farmer sold 5 chickens. We can substitute this value of h back into the first equation to find c:

c + h = 11

c + 5 = 11

c = 6

Therefore, the farmer sold 6 cows and 5 chickens.

The function f(x) = x⁴ - 2x³ + 5 is concave downwards on the interval (0,1) and concave upwards everywhere else.

To determine the concavity of the function f(x) = x⁴ - 2x³ + 5, we need to find its second derivative f''(x).

f(x) = x⁴ - 2x³ + 5

f'(x) = 4x³ - 6x²

f''(x) = 12x² - 12x

The concavity of the function is determined by the sign of its second derivative.

If f''(x) > 0 for all x, the function is concave upwards.

If f''(x) < 0 for all x, the function is concave downwards.

Let's solve for f''(x) = 12x² - 12x:

12x² - 12x = 0

12x(x - 1) = 0

x = 0 or x = 1

When x < 0, f''(x) > 0, so the function is concave upwards.

When 0 < x < 1, f''(x) < 0, so the function is concave downwards.

When x > 1, f''(x) > 0, so the function is concave upwards.

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Decibel is a relative measure of signal loss or gain and is used to measure the logarithmic loss or gain of a signal.

What is a decibel?

Decibel, also known as dB, is a logarithmic unit that measures the intensity of a sound or the strength of an electrical or electromagnetic signal. A decibel measures the relative amplitude of a sound or signal, rather than its absolute magnitude. Because decibels are logarithmic, they are used to express both large and small differences in amplitude. A difference of 1 decibel corresponds to a power ratio of approximately 1.26 to 1.

Logarithmic measure: A logarithmic scale is a scale that has a constant ratio between successive values. Decibels, for example, are a logarithmic scale. The decibel scale is used to measure the amplitude of sound waves and electrical or electromagnetic signals. Because decibels are logarithmic, they can be used to express a wide range of signal levels, from very weak to very strong.

Relative measure: Relative measure is a measure that compares one value to another. It is used in a variety of fields, including statistics, physics, and engineering. Decibels are a relative measure because they compare one signal to another. They are used to express the relative gain or loss of a signal, rather than its absolute magnitude.

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Solving for Cartesian product n(B), we have n(B) = 72 / 24 = 3.

The Cartesian product is a mathematical operation that takes two sets and produces a set of all possible ordered pairs of elements from both sets.

In other words, if A and B are two sets, their Cartesian product (written as A × B) is the set of all possible ordered pairs (a, b) where a is an element of A and b is an element of B.

For example, if A = {1, 2} and B = {3, 4}, then A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}.

By the question.

We know that n (Ax B) represents the number of elements in the set obtained by taking the Cartesian product of sets A and B.

Using the formula for the size of the Cartesian product, we have:

n (Ax B) = n(A) x n(B)

Substituting the given values, we get: 72 = 24 x n(B)

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The required answers are 19) c = 8√2 units 20) a = b =a = [tex]\frac{\sqrt{10} }{14}[/tex].

A right triangle or right-angled triangle, or more formally an orthogonal triangle, formerly called a rectangled triangle, is a triangle in which one angle is a right angle, i.e., in which two sides are perpendicular. The relation between the sides and other angles of the right triangle is the basis for trigonometry.

19) Using Pythagoras theorem

[tex]c^2=a^2+a^2[/tex]

[tex]c^2=\sqrt{a^2+a^2}[/tex]

[tex]c=\sqrt{8^2+8^2}[/tex]

c = 8√2 units

20) It is isosceles triangle,So a = b

Now

[tex]$Sin(45) =\frac{a}{\frac{\sqrt{5}}{7} }[/tex]

a = [tex]\frac{\sqrt{10} }{14}[/tex]

Thus, required answers are 19) c = 8√2 units 20) a = b =a = [tex]\frac{\sqrt{10} }{14}[/tex].

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The given statements are True , False, False, True and True.

They are elaborated in points below:

(a) True:

Let x and y be even integers.

Then, by definition, x = 2a and y = 2b for some integers a and b.

Adding these equations, we get x + y = 2a + 2b = 2(a + b), which is an even integer. Therefore, the statement is true.

(b) False:

A counterexample is x = 1 and y = 3. Then x + y = 4, which is even, but neither x nor y is even.

Therefore, the statement is false.

(c) False:

A counterexample is x = -2 and y = 4. Then x² = (-2)²= 4 and y = 4² = 16, but x ≠ y. Therefore, the statement is false.

(d) True:

Since x < y, we have y - x > 0. Multiplying both sides by y + x, we get y² - x² > 0.

Rearranging the terms, we have x² < y².

Since x and y are both non-negative, we can take the square root of both sides to get x < y.

Therefore, the statement is true.

(e) True:

Since x and y are positive, we can take the square root of both sides of the inequality x < y to get √x < √y.

Then, multiplying both sides by √x + √y, we get x + 2√xy + y > x + y.

Since x + y is positive, we can divide both sides by x + y to get 1 + 2√(xy)/(x+y) > 1.

Rearranging the terms, we get √(xy) < (x + y)/2. Squaring both sides, we get xy < (x + y)²/4.

Simplifying, we get x²/4 - xy + y²/4 > 0. Rearranging the terms, we get (x - y/2)² > 0, which is always true since the square of any non-zero number is positive. Therefore, the statement is true.

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The equation that can be used to find d, the diameter of a new soccer ball, is: |d - 8.65| = 0.05

What is diameter ?

Diameter is the distance across a circle passing through its center, and it is defined as the length of a straight line passing through the center of a circle and connecting two points on the circumference of the circle. In the case of a soccer ball, the diameter is the distance between two opposite points on the surface of the ball, passing through its center.

Radius is a term used in geometry to refer to the distance from the center of a circle to any point on its circumference. It is defined as half of the diameter of a circle. In the case of a soccer ball, the radius is the distance from the center of the ball to its surface. The radius is related to the diameter by the formula:

radius = diameter / 2

The minimum possible diameter of a soccer ball is:

d = 8.65 - 0.05 = 8.60 inches

The maximum possible diameter is:

d = 8.65 + 0.05 = 8.70 inches

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If triangle ABC has points A(2, -4) B(-3, 1) C(-2, -6) and you perform the following transformations, B' would be located at B' (3, 2).

In Geometry, a reflection over or across the y-axis is represented and modeled by this transformation rule (x, y) → (-x, y). This ultimately implies that, a reflection over or across the y-axis would maintain the same y-coordinate (y-axis) while the sign of the x-coordinate (x-axis) would change from positive to negative or negative to positive.

By applying a reflection over the y-axis to the coordinate of the given point B (-3, 1), we have the following coordinates:

Coordinate B = (-3, 1)   →  Coordinate B' = (-(-3), 1) = (-3, 1).

Next, we would apply a rotation of 90° clockwise as follows;

(x, y)                               →            (y, -x)

Coordinate B' = (-3, 1) → Coordinate B' = (1, (-3)) = (1, 3)

Lastly, we would apply a translation (x + 2, y - 1) as follows:

Coordinate B' = (1, 3) → (1 + 2, 3 - 1) = B' (3, 2).

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Let n be a positive integer. We can use the properties of modular arithmetic to calculate this remainder. Let's start with a = (32n + 4)-1 mod 9. We can rewrite this as a = 9 - (32n + 4)-1 because 9 = 0 mod 9.

We can use Fermat's Little Theorem to calculate (32n + 4)-1. This theorem states that (32n + 4)-1 mod 9 = (32n + 4)8 mod 9.

Using the identity (a + b)n mod m = ((a mod m) + (b mod m))n mod m, we can simplify the equation to (32n mod 9 + 4 mod 9)8 mod 9.

32n mod 9 = 0, so (32n mod 9 + 4 mod 9)8 mod 9 = 48 mod 9 = 1.

Finally, a = 9 - 1 = 8 mod 9, so the remainder when a is divided by 9 is 8.

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Jerry will have $1825 in his account after 3 years and Jerry will have $6125 in his account after 10 year when compounded.

Simple interest is computed just using the principle, which is the initial sum borrowed or put into an investment. The interest rate is constant throughout time and solely applies to the principal sum. Short-term loans or investments frequently employ simple interest.

The given situation can be modeled as a linear equation given by:

y = mx + c

For Jerry we have:

y = 50x + 125

For 3 years = 36 months we can substitute x = 36:

y = 50(36) + 125

y = 1825

For x = 10:

y = 50(120) + 125

y = 6125

Hence, Jerry will have $1825 in his account after 3 years and Jerry will have $6125 in his account after 10 year.

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Answer:

not sure if this sign � was important did it the best way I could

Step-by-step explanation:

To solve this problem graphically, we will first set up a system of inequalities based on the given information:

x ≥ 8 (Fatoumata must work at least 8 hours lifeguarding)

y ≤ 12 - x (Fatoumata can work at most 12 total hours)

15x + 10y ≥ 140 (Fatoumata must earn a minimum of $140)

To graph these inequalities, we can plot the points (8,0), (12,0), and (0,14) on a coordinate plane and draw lines connecting them. The line between (8,0) and (12,0) represents the constraint on the number of hours Fatoumata can work, while the line between (8,0) and (0,14) represents the constraint on the amount of money she must earn. The shaded region that satisfies all three inequalities is the feasible region.

To find one possible solution, we can pick any point within the feasible region. One such point is (8,6), which represents working 8 hours lifeguarding and 6 hours tutoring. This point satisfies all three inequalities:

x ≥ 8 is true since x = 8

y ≤ 12 - x is true since y = 6 ≤ 12 - 8

15x + 10y ≥ 140 is true since 15(8) + 10(6) = 180 ≥ 140

Therefore, one possible solution is for Fatoumata to work 8 hours lifeguarding and 6 hours tutoring to earn at least $140 while not exceeding 12 total hours worked.

The approximate slope and y-intercept of the line of best fit is y = -0.17x + 33.3.

In Mathematics, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁) or [tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)[/tex]

Where:

At data point (20, 20), a linear equation in slope-intercept form for this line can be calculated by using the point-slope form as follows:

[tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)\\\\y - 20 = \frac{(10- 20)}{(80-20)}(x -20)[/tex]

y - 20 = -1/6(x - 20)

y = -x/6 + 20/6 + 20.

y = -x/6 + 200/6

y = -0.17x + 33.3

In this context, we can reasonably infer and logically deduce that a linear function that represent the line of best fit in slope-intercept form is y = -0.17x + 33.3.

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The solution to the equation √(1 + 25/144) = 1 + x/12 is x = 5/6. This was achieved by simplifying the left side of the equation and isolating x on one side.

To solve the equation √(1 + 25/144) = 1 + x/12, we start by simplifying the left side of the equation. The expression inside the square root can be simplified to (144 + 25)/144 = 169/144. Taking the square root of this fraction gives us √(169/144) = (13/12).

Next, we subtract 1 from both sides of the equation to isolate x on one side: (13/12) - 1 = x/12. This simplifies to 1/12 = x/12.

Finally, we multiply both sides by 12 to solve for x: x = (1/12)*12 = 5/6.

So the solution to the equation √(1 + 25/144) = 1 + x/12 is x = 5/6.

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Therefore, the transfer function of the system is:

G(s) = Y(s) / U(s) = U(s) / (s^3 + 0.25s + 1.25)

In summary, the differential equation depicting the dynamics of a fourth Single-Input-Single-Output (SISO) system is y'''(t) + 0.25y'(t) + 1.25y(t) = u^2(t), and the transfer function of the system is G(s) = U(s) / (s^3 + 0.25s + 1.25).

The differential equation describing the dynamics of a single-input-single-output (SISO) system is given by ddy(t) + ay'(t) + by(t) = cdu(t), where y(t) is the output of the system, u(t) is the input of the system, and d, a, b, and c are constant coefficients representing the system dynamics.

For the given differential equation, which is the dynamics of four Single-Input-Single-Output (SISO) systems, we have:

[tex]y(t) + ay'(t) + by''(t) + cy'''(t) = u(t)[/tex]
Here, we have the third-order differential equatior : [tex]y'''(t) + 0.25y'(t) + 1.25y(t) = u^2(t)[/tex]

The given equation is a third-order, linear, time-invariant (LTI) differential equation. In control theory, a differential equation that relates the input u(t) to the output y(t) of a system is known as the transfer function.

The transfer function of the given system can be found by applying the Laplace transform to the differential equation:

s^3Y(s) + 0.25sY(s) + 1.25Y(s) = U^2(s)

where Y(s) and U(s) are the Laplace transforms of y(t) and u(t), respectively.

Solving for Y(s), we get:

Y(s) = U^2(s) / (s^3 + 0.25s + 1.25)

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In order for each of the 25 students in the class to get a full glass of milk during the break, six bottles of milk are required.

Each student in a class of 25 drinks a full 210 millilitre glass of milk, hence the amount of milk consumed overall during the break is:

25 students times 210 millilitres each equals 5250 millilitres.

Milk comes in 1000 millilitre bottles, thus to determine how many bottles are needed, divide the entire amount eaten by the volume of milk in each bottle.

5.25 bottles are equal to 5250 millilitres divided by 1000 millilitres.

We must round up to the nearest whole number because we are unable to have a fraction of a bottle. This results in:

6 bottles in 5.25 bottles

In order for each of the 25 students in the class to get a full glass of milk during the break, six bottles of milk are required.

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Hello,

3m + 5 > 2(m - 7) =

3m + 5 > 2m - 14

3m - 2m > - 14 - 15

x > - 29

Step-by-step explanation:

3m±2m>-14-5

5m>-19

m>-19/5

m>3.8

The answer is option C: 109 shares. If the broker charges a set free of $7 per transaction. If the shares are worth $7.69/sh and you have $850 to invest 109 shares can you purchase.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

To calculate the number of shares you can purchase, you need to subtract the broker's fee from the total amount you have to invest and then divide that by the share price:

$850 - $7 = $843 (the amount available to purchase shares)

$843 ÷ $7.69/share = 109.48

Since you cannot purchase a fraction of a share, you would be able to purchase 109 shares.

Therefore, the answer is option C: 109 shares.

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The probability of 0.0475 is less than 0.05, so it is unlikely to happen. Hence, it would be unusual if less than 30% of the sampled teenagers owned smartphones.

a) Find the probability that the proportion of the sampled teenagers who own a smartphone is between 0.35 and 0.45b) Find the probability that less than 45% of sampled teenagers own smartphonesc) Would it be unusual if less than 30% of the sampled teenagers owned smartphones?a)Find the probability that the proportion of the sampled teenagers who own a smartphone is between 0.35 and 0.45We know that the proportion of teenagers aged 12-17 who own smartphones is 0.38. Now, we need to find the probability that the proportion of sampled teenagers who own smartphones is between 0.35 and 0.45.The sample size is 120.Let P be the probability that a teenager has a smartphone. The population mean is P = 0.38, while the standard deviation is σ =√((P (1 - P)) /n).P (0.38) and n (120) are given, so σ is calculated as follows:σ=√((P (1 - P)) /n) =√((0.38 (1 - 0.38)) /120) =√((0.38 × 0.62) /120) =0.048Now, let us define the Z scores for both sides, i.e. for 0.35 and 0.45.z1 = (0.35 - 0.38) /0.048 = -0.625z2 = (0.45 - 0.38) /0.048 = 1.458Hence, P(0.35 < p < 0.45) is equal to P(-0.625 < z < 1.458).The probability can be calculated using standard normal tables, which gives 0.5763. Therefore, the probability that the proportion of the sampled teenagers who own a smartphone is between 0.35 and 0.45 is 0.5763.b) Find the probability that less than 45% of sampled teenagers own smartphonesLet P be the probability that a teenager has a smartphone. The population mean is P = 0.38, while the standard deviation is σ =√((P (1 - P)) /n).P (0.38) and n (120) are given, so σ is calculated as follows:σ=√((P (1 - P)) /n) =√((0.38 (1 - 0.38)) /120) =√((0.38 × 0.62) /120) =0.048Now, let us define the Z score as follows:z = (0.45 - 0.38) /0.048 = 1.458We must find the probability that a randomly selected teenager from the sample has less than 0.45 probability of owning a smartphone. This probability can be calculated using a standard normal table. P(z < 1.458) = 0.9265, thus, the probability that less than 45% of sampled teenagers own smartphones is 0.9265.c) Would it be unusual if less than 30% of the sampled teenagers owned smartphones?For this, we need to find the Z-score of 0.30.z = (0.30 - 0.38) /0.048 = -1.667The probability is obtained using a standard normal table. The probability is 0.0475. The probability of 0.0475 is less than 0.05, so it is unlikely to happen. Hence, it would be unusual if less than 30% of the sampled teenagers owned smartphones.

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Answer:

A. Scale factor is 2

B. Scale factor is 1/2

Step-by-step explanation:

Calculating the expected value of payout using the probabilities given, we can bet on blue for all 10 races, bet on red for all 10 races and bet on purple for all 10 races

To determine the most efficient bets to increase your money during the 10 rounds, we need to consider the probability and potential payout for each racecar. We can start by calculating the expected value of each racecar's payout:

Blue: 0.3 x 2 = 0.6

Red: 0.25 x 2.5 = 0.625

Purple: 0.18 x 3 = 0.54

Yellow: 0.1 x 5 = 0.5

Black: 0.08 x 7 = 0.56

Green: 0.05 x 10 = 0.5

Orange: 0.03 x 15 = 0.45

Pink: 0.01 x 50 = 0.5

Based on the expected value of payout, the most efficient bets would be on the racecars with the highest expected value. However, we also need to consider the probability of each racecar winning or placing in the top 3.

To maximize our chances of winning, we should bet on the racecars with the highest probability of winning or placing in the top 3, even if their expected payout is not the highest.

Assuming that if one racecar is likely to win but doesn't win, they are statistically most likely to come second, and so on, we can rank the racecars in terms of their probability of winning or placing in the top 3:

1. Blue - 30% chance of winning

2. Red - 25% chance of winning

3. Purple - 18% chance of winning

4. Black - 8% chance of winning

5. Yellow - 10% chance of winning

6. Green - 5% chance of winning

7. Orange - 3% chance of winning

8. Pink - 1% chance of winning

Based on this ranking, the most efficient bets would be as follows:

1. Bet on Blue for all 10 races. The expected payout is not the highest, but the probability of winning is the highest at 30%.

2. Bet on Red for all 10 races. The probability of winning is the second-highest at 25%.

3. Bet on Purple for all 10 races. The probability of winning is the third-highest at 18%.

Betting on these three racecars will give you the highest chance of winning or placing in the top 3, and therefore, the most efficient bets to increase your money. However, keep in mind that gambling always carries risks, and there is no guarantee of winning.

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The probability that more heads are tossed using coin A than coin B is 5/16.

The given data is: Coin A is tossed three times and coin B is tossed two times. We have to find the probability that more heads are tossed using coin A than coin B.

P(E) = Number of favorable outcomes/ Total number of possible outcomes

Coin toss:

There are two possible outcomes in a coin toss, Head or Tail. The probability of getting a head in a coin toss is

1/2 = 0.5.

Therefore, the probability of getting a tail in a coin toss is also 1/2 = 0.5.

Let's calculate the possible outcomes when coin A is tossed three times.

There are 2 possible outcomes when one coin is tossed.

Number of possible outcomes when three coins are tossed = 2 * 2 * 2 = 8

Likewise, the possible outcomes when coin B is tossed two times are:

The number of possible outcomes = 2 * 2 = 4

Therefore, the total number of possible outcomes = 8 * 4 = 32

Now, we will find out the cases where the number of heads is more when coin A is tossed three times.

HHH HHT HTH HTT THH THT TTH TTT HHT HTT THT TTT TTH TTT HTT TTT THT TTT TTT TTT

Therefore, the number of times when more heads are obtained when coin A is tossed three times is 10. (We have to exclude the case when there is an equal number of heads.)

Therefore, the required probability is: P = Number of favorable outcomes/ Total number of possible outcomes

P = 10/32P = 5/16

Therefore, the probability that more heads are tossed using coin A than coin B is 5/16.

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Answer:

Step-by-step explanation:

42

Therefore, the compound inequality for the diameter of the washers is: 3.150 ≤ d ≤ 3.240.

In mathematics, an inequality is a statement that compares two values or expressions, indicating that one is greater than, less than, or equal to the other. The symbols used to represent inequalities are:

">" which means "greater than"

"<" which means "less than"

"≥" which means "greater than or equal to"

"≤" which means "less than or equal to"

Inequalities can be solved by applying algebraic techniques, such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same number. The solution to an inequality is a range of values that satisfy the inequality.

Here,

The formula for the circumference of a circle in terms of its diameter is:

C = πd

where π (pi) is approximately 3.14.

We are given that the acceptable range for the circumference of the washer is 9.9 ≤ C ≤ 10.2 centimeters. Substituting C = 3.14d into this inequality, we get:

9.9 ≤ 3.14d ≤ 10.2

Dividing all sides of the inequality by 3.14, we obtain:

3.15 ≤ d ≤ 3.24

Rounding to three decimal places, the corresponding interval for the diameters of the washers is:

3.150 ≤ d ≤ 3.240

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In the following question, among the conditions given, It can be given by Ymax = H(N*)So the maximum sustainable yield for a population governed by a Gompertz model and subjected to harvesting (either constant yield or constant effort) can be calculated using the above formula.

To find the maximum sustainable yield for a population governed by a Gompertz model and subjected to harvesting (either constant yield or constant effort) we will start with the formula of the Gompertz model that is, IN = r Nlog(K)Solution: Given that the population is governed by a Gompertz model = r Nlog(K)We know that the harvesting term that is responsible for the overexploitation of the resources, hence we can write the Gompertz model as dN/dt = rN[1 - N/K] - H(x)WhereH(x) can be a constant yield or constant effort in the harvesting term.

To find the maximum sustainable yield, we will use the concept of steady-state or equilibrium population level. Steady-state condition:dN/dt = 0Solving this equation gives us the steady-state population level, say N*.Now we need to calculate the maximum amount of resources that can be harvested sustainably. This maximum amount of resources harvested sustainably is called the maximum sustainable yield (MSY). It can be given by Ymax = H(N*)So the maximum sustainable yield for a population governed by a Gompertz model and subjected to harvesting (either constant yield or constant effort) can be calculated using the above formula.

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Answer:

Step-by-step explanation:

To find the normal plane and osculating plane, we first need to find the required derivatives.

x = 5t, y = t^2, z = t^3

dx/dt = 5, dy/dt = 2t, dz/dt = 3t^2

So, the velocity vector v and acceleration vector a are:

v = <5, 2t, 3t^2>

a = <0, 2, 6t>

Now, let's evaluate them at t = 1 since the point (5, 1, 1) is given.

v(1) = <5, 2, 3>

a(1) = <0, 2, 6>

The normal vector N is the unit vector in the direction of a:

N = a/|a| = <0, 1/√10, 3/√10>

Using the point-normal form of the equation for a plane:

normal plane equation = 0(x-5) + 1/√10(y-1) + 3/√10(z-1) = 0

Simplifying this equation we get:

5x + 2y + 3z = 30

The osculating plane can be found using the formula:

osculating plane equation = r(t) · [(r(t) x r''(t))] = 0

where r(t) is the position vector, and x is the cross product.

At t = 1, the position vector r(1) is <5, 1, 1>, v(1) is <5, 2, 3>, and a(1) is <0, 2, 6>.

r(1) x v(1) = <-1, 22, -5>

r(1) x a(1) = <12, -6, -10>

v(1) x a(1) = <-12, 0, 10>

Substituting these values into the formula, we get:

osculating plane equation = (x-5, y-1, z-1) · <12, -6, -10> = 0

Simplifying this equation we get:

3x - 15y + 5z = 5

Therefore, the equations for the normal plane and osculating plane at (5, 1, 1) are:

(a) 5x + 2y + 3z = 30

(b) 3x - 15y + 5z = 5

Using trigonometry, the height of the lighthouse from the top of the cliff is approximately 617.38 meters.

Trigonometry is a tool we can utilize to tackle this issue. Let's use "h" to represent the lighthouse's height and "x" to represent the distance between both the boat and the cliff's edge. Next, we have:

tan(24.7º) = h/x (1)

tan(28.4º) = (h + y)/x (2)

where "y" is the height of the cliff.

We want to find "h + y". To eliminate "x", we can use equation (1) to express "x" in terms of "h", and substitute it into equation (2):

x = h/tan(24.7º)

tan(28.4º) = (h + y)/(h/tan(24.7º))

Simplifying and solving for "h", we get:

h = y/tan(28.4º - 24.7º)

Now we can substitute this expression for "h" into equation (1) to find "y":

tan(24.7º) = y/(y/tan(28.4º - 24.7º))

Simplifying and solving for "y", we get:

y = 750 tan(24.7º) / (tan(28.4º - 24.7º))

y ≈ 237.78 meters

Therefore, the height of the lighthouse from the top of the cliff is:

h + y ≈ y/tan(28.4º - 24.7º) + y ≈ 617.38 meters.

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The question is -

A lighthouse is located at the edge of a cliff, as shown. A boat at sea level is 750 meters away from the base of the cliff. The angle of elevation between sea level and the base of the lighthouse measures 24.7º. The angle of elevation between sea level and the top of the lighthouse measures 28.4°. Find the height of the lighthouse from the top of the cliff.