A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 100cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds.
Determine the spring constant k.
k = ? Newtons / meter
Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y,y′,y′′,t.)
Differential equation: ?
Initial conditions: y(0) = ? and y′(0) = ?
Solve the initial value problem for y(t).
y(t) = ?
Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 ≤ t < [infinity]. If there is no such maximum, enter NONE.
maximum excursion = ? meters

Check the picture below.

so if we just set h = 0, we'll get the "t" when that happened

[tex]\stackrel{h}{0}=-16t^2+29t+6\implies 0=-(16t^2-29t-6)\implies 0= 16t^2-29t-6 \\\\\\ 0=(t-2)(16t+3)\implies t= \begin{cases} ~~ ~ 2 ~~ \checkmark\\ -\frac{3}{16} ~~ \bigotimes \end{cases}[/tex]

now, let's notice that we get two valid values for "t", however the negative doesn't apply in this case, because we can't quite have negative seconds for the object in motion.

Answer:

2

Step-by-step explanation:

h(t) = -16t² + 29t + 6

h(2) = -16 * 2² + 29 * 2 + 6

h(2) = 0

t = 2 second's

As a result, the Styrofoam collar has a volume of roughly 179.594 cubic inches.

The quantity of space occupied by a three-dimensional object is measured by its volume. Units like cubic meters (m3), cubic centimeters (cm3), or cubic inches (in3) are frequently used to quantify it. Depending on the shape of the item, different formulas can be used to determine its volume. For instance, the volume of a cube can be calculated by multiplying its length, breadth, and height, while the volume of a cylinder can be calculated by dividing the base's area (typically a circle) by the cylinder's height.

given

We must apply the calculation for the volume of a cone's frustum in order to determine the volume of the Styrofoam collar:

[tex]V = (1/3)\pi h(R^2 + Rr + r^2)[/tex]

where h is the height of the frustum, r is the small radius, and R is the large radius.

Given the numbers, we can determine:

R = 5 in.

3 centimeters is r.

24 inches tall

With these numbers entered into the formula, we obtain[tex]V = (1/3)\pi (24)(5^2 + 5*3 + 3^2)\\\\ 179.594 cubic inches[/tex]

As a result, the Styrofoam collar has a volume of roughly 179.594 cubic inches.

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Grace need to buy the jump rope, the game, and the ball $8.

$7 will get you a jump rope.

$10 will get you a board game.

$4 will get you a playground ball.

total amount to be spent: $7 + $10 + $4 = $21

She has ten dollars.

$3 was all her uncle gave her.

13 dollars are all she has.

She needed $8, therefore 21 - 13 = $8.

a sum of money awarded as compensation, a bounty, or to cover costs. a wage that comes with a cost-of-living supplement. especially: a regular amount set aside for household or personal costs.

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Answer:

2p - 12

Step-by-step explanation:

(24p - 10) + (-22p - 2)

24p - 10 - 22p - 2

2p - 12

So, the answer is 2p - 12

The proportion of students who can roll their tongues will be estimated and the margin of error for a 95 percent confidence interval for the true proportion of tongue rollers among students will be determined. There were 317 tongue rollers out of a sample of 400 students.

      As a result, the sample proportion is 317/400 = 0.7925.

      We'll compute the margin of error next. The margin of error (E) for a 95 percent confidence interval is:

                E = zα/2 * sqrt[p(1 - p) / n]

                              where zα/2 is the z-score that corresponds to the level of confidence α/2, p is the sample proportion, and n is the sample size.

               E = 1.96 * sqrt[0.7925 * (1 - 0.7925) / 400]E

                  = 1.96 * sqrt[0.7925 * 0.2075 / 400]E

                  = 1.96 * sqrt(0.00040875)E

                  = 1.96 * 0.0202E

                  = 0.0395

    The margin of error is approximately 0.04 or 4 percent. Hence, the correct option is 0.04.

The margin of error for a 95% confidence interval for the true proportion of tongue rollers among students is closest to 0.04

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Victor bought 10 packages of the largest-grain sandpaper.

Total money spent (20+15+9+6) = 50; total money left over (30+15+6+0) = 51. Balance plus spent always equals 50, but balance added to balance does not always equal 50. So, it is always necessary to include simply the amount spent or the expenditure rather than the balance.

Thus, we know how much he spent:

2S + (61 - 2S) = 61 - S

$1 on sandpaper with the biggest grit. We can condense this phrase as follows:

61 - S = 61 - 2S

Adding S to both sides, we get:

S = 0

Then we know that he spent:

2L + Ly = 61

Spending money on sandpaper. Additionally, since he purchased two packages of the finest sandpaper, the price of those two packages is:2S = 2L

We can substitute 2L for 2S in the first equation:

2L + Ly = 61

Simplifying, we get:

2L + L(2/3)L = 61

Multiplying both sides by 3/2, we get:

3L² = 91.5

Taking the square root of both sides, we get:

L ≈ 5.27

determine how many packages of the coarsest sandpaper Victor purchased:

2L + Ly = 61

2(5.27) + 5.27y = 61

10.54 + 5.27y = 61

5.27y = 50.46

y ≈ 9.56

Rounding to the nearest whole number, we get:

y = 10

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Answer:

54

Step-by-step explanation:

Answer:

y=54

as x=3

so y=2*x^3

y= 2*3^3

y=2*27

y=54

Therefore, the expression Marie should have used in Step 3 is option B: 2/21 ÷ ¾, which represents the division of the fraction 21/2 by 3/4.

A fraction is a way of expressing a quantity that represents a part of a whole or a ratio between two quantities. It is usually written as one number (the numerator) over another number (the denominator), separated by a horizontal line or slash. For example, the fraction 2/3 represents two parts out of three equal parts of a whole, or a ratio of two to three. Fractions can be expressed in various forms, such as proper fractions (where the numerator is smaller than the denominator), improper fractions (where the numerator is greater than or equal to the denominator), and mixed numbers (where a whole number and a proper fraction are combined). Fractions are an important concept in many areas of mathematics, including arithmetic, algebra, geometry and calculus.

by the question.

To solve this, Marie can use the following steps:

Step 1: 10 ½ ÷ ¾

Step 2: 21/2 ÷ ¾

Step 3: (21/2 ÷ 3/4) (finding the quotient of two fractions)

Step 4: (21/2) × (4/3) (dividing by a fraction is the same as multiplying by its reciprocal)

Step 5: 14

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Answer:

50 years

Step-by-step explanation:

Assuming it doesn't compound, 5,000*3 = 15,000.

5,000 * 6% = 5,000 * 0.06 = 300.

15,000 / 300 = 50 years.

The calculation shows that 9,000 scientists have seen the deep ocean through Alvin's windows; and

a total of 13,500 people have traveled in Alvin over the course of its 4,500 dives.

1) If on each of the 4,500 dives Alvin carried a new pilot and two new scientists, then the total number of scientists who have seen the deep ocean through Alvin's windows is:

4,500 dives x 2 scientists per dive = 9,000 scientists

Therefore, 9,000 scientists have seen the deep ocean through Alvin's windows.

2) To calculate the total number of people who traveled in Alvin, we can add the number of pilots and scientists on each dive and multiply by the number of dives:

4,500 dives x (1 pilot + 2 scientists)

= 4,500 x 3

= 13,500 people

Therefore, a total of 13,500 people have traveled in Alvin over the course of its 4,500 dives.

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Answer:3.79*10^14

Step-by-step explanation:

370000000000000+9000000000000=379000000000000

=3.79 x 10^14

Answer:

(3.79×10^14)

Step-by-step explanation:

sjskakakzks

The unit rate is a helpful tool for comparing speeds and calculating distances traveled in a given amount of time.

The formula for speed is: speed = distance / time where "distance" is the distance traveled by an object and "time" is the duration of travel. This formula can be used to calculate the speed of an object if the distance it has traveled and the time it took to travel that distance are known. It can also be used to calculate the distance traveled by an object if its speed and the time it traveled at that speed are known.

In the given question,

The unit rate describes Sergio's cycling speed by giving the distance he travels in a given amount of time, which is 6 miles per hour. This means that for every hour he cycles, he travels a distance of 6 miles.

By expressing Sergio's cycling speed as a unit rate, we can easily compare it to other speeds and determine how long it will take him to travel a certain distance.

For example, if Sergio increases his target speed to 8 miles per hour, we can use the unit rate to calculate how much farther he must cycle in a given amount of time.

If he wants to cycle for 2 hours, we know that he will travel 6 x 2 = 12 miles at his original speed of 6 miles per hour.

If he wants to cycle for the same 2 hours at a speed of 8 miles per hour, we can use the unit rate to calculate that he will travel 8 x 2 = 16 miles.

This means that he must cycle an additional 4 miles to reach his target distance.

Overall, the unit rate is a helpful tool for comparing speeds and calculating distances traveled in a given amount of time.

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The number of bicycles they need to sell to make equal money is 5. The amount they would make is $500.

In contrast to variable costs, which change depending on the volume of production or sales, fixed costs are expenses that remain constant. Rent, insurance, and salary are examples of fixed costs that are constant regardless of the volume of output or sales. Materials, labour, and shipping expenses are examples of variable costs that change according to the volume of output or sales.

Let us suppose the number of bicycle sold = x.

Then the equation for Jimmy is:

$250 + 50x

The equation for Tom is:

$400 + 20x

To make the same amount we equate the two equations as follows:

250 + 50x = 400 + 20x

30x = 150

x = 5

Substituting the value of x in equation 1 we have:

J = 250 + 50(5)

J = 250 + 250

J = 500

Hence, the number of bicycles they need to sell to make equal money is 5. The amount they would make is $500.

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Answer:

uh I don't even know what this is

Answer: x = 500

Step-by-step explanation:

Use the Pythagorean Theorem

a²+b²=c²

300² + 400² = x²

90000 + 160000 = x²

250000 = x²

√250000 = √x²

500 = x

hope i explained it :)

The answer is c): 17 and one-half quarts

To answer the question, we need to find the total amount of juice Sam made by adding the given quantities. However, the given quantities are in different units, which makes the addition difficult. Therefore, we need to convert all quantities to the same unit before adding them.

1 gallon = 4 quarts (since 1 gallon is equal to 128 ounces, and 1 quart is equal to 32 ounces,

thus 1 gallon = 128/32 = 4 quarts)

1 quart = 2 pints

1 pint = 2 cups

Therefore, 3 gallons = 3 x 4 = 12 quarts

2 quarts = 2 x 1 = 2 quarts

4 pints = 4 / 2 = 2 quarts

6 cups = 6 / 4 = 1.5 quarts

Now, we can add all the quantities in quarts to get the total amount of juice that Sam made:

12 + 2 + 2 + 1.5 = 17.5 quarts

Therefore, Sam made 17 and one-half quarts in all. Thus, the correct option is (c) seventeen and one half.

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Draw rectangles for each interval with a height equal to the frequency. The rectangles should be adjacent to each other and have equal width.

What is rectangle?

A rectangle is a two-dimensional shape with four sides, where the opposite sides are equal and parallel, and all angles are right angles (90 degrees).

To create a histogram, you need to first group the data into intervals or bins. You can use the intervals given in the table as a guide, or you can create your own intervals.

For this example, I will use the intervals [0-10), [10-20), [20-30), [30-40), [40-50), [50-60), [60-70), [70-80), [80-90), and [90-100).

Next, count the number of scores that fall into each interval. For example, there are no scores between 0 and 10, one score between 10 and 20, two scores between 20 and 30, and so on.

Therefore, plot the intervals on the x-axis and the frequency (or count) on the y-axis. Draw rectangles for each interval with a height equal to the frequency. The rectangles should be adjacent to each other and have equal width.

below is the histogram -

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The simplified square expression for[tex]\sqrt{3a^2/10b^6}[/tex] is [tex]|3a| / (\sqrt{10} * b^{3})[/tex].

What is square root ?

In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 x 3 = 9.

The square root is denoted by the symbol √, also known as the radical symbol. For instance, the square root of 16 is written as √16 = 4.

The square root can be used to solve various types of equations, including quadratic equations and problems involving areas and volumes. It is also used in various fields such as physics, engineering, and finance.

According to the question:
To simplify the expression [tex]\sqrt{3a^{2}/10b^6}[/tex], we can first separate the numerator and denominator inside the square root:

[tex]\sqrt{3a^2/10b^6} = \sqrt{3a^2}/\sqrt{10b^6}[/tex]

Next, we can simplify the square root of the numerator:

[tex]\sqrt{3a^2} = |3a|,[/tex] where |За| represents the absolute value of За.

Finally, we can simplify the square root of the denominator by factoring out the perfect square[tex]b^2[/tex]:

[tex]\sqrt{10b^6} = \sqrt{10} * \sqrt{b^6} = \sqrt{10} * b^{3}[/tex]

Substituting these values back into the original expression, we get:

[tex]\sqrt{3a^2/10b^6} = |3a| / \(sqrt{10} * b^3[/tex]

Therefore, the simplified expression for[tex]\sqrt{3a^2/10b^6}[/tex] is [tex]|3a| / (\sqrt{10} * b^{3})[/tex].

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Answer: x = -2

Step-by-step explanation:

Since both equations are in the y-slope form,

we can use substitution for y in finding x.

Hence,

4x=x-6

4x-x=x-x-6

Subtract x from both sides to get x on one side and integer on one side.

[tex]\frac{3x}{3} =\frac{-6}{3}[/tex]

Divide 3 to find the value of x

x=-2

Answer:

Every day, the chef consumes 5443.20 grams of butter, which is calculated using the conversion coefficients 16 oz/1 pound and 28.35 grams/1oz.

A restaurant chef uses 12 pounds of butter every day, as specified in the question.

We need to figure out how much butter the chef uses each day in grams.

Applying the conversion parameters provided, 16 oz/1 lb and 28.35 grams/1oz

According to the data provided, the needed solution is as follows: 12 lb 16 oz/1 lb 28.35 g/1 ounce 5443.20 grams

As a result, the chef consumes 5443.20 grams of butter every day.

Step-by-step explanation:

Brainliest pls

Answer:

345.6

Or 345 full cubes

Step-by-step explanation:

To answer this question we first need to find the volume of the cuboid!

To find volume we use the equation...

For the cuboid we are given the dimensions 60, 24 and 30 so we just need to multiply them...

We now need to the the volume of the cube which we can just do by cubing the value given

We now need to divide the two results together to find out how many cubes would fit...

Hope this helps, have a lovely day!

If an angle of measurement of 140° then; a circle is centered at the vertex of the angle, then the arc subtended by the angle's rays is 0.0233 cm times as long as 1/360th of the circumference of the circle. Also if a circle is centered at the vertex of the angle, and 1/360th of the circumference is 0.06 cm long then length of the arc subtended by the angle's rays 8.4 cm. Another circle is centered at the vertex of the angle then arc subtended by the angle's rays is 70 cm long,Therefore the circumference of the circle is 180 cm.

a.) To find the fraction of the circle's circumference subtended by the angle's rays, we divide the angle measure by 360 degrees:

fraction of circle's circumference = 140/360

Simplifying this fraction, we get:

fraction of circle's circumference = 7/18

To find the length of the arc subtended by the angle's rays, we multiply the fraction of the circle's circumference by the circumference of the circle. Let's call the circumference of the circle "C":

length of arc = (7/18)*C

We're also told that the length of 1/360th of the circumference is equal to 0.06 cm. So, we can write:

(1/360)*C = 0.06

Multiplying both sides by 360, we get:

C = 360*0.06 = 21.6 cm

Now, we can substitute this value of C into the expression for the length of the arc:

length of arc = (7/18)*C

length of arc = (7/18)*(21.6)

length of arc = 8.4 cm (rounded to one decimal place)

Therefore, the length of the arc subtended by the angle's rays is 8.4 cm.

b.) We're given that 1/360th of the circumference of the circle is 0.06 cm long. To find the length of the arc subtended by the angle's rays, we need to multiply 140/360 by 0.06:

length of arc = (140/360)*0.06

length of arc = 0.0233 cm (rounded to four decimal places)

Therefore, the length of the arc subtended by the angle's rays is approximately 0.0233 cm.

c.) We're told that the length of the arc subtended by the angle's rays is 70 cm. To find the circumference of the circle, we need to find the length of 1/360th of the circumference first. We can do this by dividing 70 by 1/360:

(1/360)*C = 70

Multiplying both sides by 360, we get:

C = 70*360 = 25,200 cm

Therefore, the circumference of the circle is 25,200 cm. We can also verify this by dividing the length of the arc by the fraction of the circumference subtended by the angle's rays:

length of arc = (7/18)*C

C = (18/7)*length of arc

C = (18/7)*70

C = 180 cm (rounded to one decimal place)

This is a different value than we got earlier, so we need to check our calculations. It turns out that the previous calculation was incorrect - we made a mistake when multiplying 7/18 by 21.6. The correct calculation gives us:

length of arc = (7/18)*C

length of arc = (7/18)*(21.6)

length of arc = 8.4 cm (rounded to one decimal place)

Now, we can calculate the circumference of the circle:

length of arc = (7/18)C

C = (18/7) *length of arc

C = (18/7) *70

C = 180 cm (rounded to one decimal place)

Therefore, the circumference of the circle is 180 cm.

Also, If an angle of measurement of 140° then; a circle is centered at the vertex of the angle, then the arc subtended by the angle's rays is 0.0233 cm times as long as 1/360th of the circumference of the circle.

b. A circle is centered at the vertex of the angle, and 1/360th of the circumference is 0.06 cm long.The length of the arc subtended by the angle's rays 8.4 cm

c. Another circle is centered at the vertex of the angle.

The arc subtended by the angle's rays is 70 cm long,Therefore the circumference of the circle is 180 cm.

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The value of the expression z + 3x4 using arithmetic operation where z = 15, is 27. The answer is A) 27.

The given expression is z + 3x4, where z = 15. To evaluate this expression, we substitute 15 for z and perform the multiplication. First, we multiply 3 and 4, which gives us 12. Then, we add 15 and 12 to get the final result of 27.

z + 3x4 = 15 + 3x4

= 15 + 12

= 27

Therefore, the value of the expression when z = 15, is 27. In other words, using arithmetic operation of multiplication and addition, which gives us the final answer of 27. So, the correct answer is option A).

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____The given question is incomplete , the complete question is given below:

Evaluate the expression, where z = 15

z + 3x4

A. 27

B. 32

C. 56

D. 1,304

To determine the degree of the Maclaurin polynomial required for the error in the approximation of the function f(x) = -x+1 at the indicated value of x to be less than 0.001, we can use the formula: N ≥ ln(error)/ln(absolute value of x) + 1.

For our given function, the error is 0.001, and the value of x is 0.2. Plugging these values into the formula, we get: N ≥ ln(0.001)/ln(0.2) + 1, which is equivalent to N ≥ 6.64 + 1 = 7.64. Therefore, we need the degree of the Maclaurin polynomial to be 7.64 in order for the error in the approximation of the function at the indicated value of x to be less than 0.001.

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First before two trains were [tex]990[/tex] kilometers apart, it will require [tex]5.5[/tex] hours.

Train speed is calculated as total distance traveled divided by travel time. The time it takes for two trains to pass each other is equal to (a+b) / (x+y) if the lengths of the trains, say a or b, are known and they are going at speeds of y and x, respectively.

Typically, a locomotive fueled by electricity or diesel powers trains. If there are several route networks, complicated signaling methods are used. One of the quickest forms of land transportation is rail.

[tex]distance = rate * time[/tex]

distance between trains [tex]= (85 km/h) * t + (95 km/h) * t[/tex]

distance between trains [tex]= (85 + 95) km/h * t[/tex]

distance between trains [tex]= 180 km/h * t[/tex]

Now, we can set up an equation to solve for the time it takes for the trains to be [tex]990[/tex] km apart:

[tex]180 km/h * t = 990 km[/tex]

[tex]t = 990 km / 180 km/h[/tex]

[tex]t = 5.5[/tex] hours

Therefore, it will take [tex]5.5[/tex] hours before the two trains are [tex]990[/tex] km apart.

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The answer of the given factor and multiplication term for the positive integers are 3.

Positive, negative, and zero digit whole numbers are all included in the category of integers. They can be stated without any fractional or decimal components and are a subset of the real numbers. On a number line, integers can be visualized as positive numbers to the right of zero and negative numbers to the left.

In mathematics, a multiplication term is a mathematical expression that involves the multiplication of two or more factors. The factors can be numbers, variables, or a combination of both. Multiplication terms are commonly represented using the multiplication symbol "×" or the dot "." symbol.

For example, in the expression 2x × 3y, the multiplication term is 2x × 3y, which involves the multiplication of the factors 2x and 3y.

If you want to find the smallest possible value of the given term we have that,The smallest factor of 15 is 1.

The smallest multiple of 3 is 3so, the smallest answer is 1x3 = 3.

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Answer: 5 ≥ y-2 is equivalent to y ≤ 7.

Step-by-step explanation: To check if 5 is greater than or equal to y-2, we need to isolate the variable y on one side of the inequality sign.

5 ≥ y - 2

First, we can add 2 to both sides to get rid of the subtraction of 2 on the right side:

5 + 2 ≥ y - 2 + 2

7 ≥ y

Therefore, we can see that y is less than or equal to 7 for this inequality to hold true.

Yes, every odd integer can be written as the difference of two squares.

To prove this, let k be a positive integer. Then the difference of the squares of k+1 and k is (k+1)² - k² = (k+1)(k+1) - k(k) = k² + 2k + 1 - k² = 2k + 1, which is an odd integer. Thus, every odd integer can be written as the difference of two squares.

To prove this, we first chose a positive integer, k. We then found the difference of the squares of k+1 and k to be (k+1)² - k² = (k+1)(k+1) - k(k) = k² + 2k + 1 - k² = 2k + 1. Since 2k + 1 is an odd integer, it follows that every odd integer is the difference of two squares.

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Using pythagoras' theorem in the right-angled triangle

A right-angled triangle is a polygon with 3 sides in which one angle is a right angle

Now, since we have 3 triangles, using Pythagoras' theorem in all three triangles, we have

h² + (12 - x)² = 12² - 6² (1)

Also, h² + x² = 6²  (2)

So, h² + (12 - x)² = 12² - 6²

h² + (12 - x)² = 144 - 36

h² + (12 - x)² = 108  (3)

From equation (2), h² = 36 - x²

Substituting this into equation (3), we have that

h² + (12 - x)² = 108  (3)

36 - x² + (12 - x)² = 108  (3)

Expanding the brackets, we have that

36 - x² + 144 - 24x + x² = 108

36 + 144 - 24x = 108

180 - 24x = 108

-24x = 108 - 180

-24x = -72

x = -72/-24

x = 3

Since  h² = 36 - x²

h = √(36 - x²)

So, substituting the value of x = 3 into the equation, we have that

h = √(36 - x²)

h = √(36 - 3²)

h = √(36 - 9)

h = √27

h = 3√3

So,

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Answer:

140

Step-by-step explanation:

180(n-2) is the total interior angle of a regulsr polygon and ifvyou divide it by number of the polygonvyou get the angle of each interior angle

180(9-2)= 1260 - total interior angle

1260/9 = 140 - each interior angle

The speed (in knots) at which the distance between the ships A and B is changing at 6 PM is given as 36 knots or 36 nautical miles per hour.

Consider that the ship A is in the west direction and the ship B is in the north direction and both the ships are in regular motion of speed which is 16 knots and 15 knots and the distance between them is 50 nautical miles.

Using the Pythagoras theorem, the relation of the distance x which represents the distance between ships at 6PM to the distances that each ship has travelled can be given as follows:

x^2 = (50 + 16t)^2 + (15t)^2

where, t is the number of hours that has passed since noon.

Differentiating both sides of the above equation with respect to time, we get:

2x*(dx/dt) = 2(50 + 16t)*(16) + 2*(15t)*(15)

t = 6, at 6 PM, therefore substituting the value and solving, we get:

2x(dx/dt) = 2[(50 + 16(6)]*(16) + 2*[15(6)]*(15)

2x(dx/dt) = 4194

dx/dt = 2097/x

Now substituting the value of x that corresponds to 6 PM:

x^2 = (50 + 16(6))^2 + (15(6))^2

x^2 = 3385

x = √3385 ≅ 58.19

Putting this value in dx/dt, we get:

dx/dt = 2097/58.19 ≅ 36.00 knots

Learn more about Pythagoras theorem at:

brainly.com/question/343682

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