In 1680, Isaac Newton, scientist astronomen, and mathematician, used a comet visible from Earth to prove that some comers follow a parabolic path through space as they travell around the sun. This and other discoveries like it help scientists to predict past and future positions of comets. ​

Mr. Ali filled 6 of the 400 ml juice bottles and 8 of the 1 liter juice bottles.

First, convert 15.8 liters to milliliters:

15.8 L = 15,800 mL

Let x be the number of 400 ml juice bottles filled, and let y be the number of 1 liter juice bottles filled.

The total amount of juice filled can be represented as:

400 ml/bottle * x + 1000 ml/bottle * y = 15,800 ml

Simplifying, we get:

4x + 10y = 158

We also know that there are 3200 ml of juice left:

400 ml/bottle * (x - 3200/400) + 1000 ml/bottle * y = 0

Simplifying, we get:

x + 2.5y = 28

We now have two equations with two variables. Solving for x and y, we get:

x = 6

y = 8

Therefore, Mr. Ali filled 6 of the 400 ml juice bottles and 8 of the 1 liter juice bottles.

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To solve the trigonometric equation 2cos^2(x) + cos(2x) = 0 in the interval [0, 2π), we will first use the double angle formula for cos(2x) and then solve for x. Recall that cos(2x) = 2cos^2(x) - 1.

Substitute this into the equation: 2cos^2(x) + (2cos^2(x) - 1) = 0 Combine the terms: 4cos^2(x) - 1 = 0 Now, isolate cos^2(x): cos^2(x) = 1/4 Take the square root of both sides: cos(x) = ±√(1/4) = ±1/2 Now, find the values of x in the interval [0, 2π) that satisfy the equation: For cos(x) = 1/2: x = π/3, 5π/3 For cos(x) = -1/2: x = 2π/3, 4π/3 Combine the answers as a comma-separated list: x = π/3, 2π/3, 4π/3, 5π/3

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The objects used to measure mass are Balances and Scales. The objects used to measure Volume are Manometer, Pipettes, Graduated cylinders, Flasks, Volumetric flasks and Beakers. Here Density formula can be used to measure both mass and volume.

The problem is asking to match different measuring tools with the measurements they are used for, i.e., mass or volume.

The first tool is a manometer. A manometer is used to measure pressure and not mass or volume, so it does not belong in either category.

The next set of tools are pipettes, graduated cylinders, and volumetric flasks. These tools are all used to measure volume, so they belong in the volume category.

The next set of tools are scales and balances. These tools are used to measure mass, so they belong in the mass category.

The formula for density can be used to calculate the mass of an object given its volume and density, or the volume of an object given its mass and density, so it belongs in both categories.

Finally, a beaker or a graduated cylinder can be used to measure volume, so it belongs in the volume category.

Therefore, the correct categorization of the measuring tools are as follows

Volume

Pipettes

Graduated cylinders

Volumetric flasks

Beaker or graduated cylinder

Mass

Scales

balances

Both

Formula for density

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Answer:

100π or ≈ 314.1592

Step-by-step explanation:

Area of a circle formula: A = πr²

A = π(10)²

A = π(100)

A = 100π ≈ 314.1592

Answer:

100pi

Step-by-step explanation:

pi*radius^2=area

pi*10^2=100pi

The equilibrium quantity is approximately 34.47 items and the producer surplus at the equilibrium quantity is approximately 396.11.

To find the equilibrium quantity, we need to find the quantity at which the demand and supply functions are equal:

Demand function: d(x) = 588.7 – 0.4x^2

Supply function: s(x) = 8(x) = 0.322

Setting these two functions equal to each other, we get:

588.7 – 0.4x^2 = 0.322x

Simplifying this equation, we get:

0.4x^2 + 0.322x - 588.7 = 0

Using the quadratic formula, we get:

x = (-0.322 ± √(0.322^2 + 40.4588.7)) / (2*0.4)

x ≈ 34.47 or x ≈ -43.67

Since we cannot have a negative quantity, the equilibrium quantity is approximately 34.47 items.

To find the producer surplus at the equilibrium quantity, we need to calculate the area between the supply curve and the equilibrium price, which is the price that corresponds to the equilibrium quantity. We can find the equilibrium price by plugging the equilibrium quantity into either the demand or supply function:

s(34.47) = 8(34.47) = 11.58

So the equilibrium price is approximately 11.58.

Now we can find the producer surplus by integrating the supply function from 0 to the equilibrium quantity, and subtracting the result from the area of a rectangle with height equal to the equilibrium price and width equal to the equilibrium quantity. The formula for producer surplus is:

Producer Surplus = (Equilibrium Price * Equilibrium Quantity) - ∫[0, Equilibrium Quantity] Supply Function dx

Plugging in the values we found, we get:

Producer Surplus = (11.58 * 34.47) - ∫[0, 34.47] 0.322 dx

Integrating the supply function, we get:

∫[0, 34.47] 0.322 dx = 0.322 * 34.47 ≈ 11.10

So the producer surplus is:

Producer Surplus ≈ (11.58 * 34.47) - 11.10 ≈ 396.11

Therefore, the equilibrium quantity is approximately 34.47 items, and the producer surplus at the equilibrium quantity is approximately 396.11.

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The measure of angle W is 30 degrees.

The figure in the image is a right triangle.

Angle W = ?

Adjacent to angle W = 12

Opposite to angle W = 4√3

To determine the measure of angle W, we use the trigonometric ratio.

Note that: tangent = opposite / adjacent

Plug in the values:

tan(W) = 4√3 / 12

Take the tan inverse

W = tan⁻¹( 4√3 / 12 )

W = 30°

Therefore, angle W measure 30 degrees.

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Answer:2.78

Step-by-step explanation:

the probability of rolling a 5 on a six-sided die and then rolling a 2? The chances of rolling a 5 on the first die is 1 in 6. The chances of rolling a 2 on the second roll in 1 in 6. So the probability of a 5, then a 2, is 1 in 36.

We can see that this gives us the exact same answer as the first method: 1/36 as a percentage is 2.78%.

Answer:

A) [tex]2.78\%[/tex]

Step-by-step explanation:

Assuming that Chris throws two standard dices (meaning 6 sides, with the sides numbered as 1, 2, 3, 4, 5, 6, respectively). There is one chance of the dice landing on a 5 for both dices, and presuming that the dices do not hit each other, would be independent variables. This means that there is a 1/6 chance for both:

[tex]\frac{1}{6} * \frac{1}{6} = \frac{1}{36}[/tex]

Change the fraction to a percentage. Divide and solve for a decimal. Then, move the decimal point to the left two place values, and you will obtain your percentage:

[tex]\frac{1}{3} = 0.02777 = 2.777\%[/tex]

You are rounding to the nearest tenth place value (2nd place value right of the decimal point):

[tex]2.777\% = 2.78\%[/tex]

[tex]2.78\%[/tex] is your answer, or A).

~

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To evaluate the expression 37 * arcsin(sin(-3π/8)), follow these steps:

1. First, identify the expression: 37 * arcsin(sin(-3π/8))
2. Calculate the value of sin(-3π/8) using the sine function: sin(-3π/8)
3. Apply the arcsin function to the result from step 2: arcsin(sin(-3π/8))
4. Multiply the result from step 3 by 37: 37 * arcsin(sin(-3π/8))

Let's solve each step:

2. sin(-3π/8) = -0.3826834324 (rounded to 10 decimal places)

3. arcsin(-0.3826834324) = -π/8 (in exact form, since the input is the sine of a known angle)

4. 37 * (-π/8) = -37π/8

So, the expression 37 * arcsin(sin(-3π/8)) evaluates to -37π/8 in exact form.

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Answer: B

Step-by-step explanation:

Your original matrix:

1 0

0 1

after dilation

3 0

0 3

after reflection... it only changes sign of y because its reflected across x-axis

3 0

0 -3

B

The total surface area of the solid is approximately 1818 [tex]cm^{2}[/tex]. Let's call the radius of the hemisphere and cylinder "r".

The volume of the solid is the sum of the volumes of the hemisphere and cylinder: V = (2/3)π[tex]r^{3}[/tex] + π[tex]r^{2}[/tex]h. Substituting in the given values, we get: 2707 = (2/3)π [tex]r^{3}[/tex] + π[tex]r^{2}[/tex](80)

To solve for r, we can rearrange the equation and use a numerical method or calculator. We get: r ≈ 11.6 cm

Now, we can use the radius to find the surface area of the solid. The surface area is the sum of the curved surface areas of the hemisphere and cylinder, plus the area of the circular base of the cylinder: A = 2π[tex]r^{2}[/tex] + 2πrh + π[tex]r^{2}[/tex].

Substituting in the given values and solving for A, we get: A ≈ 1818 [tex]cm^{2}[/tex]. Therefore, the total surface area of the solid is approximately 1818 [tex]cm^{2}[/tex].

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Option B is correct. In this context, the meaning is: Among hot dogs with the same number of calories, the sodium content varies, with a standard deviation of about 75 milligrams

The calculated test statistic is 3.75

The test statistics can be gotten from the data that we already have available in this question

The coefficient is given as 2.235

The Standard error of the coefficient is given as 0.596

The formula used is given as

Such that t = coefficient /  Standard error

where the coefficient = 2.235

The standard error = 0.596

We have to apply the values to formula:

= 2.235 / 0.596

= 3.75

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Answer:

Supplementary angles are angles that have a sum of 180°. When two angles are supplementary, we say that one angle is the supplement of the other.

Complementary angles are angles that add up to 90°. When two angles are complementary, we can say that one angle is the complement of the other

Step-by-step explanation:

The median weight of the black bass caught by the fisherman is 10.5 pounds.

To find the median, we need to arrange the weights in order from smallest to largest: {6, 7, 8, 12, 13, 14}. Since there are an even number of weights, the median is the average of the two middle values, which are 8 and 12. Therefore, the median weight is (8+12)/2 = 10.5 pounds.

The median is a measure of central tendency that represents the middle value in a dataset. It is less sensitive to extreme values than the mean and is useful for describing the typical value in a skewed distribution.

In this case, the median weight of 10.5 pounds indicates that half of the black bass caught by the fisherman weighed less than 10.5 pounds, and half weighed more than 10.5 pounds.

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Since 1.1584 > 1, the truck cannot pass under the archway without going into the other lane.

A semi-elliptic archway with a height of 15 feet at the center and a width of 50 feet can be visualized as half of an ellipse.

The major axis of the ellipse corresponds to the width, while the minor axis corresponds to the height. In this case, the major axis (a) is 25 feet, and the minor axis (b) is 15 feet.

To determine if a truck that is 12 feet high and 14 feet wide can drive under the archway without going into the other lane, we can use the equation of an ellipse: (x²/a²) + (y²/b²) = 1.

The truck will occupy half of the road width, which is 25 feet, so its horizontal distance from the center (x) is 25 - 7 = 18 feet, and its height (y) is 12 feet.

Plugging these values into the equation, we get: (18²/25²) + (12²/15²) = (324/625) + (144/225) ≈ 0.5184 + 0.64 = 1.1584.

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The data set is 2, 7, 7, 8, 9, and 9, with a mean of 7. The outlier is 2, and the mean without the outlier is 6.6. The outlier pulls the mean lower than the center, but once removed, the mean becomes more representative of the data set.

To find the mean of a set of values, we add up all the values and divide by the total number of values.

In this case, we know that the mean of six values is 7, so we can set up the following equation

(2 + 7 + 7 + 8 + 9 + 9) / 6 = 7

Simplifying the equation, we get

42 / 6 = 7

So, the sum of the six values is 42.

Now, we know that there is one outlier that pulls the mean higher than the center. In other words, one of the values is much larger than the others. Let's assume that the outlier is 20.

So, the new sum of the six values would be

2 + 7 + 7 + 8 + 9 + 20 = 53

To find the mean without the outlier, we need to subtract the outlier from the sum and divide by the remaining number of values. In this case, there are five values remaining. So, we get

(2 + 7 + 7 + 8 + 9) / 5 = 33 / 5 = 6.6

Therefore, the possible data set is 2, 7, 7, 8, 9, and 9, and the mean without the outlier is 6.6.

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--The given question is incomplete, the complete question is given

" The mean of six values is 7. There is one outlier that

pulls the mean higher than the center. What could the

data set be? What is the mean without the outlier?  

The possible data set is 2, 7, 7, 8, 9, and 9. "--

The probabilities are given as follows:

a) Square: 1/6.

b) Not the triangle: 43/48.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The total area of the figure is given as follows:

12 x 8 = 96 units². (rectangle).

The area of the square is given as follows:

4² = 16 units² (square of the side lengths).

Hence the probability of the square is given as follows:

p = 16/96

p = 1/6.

The area of the triangle is given as follows:

A = 0.5 x 4 x 5 = 10 units². (half the multiplication of the side lengths).

Hence the complement of the area of the triangle is of:

96 - 10 = 86 units².

And the probability of the complement is of:

86/96 = 43/48.

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The unit multiplier for the conversion is 24 ft/min = (24 ft² / 1 min) * (12 inch / ft) * (12 inch / ft) * (1 min / 60 sec)

An exponential equation is an expression that shows how numbers and variables using mathematical operators.

1 minutes = 60 seconds

1 foot = 12 inch

The unit multiplier for the conversion of 24 square feet per minute to square inches per second

24 ft/min = (24 ft² / 1 min) * (12 inch / ft) * (12 inch / ft) * (1 min / 60 sec)

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Mrs. Davis travelled a total of 553 miles.

This can be calculated as:

Distance covered when she flew from Atlanta to Washington d.c.= 547 miles

Distance covered when she rented a car and drove to Baltimore = 6 miles.

Therefore, total miles can simply be calculated as:

Distance covered when she flew from Atlanta to Washington d.c + Distance covered when she rented a car and drove to Baltimore = 547miles+ 6 miles

                                                                                             = 553 miles

Hence, the total miles Mrs. davis travelled is equal to 553 miles.

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The population in 4 year is 160 and in 6 year is 320.

The population is grows by a factor of 2 every two years.

We can use the following formula to get the rate of population growth every two years:

Growth factor: (population after n years / (population after (n-2) years) ^ 1/2

This formula can be used to determine the growth factor as follows:

Growth factor is equal to (40/10)*(1/2) = 2

This indicates that every two years, the population increases by a factor of 2.

Then, for 4 year the population

= 10 x 2⁴

= 10 x 16

= 160

For 6 year the population is

= 10 x 2⁶

= 10 x 32

= 320

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Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

The values of the letters are:  a = k / (648b), b = k / (1296c), c = k / (1296b) and 24c = k / (576a).

To determine the value of the letters in the given sequences, we need to first recall the formula for inverse proportionality, which states that the product of the terms in one sequence is equal to the constant value of the product of the terms in the other sequence. Mathematically, we can represent this as:

4 x 8 x a x 18 = k = 54 x b x 24 x c

Here, k is the constant of proportionality. To find the value of the letters, we can solve for them algebraically. First, we can simplify the equation by dividing both sides by 4 x 18 x 24:

a = k / (4 x 8 x 18 x 24 / 54 x b x c)

a = k / (6b c)

Next, we can substitute the given values of the sequence into the equation and simplify:

a = k / (6b c) = k / (648b)

Multiplying both sides by 648b, we get:

648b a = k

Similarly, we can solve for the values of the other letters as follows:

b = k / (54 x 24 x c) = k / (1296c)

24c = k / (4 x 8 x a x 18) = k / (576a)

c = k / (54 x b x 24) = k / (1296b)

Therefore, the values of the letters are:

a = k / (648b)
b = k / (1296c)
c = k / (1296b)
24c = k / (576a)

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Answer: 15.6 units

Step-by-step explanation:

The surface areas of the pyramids are listed below:

In this problem we find six cases of square pyramids, whose surface areas shall be found by means of the following formulas:

A = 4 · 0.5 · b · s + b²

Where:

Now we proceed to determine the surface area of the square pyramid:

Case 1:

A = 4 · 0.5 · (5 in) · (10 in) + (5 in)²

A = 125 in²

Case 2:

A = 4 · 0.5 · (8 ft) · (15 ft) + (8 ft)²

A = 304 ft²

Case 3:

A = 4 · 0.5 · (10 yd) · (16 yd) + (10 yd)²

A = 420 yd²

Case 4:

A = 4 · 0.5 · (3 yd) · (8 yd) + (3 yd)²

A = 57 yd²

Case 5:

A = 4 · 0.5 · (8 in) · (17 in) + (8 in)²

A = 336 in²

Case 6:

A = 4 · 0.5 · (4 ft) · (11 ft) + (4 ft)²

A = 104 ft²

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To determine how many seconds longer Nicholas cleans his room than Gloria, we first need to convert both their cleaning times into seconds.

1. Convert Nicholas's cleaning time to seconds:
Nicholas cleans his room in 10 hours. There are 60 minutes in an hour and 60 seconds in a minute. So, we multiply 10 hours by 60 minutes and then by 60 seconds:
10 hours * 60 minutes/hour * 60 seconds/minute = 36,000 seconds

2. Convert Gloria's cleaning time to seconds:
Gloria cleans her room for 3 minutes and 18 seconds. We convert 3 minutes into seconds by multiplying it by 60 seconds/minute:
3 minutes * 60 seconds/minute = 180 seconds
Now, add the 18 seconds to the 180 seconds:
180 seconds + 18 seconds = 198 seconds

3. Subtract Gloria's cleaning time from Nicholas's cleaning time:
36,000 seconds (Nicholas) - 198 seconds (Gloria) = 35,802 seconds

In conclusion, Nicholas cleans his room for 35,802 seconds longer than Gloria.

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A open box without a top 1260 sq. in. wood will Gabriella use.

Since the top of the box is the same area as the base, calculate the base.

B = length × width

Length of the wooden box = 18 in.

Width of the wooden box = 15 in.

B = 18(15) = 270 in.

Calculate the surface area of the box.

Surface Area = 2(B + wh + hl)

h = 15

w × h = 15(15) = 225

h × l = 15(18) = 270

Surface Area of the wooden box = 2(270 + 225 + 270) = 2(765) = 1,530 sq. inches

Subtract the base from the surface area: 1,530 - 270 = 1,260sq. in.

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The given question is incomplete, complete question is:

Gabriella is beauty a wooden box with a rectangular base that is 18 in by 15 in and is 15 in tall if she wants a open box without a top how much wood will Gabriella use

The library is about 26.1 miles from the courthouse.

To solve this problem, we will apply the Pythagorean theorem, which tells us that during a right triangle, the square of the period of the hypotenuse( the longest side) is same to the total of the places of the lengths of the different two sides.

In this instance, the park is on the right angle, and the library and courthouse are the opposite two factors. we can consider the distance among the library and the park because the length of 1 leg of the triangle, and the space among the courthouse and the park as the length of the other leg.

So, using the Pythagorean theorem, we're suitable to calculate the period of the hypotenuse( the distance among the library and the courthouse)

library- to- park distance2 courthouse- to- park distance2 = library- to- courthouse distance2

[tex](14.5)^2 + (21.7)^2 = library-to-courthouse distance^2[/tex]

[tex]210.25 + 471.29 = library-to-courthouse distance^2[/tex]

[tex]681.54 = library-to-courthouse distance^2[/tex]

Taking the square root of both aspects, we get

library- to- courthouse distance[tex]= sqrt(681.54) \approx26.1[/tex]

Accordingly, the library is about 26.1 miles from the courthouse.

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The time that it will take the bacteria to grow to 2097152 is: 2 hours 48 minutes

An exponential function is defined as a Mathematical function in the form f(x) = [tex]a^{x}[/tex].

where:

“x” is a variable.

“a” is a constant which is called the base of the function and it should be greater than 0

However; the exponential function is:

1([tex]2^{x}[/tex]) = 2097152

log 2x  = log 2097152

x log 2 = log 2097152

x = log 2097152/log 2

x = 6.32162/0.3010

x = 21 times the population doubles

21(8) = 168 minutes = 2 hours 48 minutes

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The amount made by Caroline is $593.75 and 200 + 0.25x when she made $1574 and $x in sales respectively.

The total amount will be given by the formula using percentage -

Total amount = Base salary + 25% × her total sales

Keep the value in equation when she made $1575

Total amount = 200 + 25% × 1575

Total amount = 200 + 393.75

Total amount = $593.75

Keep the value in equation when she made $x

Total amount = 200 + 25% × x

Total amount = 200 + 0.25x

Hence, the earned amount is $593.75 and 200 + 0.25x.

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The complete question is -

Caroline works in a department store selling clothing. She makes a guaranteed salary of $200 per week, but is paid a commision on top of her base salary equal to 25% ofher total sales for the week. How much would Caroline make in a week in which she made $1575 in sales? How much would Caroline make in a week if she made x dollars in sales?

Length of the platform at the front of the room whose area is 3x² + 10x - 8 and width is (x+4) m is (3x - 2) m

Area of the rectangular platform = 3x² + 10x - 8

Width of the rectangular platform = x+4

Area = length × width

Length = area/width

Length = [tex]\frac{3x^{2} + 10x - 8}{x+4}[/tex]

By splitting the middle term we get

Length = [tex]\frac{3x^{2} + 12x -2x -8 }{x+4}[/tex]

By taking common we get

Length = [tex]\frac{3x(x+4) - 2(x+4)}{x+4}[/tex]

By taking x+4 common we get

Length = [tex]\frac{(3x-2)(x+4)}{x+4}[/tex]

Cutting the x+4 from denominator and numerator we get

Length = 3x-2

Length of the platform at the front of room is 3x-2

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