A box contains 10 red buttons, 30 green buttons, 40 blue buttons, and 20 yellow buttons. ten buttons were removed, tallied, and then returned to the box. three additional samples were taken in the same way. which sample is the best representation of the buttons in the box?

The distance which this light travel per second is equal to 3 × 10⁸ meters per seconds.

In Mathematics and Science, speed is the distance covered by a physical object per unit of time.

In Mathematics and Science, the speed of any a physical object can be calculated by using this formula;

Speed = distance/time

By making distance the subject of formula, we have:

Distance, d(t) = speed × time

Distance = (9.45 × 10¹⁵ meters per year) × (1 year/ 3.15 × 10⁷ seconds)

Distance = 3 × 10⁸ meters per seconds.

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Complete Question:

Light travels 9.45 × 10¹⁵ meters in a year. There are about 3.15 × 10⁷ seconds in a year. How far does light travel per second?

Answer:

Step-by-step explanation:

I assume the question is what trig function do you use to find x?  If that's correct then answers, from TOP to BOTTOM, are:

tan

cos

sin

Answer:

Triangle 1: tangent

Triangle 2: cosine

Triangle 3: sine

Step-by-step explanation:

First, some definitions before working the problem:

The three standard trigonometric functions, cosine, tangent, and sine, are defined as follows for right triangles:

[tex]sin(\theta)=\dfrac{opposite}{hypotenuse}[/tex]

[tex]tan(\theta)=\dfrac{opposite}{adjacent}[/tex]

[tex]cos(\theta)=\dfrac{adjacent}{hypotenuse}[/tex]

One memorization tactic is "Soh Cah Toa" where the first capital letter represents one of those three trigonometric functions, and the "o" "a" and "h" represent the "opposite" "adjacent" and "hypotenuse" respectively.

The triangle must be a right triangle, or there wouldn't be a "hypotenuse", because the hypotenuse is always across from the right angle.  

Triangle 1

For the first triangle, the known acute angle is in the bottom left.  The two sides of the triangle that are known or are a "goal to find" are not the hypotenuse, so they are the "opposite" & "adjacent".

Specifically, the side of length "13" is touching the known acute angle AND the right angle, so it is the adjacent side.  The unknown side of length "x" is is touching the right angle but is NOT touching the known acute angle, so it is the "opposite" (across from the angle).

Out of "Soh Cah Toa," the part that uses o & a is "Toa".  The "T" in "Toa" stands for Tangent.  So, the desired function to use for the first triangle is the Tangent function.

Triangle 2

For the second triangle, the known acute angle is in the top left.  This time, the "adjacent" side is unknown, labeled as x, so it is the "goal to find" side.  The "hypotenuse" is known.

Therefore, the two sides of the triangle that are known or are a "goal to find" are the "adjacent" & "hypotenuse".

Out of "Soh Cah Toa," the part that uses "a" & "h" is "Cah".  So, the desired function to use for the first triangle is the Cosine function.

Triangle 3

For the third triangle, the known acute angle is in the top right.

This time, the side (at the bottom) across from the angle (at the top right) is known -- the "opposite" leg.  Additionally, the "hypotenuse" is unknown and is our "goal to find" side.  

Therefore, the two sides of the triangle that are known or are a "goal to find" are the "opposite" & "hypotenuse".

Out of "Soh Cah Toa," the part that uses "o" & "h" is "Soh".  So, the desired function to use for the first triangle is the Sine function.

Answer:

Slope= -1/3

Step-by-step explanation:

The slope is found using (y₂ - y₁) / (x₂ - x₁)

(y₂ - y₁)

So let's do the numerator first with the y. -52-(-32). The two negative signs  make 32 positive so -52 + 32= -20

(x₂ - x₁)

Now the denominator, x. -10-(-70). Same thing here, the two negative signs  make 70 positive so -10 + 70 = 60

(y₂ - y₁) / (x₂ - x₁)

Now put them together so -20/60 which equals -1/3 which the slope

The expression (2x-3)(2x-3) is equal to (2x-3)^2.

To expand the expression (2x-3)(2x-3), we can use the FOIL method (which stands for First, Outer, Inner, Last).

Multiplying the first terms of each binomial, we get 2x times 2x, which is 4x^2.

Multiplying the outer terms, we get -3 times 2x, which is -6x.

Multiplying the inner terms, we get -3 times 2x again, which is also -6x.

Multiplying the last terms of each binomial, we get -3 times -3, which is 9.

Combining like terms, we get 4x^2 - 12x + 9.

Therefore, (2x-3)(2x-3) is equal to (2x-3)^2, which is equivalent to 4x^2 - 12x + 9.

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The area of the squares are;

1.  9x²ft². Option D

2.  6x² - 7x - 3 in². Option C

The formula for calculating the area of a square is expressed as;

A = a²

Such that the parameters of the formula are;

From the information given, we have that;

Area = (3x)²

Find the square of the expression, we have that;

Area = 9x²ft²

2. Substitute the values, we have that;

Area = (2x -3)(3x + 1)

expand the bracket, we have;

Area = 6x² + 2x - 9x - 3

collect the like terms

Area = 6x² - 7x - 3

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The number of students in each bus is 15, and the number of students in each van is 28.

To find the number of students in each van and bus for the field trip to Yellowstone National Park, we can set up a system of equations using the given information. Let x represent the number of students in each van and y represent the number of students in each bus.

For high school A, we have:
2x + 8y = 254

For high school B, we have:
6x + 11y = 398

Now, we can solve this system of equations using the substitution or elimination method. We will use the elimination method:

Step 1: Multiply the first equation by 3 to make the coefficients of x the same in both equations:
6x + 24y = 762

Step 2: Subtract the second equation from the new first equation:
(6x + 24y) - (6x + 11y) = 762 - 398
13y = 364

Step 3: Divide both sides by 13 to find the value of y:
y = 364 / 13
y = 28

Now that we have the number of students in each bus, we can find the number of students in each van:

Step 4: Substitute y back into the first equation:
2x + 8(28) = 254
2x + 224 = 254

Step 5: Subtract 224 from both sides to find the value of x:
2x = 30

Step 6: Divide both sides by 2 to find x:
x = 15

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The best inequality that represents the relationship between the amount of hydrochloric acid (x) and the amount of distilled water (y) in the given graph is 3y - 2x > 0, option D is correct.

The graph shows a straight line with a negative slope passing through the origin. As the amount of hydrochloric acid, x, increases, the amount of distilled water, y, decreases

To see why, let's use a point on the line, such as (2, 3), and plug it into the inequality. We get:

3(3) - 2(2) > 0

9 - 4 > 0

This is true, so the point (2, 3) is a solution to the inequality. Any point on the line will also satisfy this inequality since it represents all possible combinations of x and y that Corrine can use in her experiment.

Alternatively, we can rewrite the inequality in slope-intercept form:

y < (2/3)x

This means that the y-values on the line are less than the corresponding values of (2/3)x. So as x increases, y must decrease to stay below the line. This confirms that 3y - 2x > 0 is the correct inequality.

Hence, option D is correct.

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The correct question is:

For a science experiment, Corrine is adding hydrochloric acid to distilled water. The relationship between the amount of hydrochloric acid, x, and the amount of distilled water, y, is graphed below. Which inequality best represents this graph?

A. 2y - 3x < 0

B. 3y - 2x < 0

C. 2y - 3x > 0

D. 3y - 2x > 0

Answer:

True (I think)

Step-by-step explanation:

Same pattern.

A -> B -> C.

D -> E -> F.

Would be false if either one didn't share the same pattern.

Answer:

no solution

Step-by-step explanation:

There are no values of x that make the equation true.

6. The probability that all 5 couples will have a baby of the preferred gender is 0.2373.

7. The probability that at least 4 of the 5 couples will have a baby of the preferred gender is 1 - 0.3672 = 0.6328.

Probability is a way to gauge how likely something is to happen. Many things are difficult to predict with absolute certainty.

Answer: The probability that one couple will have a baby of the preferred gender is 0.75. Assuming the gender of each baby is independent of the others, the probability that all 5 couples will have a baby of the preferred gender is 0.75⁵ = 0.2373.

Numeric value: 0.2373

In words: The probability that all 5 couples will have a baby of the preferred gender is 0.2373.

Answer: There are two ways to approach this problem. One way is to calculate the probability of each possible outcome (0 to 5 couples having a baby of the preferred gender) and then add up the probabilities for the outcomes where at least 4 couples have a baby of the preferred gender. Another way is to use the complement rule and subtract the probability that fewer than 4 couples have a baby of the preferred gender from 1.

Using the first method, we can calculate the probabilities as follows:

- 0 couples: 0.25⁵ = 0.0009766

- 1 couple: 5 x 0.75 x 0.25⁴ = 0.01465

- 2 couples: 10 x 0.75² x 0.25³ = 0.08789

- 3 couples: 10 x 0.75³ x 0.25² = 0.2637

- 4 couples: 5 x 0.75⁴ x 0.25 = 0.3955

- 5 couples: 0.75⁵ = 0.2373

The probabilities for the outcomes where at least 4 couples have a baby of the preferred gender are 0.3955 and 0.2373, so the total probability is 0.3955 + 0.2373 = 0.6328.

Using the second method, we can calculate the probability that fewer than 4 couples have a baby of the preferred gender as follows:

- 0 couples: 0.25⁵ = 0.0009766

- 1 couple: 5 x 0.75 x 0.25⁴ = 0.01465

- 2 couples: 10 x 0.75² x 0.25³ = 0.08789

- 3 couples: 10 x 0.75³ x 0.25² = 0.2637

The probability that fewer than 4 couples have a baby of the preferred gender is the sum of these probabilities: 0.0009766 + 0.01465 + 0.08789 + 0.2637 = 0.3672.

Therefore, the probability that at least 4 of the 5 couples will have a baby of the preferred gender is 1 - 0.3672 = 0.6328.

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Step-by-step explanation:

Expressions are collection of algebric equetion and equal sighn and used for expresion of mankind problems like items, money and other mankind problem.

to know length by using degree but most of the time for the archtechture. soon

For the function,

-number of complex zeros: four

-possible # of imaginary zeros: two pairs

-possible # of positive real zeros: zero

-possible # negative real zeros: 0 or 2

-possible rational zeros:  ±1, ±3, ±5, ±9, ±15, ±45

-factors to: (x + 3i)(x - 3i)(x + √5i)(x - √5i)

-zeros: x = ±3i, x = ±√5i.

The function is: F(x) = x⁴ +14x²+45.

Number of complex zeros: By the Fundamental Theorem of Algebra, the function has at most four complex zeros.

Possible number of imaginary zeros: If the complex zeros are not real, then there are at most two pairs of imaginary zeros.

Possible number of positive real zeros: The function has no positive real zeros since F(x) is always positive for x>0.

Possible number of negative real zeros: By Descartes' Rule of Signs, the function has either 0 or 2 negative real zeros.

Possible rational zeros: The rational zeros can be found using the Rational Root Theorem. They are of the form ±(a factor of 45) / (a factor of 1), which gives the following possible rational zeros: ±1, ±3, ±5, ±9, ±15, ±45.

To factor the polynomial:

F(x) = x⁴ +14x²+45

= (x² + 9)(x² + 5)

So the factors to linear factors are: (x + 3i)(x - 3i)(x + √5i)(x - √5i), where i is the imaginary unit.

Therefore, the zeros are: x = ±3i, x = ±√5i.

Note that all zeros are complex since there are no real roots.

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Option A The diameter and circumference of a circle have a proportional relationship is True .

Diameter

The diameter is the length acrοss the circle at its widest pοint, measured frοm center tο center . The radius, a related measurement, is a line that extends frοm the circle's centre tο its edge. The diameter is equivalent tο twice the radius. (A chοrd is a line that crοsses the circle but is nοt at the widest pοint.)

Circumference

The circle's perimeter, οr the distance arοund it, is knοwn as its circumference. Imagine encircling a circle with a string. Imagine taking the string οut and extending it in a straight line. This string's length, if measured, wοuld represent yοur circle's circumference.

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The value of Q, taking into account the significant figures is 30.8.

To work out the value of Q given the value of p, we can substitute the value of p into the equation Q = (1/6) × p².

Given p = 13.6, we can calculate Q as follows:

Q = (1/6) × (13.6)²

Q = (1/6) × 184.96

Q = 30.826666...

Now, let's consider the significant figures of the given value of p, which is 13.6 (3 significant figures).

Since the value of p has 3 significant figures, we should round our final answer for Q to 3 significant figures as well.

Considering the value of Q to a suitable degree of accuracy, we can round our answer to three significant figures, which gives us:

Q = 30.8

Therefore, the value of Q, taking into account the significant figures, is  30.8.

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To determine the range and sample standard deviation of tornadoes that touched down during each month of one year, we need to use the data in the table.

However, the table is not provided in the question.

Please provide the table with the number of tornadoes that touched down during each month of one year so I can help you with your question.
To determine the range and sample standard deviation of the number of tornadoes that touched down each month, you'll first need to provide the data in a table format.

Once you provide the data, I can help you calculate the range and sample standard deviation.

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Correct question:

Each​ year, tornadoes that touch down are recorded. The following table gives the number of tornadoes that touched down during each month of one year. Determine the range and sample standard deviation.

3 2 41 115 197 95

70 85 68 64 110 91

Range?

Sample Standard Deviation?

After one month, Betty will owe $407.02 on her credit card.

The amount Betty will owe after one month depends on how much of the stability she will pay off in the course of that point.

Assuming she does not make any payments in the course of the first month, here is how to calculate her balance:

The cash-withdrawal price is a one-time fee, so it does no longer affect the stability after one month.

Betty withdrew $400, so her starting balance is $406 ($400 for the lawnmower plus $6 cash-withdrawal price).

The interest rate is 3%, that's an annual price. To calculate the monthly charge, divide with the aid of 12: three% / 12 = 0.25%.

To calculate the interest charged for the first month, multiply the stability through the monthly interest rate: $406 * 0.25% = $1.02.

Add the interest to the balance: $406 + $1.02 = $407.02. that is Betty's balance after one month.

Consequently, after one month, Betty will owe $407.02 on her credit card.

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Answer:

12.8

Step-by-step explanation:

First, we can simplify each fraction separately:

64e^2/5e = 64/5e^(1-1) = 64/5

3e/8e = 3/8

Now we can multiply:

(64/5) * (3/8) = 12.8

Therefore, the product is 12.8.

The minimum height of the ladder needed to change the blown outdoor lightbulb is 5 meters.

To determine the minimum height of the ladder needed to change a blown outdoor lightbulb that is 5m up, we need to consider the following terms:

1. The bulb's height (5m)

2. Your reach when on the top rung of the ladder (1m)

3. The required space off the house for the ladder's base for stability (3m)

First, subtract your reach from the bulb's height: 5m - 1m = 4m. This means the ladder needs to reach at least 4 meters up the wall.

Next, we need to use the Pythagorean theorem to find the ladder's minimum height. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the ladder) is equal to the sum of the squares of the lengths of the other two sides (the distance from the house and the height up the wall).

Let's denote the ladder's height as L, the distance from the house as A (3m), and the height up the wall as B (4m).

According to the Pythagorean theorem, we have:
L² = A² + B²

Substitute the values for A and B:
L² = (3m)² + (4m)²
L² = 9m² + 16m²
L² = 25m²

Now, find the square root to get the minimum height of the ladder:
L = √25m²
L = 5m

So, the minimum height of the ladder needed to change the blown outdoor lightbulb is 5 meters.

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The solution to the problems using trigonometric ratios are:

12a) x = 16.09

12c) x = 7 and y = 7

13a) Time it takes to reach the ground is: 8 seconds

13b) Highest point reached is: 80 ft

12a) Using the law of sines, we can say that:

x/sin 90 = 9/sin 34

x = (9 * sin 90)/sin 34

x = 16.09

12c) Using the law of sines, we can say that:

x/sin 45 = 7√2/sin 90

x = (7√2 * 1/√2)/1

x = 7

Similarly, because it is an isosceles triangle, y = 7

13a) The equation of the height above the ground is :

h = 40t - 5t²

where:

h is height

t is time in seconds

Thus:

Time it takes to reach the ground is at h = 0.

40t - 5t² = 0

5t² = 40t

5t = 40

t = 8 seconds

b) Highest point reached:

h'(t) = 40 - 10t

h'(t) = 0

40 - 10t = 0

t = 4 seconds

Thus:

h_max = 40(4) - 5(4)²

h_max = 80 ft

c) Time at which ball was 35ft off ground is:

35 = 40t - 5t²

5t² - 40t + 35 = 0

Using quadratic equation calculator gives us:

t = 1 and 7 seconds

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The magnitude of the force of the car against the hill is approximately 13690 pounds.

a. To find the magnitude of the force required to keep the car from sliding down the hill, we need to calculate the force component perpendicular to the hill (the normal force) and the force component parallel to the hill (the force of friction). The force of friction must be equal and opposite to the component of the weight of the car parallel to the hill to keep the car from sliding.

First, we need to calculate the weight of the car in Newtons:

3100 pounds = 1406.13 kg

Weight = mg = 1406.13 kg * 9.81 m/s^2 = 13791.68 N

The force component perpendicular to the hill is equal to the weight of the car multiplied by the cosine of the angle of inclination:

F_perpendicular = Weight * cos(5.6°) = 13791.68 N * cos(5.6°) = 13689.55 N

The force component parallel to the hill is equal to the weight of the car multiplied by the sine of the angle of inclination:

F_parallel = Weight * sin(5.6°) = 13791.68 N * sin(5.6°) = 1275.02 N

The force of friction is equal to the force parallel to the hill, so:

F_friction = F_parallel = 1275.02 N

Therefore, the magnitude of the force required to keep the car from sliding down the hill is equal to the force component perpendicular to the hill plus the force of friction:

F_required = F_perpendicular + F_friction = 13689.55 N + 1275.02 N = 14964.57 N

Rounded to the nearest whole number, the magnitude of the force required to keep the car from sliding down the hill is approximately 14965 pounds.

b. To find the magnitude of the force of the car against the hill, we just need to calculate the force component perpendicular to the hill (the normal force):

F_normal = F_perpendicular = 13689.55 N

Rounded to the nearest whole number, the magnitude of the force of the car against the hill is approximately 13690 pounds.

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Based on the information on the number line, the numbers that represent the percentages are: 42 (100%), 21 (50%), 63 (150%).

To calculate the number that is equivalent to each percentage we must carry out the following procedure: Rule of three. In this case we must take into account that 42 represents 100% of the people.

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The possible value of <J is 78 degrees

It is important to note that the different trigonometric identities are;

Also, the law of sines in a triangle is expressed as;

sin A/a = sin B/b = sin C/c

Given that the angles are in capitals and the sides are in small letters.

From the information given, we have that;

sinI/i = sin J/j

Substitute the values, we get;

sin 72 /70 = sin J/72

cross multiply the values, we have;

sin J = 68. 476/70

divide the values

sin J = 0. 9782

Find the inverse of sin

<J = 78 degrees

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The TWO statements that represent the relationship between y = 5x and y = log5 x are:

1. They are the exponential and logarithmic form of the same equation.

2. They are inverses of one another.

The two equations are related because they represent the same relationship between x and y, but in different forms. The first equation is an exponential equation, where y is a power of 5 raised to the x power. The second equation is a logarithmic equation, where y is the exponent to which 5 must be raised to get x.

Because the two equations represent the same relationship, they are inverses of one another. If we take the logarithm of both sides of the exponential equation, we get the logarithmic equation. If we raise 5 to both sides of the logarithmic equation, we get the exponential equation. Therefore, the two equations are inverses of one another.

(a) The value of the constant is 1/24, (b) P(X<=1,Y<=1)  is 5/48 and (c)  P(X + Y <= 1) is also 5/48

(a) The constant C can be found by using the fact that the total probability of the joint density function over the entire space is equal to 1. Therefore, we integrate the joint density function over the region where it is defined and set it equal to 1:

∫∫f(x,y) dA = 1

∫[0,2]∫[0,4] Cx(1+y) dy dx = 1

C∫[0,2]x[(y+(y²)/2)] [0,4] dx = 1

C(24/5) = 1

C = 5/24

(b) To find P(X <= 1, Y <= 1), we integrate the joint density function over the region where X <= 1 and Y <= 1:

P(X<=1,Y<=1) = ∫[0,1]∫[0,1] (5/24) x(1+y) dy dx

= (5/24) ∫[0,1] x(1+(1/2)) dx

= (5/24) [(1/2) + (1/6)]

= 5/48

(c) To find P(X + Y <= 1), we integrate the joint density function over the region where X + Y <= 1:

P(X+Y<=1) = ∫[0,1]∫[0,1-x] (5/24) x(1+y) dy dx

= (5/24) ∫[0,1] x(1+(1-x)/2) dx

= (5/24) [(1/2) - (1/12)]

= 5/48

Therefore, P(X + Y <= 1) = 5/48.

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The simplified expression [tex](3.05 * 10^-7) (8.67 * 10^4)[/tex] in standard form to 3 decimal places is approximately 0.026

To simplify the expression[tex](3.05 * 10^-7) (8.67 *  10^4)[/tex] and provide the answer in standard form to 3 decimal places.

Step 1: Multiply the coefficients (3.05 and 8.67).
3.05 * 8.67 = 26.4445

Step 2: Use the properties of exponents to multiply the powers of 10.
[tex]10^{-7} * 10^4 = 10^{(-7+4)} = 10^-3[/tex]

Step 3: Multiply the results from Step 1 and Step 2.
[tex]26.4445 * 10^-3 = 0.0264445[/tex]

Step 4: Round the result to 3 decimal places.
0.0264445 ≈ 0.026

So, the simplified expression (3.05 x 10^-7) (8.67 x 10^4) in standard form to 3 decimal places is approximately 0.026.

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There are 39 red blocks out of 65 total blocks.

To solve this problem, we need to find the total number of blocks in each repeating pattern. The pattern is three red blocks followed by two blue blocks. So in each pattern, there are five blocks total.

To find the number of red blocks in 65 total blocks, we need to figure out how many times the pattern repeats. We can do this by dividing the total number of blocks (65) by the number of blocks in each pattern (5):

65 ÷ 5 = 13

So the pattern repeats 13 times.

In each pattern, there are three red blocks. So to find the total number of red blocks, we need to multiply the number of red blocks in each pattern (3) by the number of times the pattern repeats (13):

3 x 13 = 39

Therefore, there are 39 red blocks out of 65 total blocks.

The correct answer is option B.

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The volume of the large box is 120[tex]s^3[/tex].

If Frank can fit 15 cube-shaped containers in each layer and stack 8 layers into one box, then the total number of containers he can fit in one box is:

15 containers/layer x 8 layers = 120 containers

Since each container is cube-shaped, we can assume that it has the same length, width, and height. Let's represent the length of one side of the container as "s". Then, the volume of one container is:

Volume of one container = [tex]s^3[/tex]

The volume of 120 containers that can fit in one box is:

Volume of 120 containers = 120 x Volume of one container

Substituting the expression for the volume of one container, we get:

Volume of 120 containers = 120[tex]s^3[/tex]

Therefore, the volume of the large box that can hold 120 cube-shaped containers with side length "s" is 120[tex]s^3[/tex].

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To solve the exact differential equation (5t^2 + 8y) dy + (10yt + 9t^2) = 0, we need to check if it is exact or not. We do so by taking partial derivatives with respect to y and t:

∂/∂y (5t^2 + 8y) = 8
∂/∂t (10yt + 9t^2) = 10y + 18t

Since these partial derivatives are not equal, the equation is not exact. To make it exact, we can multiply the entire equation by a integrating factor, which is given by:

μ = e^(∫(∂/∂t)(10yt + 9t^2) dt) = e^(∫(10y + 18t) dt) = e^(10yt + 9t^2)

Multiplying both sides of the equation by μ, we get:

(5t^2 + 8y)e^(10yt + 9t^2) dy + (10yt + 9t^2)e^(10yt + 9t^2) dt = 0

Now, we can check if this equation is exact:

∂/∂y (5t^2e^(10yt + 9t^2) + 8ye^(10yt + 9t^2)) = 10te^(10yt + 9t^2)
∂/∂t ((10ye^(10yt + 9t^2)) + (9t^2e^(10yt + 9t^2))) = 10ye^(10yt + 9t^2) + 18t^2e^(10yt + 9t^2)

These partial derivatives are equal, so the equation is exact. Therefore, we can find a potential function Φ such that:

∂Φ/∂y = 5t^2e^(10yt + 9t^2) + 8ye^(10yt + 9t^2)
∂Φ/∂t = (10ye^(10yt + 9t^2)) + (9t^2e^(10yt + 9t^2))

Integrating the first equation with respect to y, we get:

Φ = ∫(5t^2e^(10yt + 9t^2) + 8ye^(10yt + 9t^2)) dy = (5t^2/10)e^(10yt + 9t^2) + (4y/10)e^(10yt + 9t^2) + C(t)

where C(t) is an arbitrary constant of integration that depends only on t.

Now, we can differentiate this expression with respect to t and compare it to the second equation:

∂Φ/∂t = (10t/10)e^(10yt + 9t^2) + C'(t)
(10ye^(10yt + 9t^2)) + (9t^2e^(10yt + 9t^2)) = (10t/10)e^(10yt + 9t^2) + C'(t)

Comparing the two expressions, we get:

C'(t) = 10ye^(10yt + 9t^2)

Integrating both sides with respect to t, we get:

C(t) = ∫10ye^(10yt + 9t^2) dt = e^(10yt + 9t^2) + K

where K is another arbitrary constant of integration.

Therefore, the solution to the exact differential equation (5t^2 + 8y) dy + (10yt + 9t^2) = 0 is given by:

(5t^2/10)e^(10yt + 9t^2) + (4y/10)e^(10yt + 9t^2) + e^(10yt + 9t^2) + K = 0

or simplifying:

y = (-5t^2/4) - (1/2)e^(-10yt - 9t^2) - (K/4)e^(-10yt - 9t^2)

where K is an arbitrary constant of integration.

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a) The price of the old model is P22,289.

b) The dimensions of the rectangle are 16 by 32.

a) Let x be the price of the old model. According to the problem, the price of the new 32'' LED television is P15,500 less than twice the price of the old model.

This can be expressed as 2x - P15,500 = P29,078. Solving for x, we can add P15,500 to both sides to get 2x = P44,578, and then divide both sides by 2 to get x = P22,289.

b) Let w be the width of the rectangle. According to the problem, the length of the rectangle is twice the width, so the length is 2w. The perimeter of a rectangle is the sum of the lengths of all four sides, which in this case is 2w + 2(2w) = 6w.

We are given that the perimeter is 96, so we can set up an equation: 6w = 96. Solving for w, we can divide both sides by 6 to get w = 16. Since the length is twice the width, the length is 2(16) = 32.

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we can simplify |x + 3| to x + 3 when x is greater than 5.

The absolute value function |x| is defined as the distance of x from zero on the number line. This means that |x| is always non-negative, so it can be expressed as a non-negative number.

In this case, we are given that x > 5, which means that x is greater than 5. If we add 3 to both sides of this inequality, we get:

x + 3 > 5 + 3

x + 3 > 8

This tells us that x + 3 is also greater than 8. Therefore, when x is greater than 5, the expression |x + 3| represents the distance of x + 3 from zero, which is equal to x + 3 itself because x + 3 is positive.

As a result, we can simplify |x + 3| to x + 3 when x is greater than 5.

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