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The words "Freight," "Stone," 
"Often," and "Canine" 
all contain embedded number words:

- Freight contains eight (`fr**eight**`)...
- Stone contains one (`st**one**`)...
- Often contains ten (`of**ten**`)...
- Canine contains nine (`ca**nine**`)...

"Answer:"
Each word contains a number word
“one,” “ten,” “nine,” or “eight”
within it.

Each of the words contains the name
of a number embedded within it:

1. Freight – Contains "eight"
2. Stone – Contains "one"
3. Often – Contains "ten"
4. Canine – Contains "nine"

"Answer:"
So, the commonality is that 
each word includes a number 
(one, eight, nine, ten) hidden 
within its letters.

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The correct answer is 'c'. 6CO2 + 12H2O + Chlorophyll + Sunlight -> C6H12O6 + 6O2 + 6H2O.

This equation accurately summarizes the process of photosynthesis, incorporating all necessary reactants and products, as well as the essential factors of chlorophyll and sunlight that drive the process.

Explanation:

Photosynthesis is the process by which green plants convert light energy into chemical energy, producing glucose (C6H12O6) and releasing oxygen (O2) as a by-product. The overall balanced chemical equation for photosynthesis can be represented as:

6CO2 + 12H2O + Chlorophyll + Sunlight -> C6H12O6 + 6O2 + 6H2O

1. Carbon Dioxide (6CO2): CO2 from the atmosphere is a raw material that provides carbon atoms for glucose synthesis.
2. Water (12H2O): Water absorbed by the roots is split during the light-dependent reactions. Some of it is used in forming glucose, and some is released back as water.
3. Chlorophyll: This pigment located in chloroplasts captures sunlight energy necessary for driving the reactions.
4. Sunlight: Provides the energy required to split water molecules and drive the synthesis of glucose.
5. Glucose (C6H12O6): The main product, a carbohydrate that stores energy for the plant.
6. Oxygen (6O2): Released as a by-product from the splitting of water molecules.
7. Water (6H2O): Some water is regenerated and released as a product.

Options a and b either omit essential components like chlorophyll and sunlight or misrepresent the stoichiometry of water and oxygen in the reaction.

Option d incorrectly shows CO2 as a product, which contradicts the essence of photosynthesis where CO2 is a reactant consumed by the plant.

'Answer:' 
c

The correct answer is:

a. 6CO₂ + 12H₂O → C₆H₁₂O₆ + 6O₂ + 6H₂O

Explanation:
Photosynthesis is the process by which green plants, algae, and some bacteria convert carbon dioxide and water into glucose and oxygen using sunlight as the energy source. The general balanced chemical equation for photosynthesis is:   

[ 6CO₂ + 6H₂O + \text{light energy} → C₆H₁₂O₆ + 6O₂ ]

However, when considering the overall process within the plant cells, water is both consumed and produced. Specifically, water is used in the light-dependent reactions, and some of it is regenerated in the Calvin cycle. This leads to a more detailed balanced equation:

[ 6CO₂ + 12H₂O → C₆H₁₂O₆ + 6O₂ + 6H₂O ]

- 6CO₂: Carbon dioxide taken in from the atmosphere.
- 12H₂O: Water absorbed by the plant from the soil.
- C₆H₁₂O₆: Glucose produced, which serves as an energy source for the plant.
- 6O₂: Oxygen released into the atmosphere as a byproduct.
- 6H₂O: Water regenerated during the process.

Why the Other Options Are Incorrect:
b. 6CO₂ + H₂O + Sunlight → C₆H₁₂O₆ + O₂ + 6H₂O
  - Incorrect Balance: The number of water molecules does not balance correctly (only 1 H₂O is initially consumed).

c. 6CO₂ + 12H₂O + Chlorophyll + Sunlight → C₆H₁₂O₆ + 6O₂ + 6H₂O
  - Extra Components: While chlorophyll and sunlight are essential for photosynthesis, the summary equation typically focuses on the reactants and products without specifying catalysts or energy sources.

d. 6CO₂ + 12H₂O + Chlorophyll → C₆H₁₂O₆ + 6CO₂ + 6H₂O
  - Incorrect Products: This equation incorrectly shows the production of additional CO₂, which is not a product of photosynthesis.

#Answer:
Option A best represents the summary of the photosynthesis process.

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To solve this problem, we need to find the maximum value of BD^2 for the rhombus ABCD inscribed in the hyperbola x^2/20 - y^2/24 = 1 with diagonals intersecting at the origin.

We start by noting the properties of a rhombus:

1. All sides are equal in length.
2. The diagonals are perpendicular bisectors of each other.
3. The diagonals intersect at the origin.

Because the diagonals intersect at the origin and are perpendicular, they must lie along lines with slopes that are negative reciprocals of each other. Let’s define one diagonal along the line y = mx and the other diagonal along y = -1/mx, where m is the slope of the first diagonal.

The points where these lines intersect the hyperbola will give us the vertices of the rhombus. Substituting y = mx into the hyperbola’s equation provides us with the x-coordinates of points A and C:

x^2/20 - (mx)^2/24 = 1.

This simplifies to:

x^2(1/20 - m^2/24) = 1.

Similarly, substituting y = -1/mx gives us the x-coordinates of points B and D:

x^2(1/20 - 1/(m^2 * 24)) = 1.

From these equations, we can solve for the coordinates of B and D in terms of m.

Next, the length of diagonal BD is twice the distance from the origin to point B, since the rhombus is symmetric about the origin. Using the distance formula and the coordinates of B, we find:

BD^2 = 4(xB^2 + yB^2) = 4 xB^2 (1 + 1/m^2).

Substituting for xB^2 from the earlier equation, we get:

BD^2 = 4 * 1/β (1 + 1/m^2),

where β = 1/20 - 1/(m^2 * 24).

Simplifying, we obtain:

BD^2 = 4 a^2 b^2 (1 + m^2)/(m^2 b^2 - a^2).

To find the maximum value of BD^2, we analyze this function. By taking the derivative with respect to m^2 and setting it to zero, we find that this expression reaches a maximum when m^2 = 1.

Plugging m^2 = 1 back into the equation, along with a^2 = 20 and b^2 = 24, we find:

BD^2 = 8 a^2 b^2/(b^2 - a^2) = 8 * 20 * 24/(24 - 20) = 8 * 20 * 24/4 = 960.

'Answer:'
Therefore, the maximum value 
of BD^2 is 960.

To solve the problem, let's analyze the geometric conditions given:

1. Hyperbola Equation:
x^2/20 - y^2/24 = 1
Points A, B, C, and D lie on this hyperbola.

2. Rhombus Properties:
Opposite sides are equal.
Diagonals bisect each other at the origin (the midpoint).

3. Parametrizing the Points:
Let's denote:
A = (x1, y1), C = (-x1, -y1)
B = (x2, y2), D = (-x2, -y2)
Since ABCD is a rhombus, the diagonals AC and BD intersect at the origin, and:
AB = BC (distance equality)
This leads to the condition:
x1x2 + y1y2 = 0

4. Minimizing BD^2:
The length of diagonal BD is:
BD = 2sqrt(x2^2 + y2^2), BD^2 = 4(x2^2 + y2^2)
Subject to the hyperbola constraints:
x2^2/20 - y2^2/24 = 1
By solving the system of equations and applying the conditions, we find that:
BD^2 >= 480
The equality BD^2 = 480 cannot be achieved but can be approached as closely as desired.

5. Conclusion:
The greatest real number that is less than BD^2 for all such rhombi is 480.

#Answer:
480

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Clever trick!
You still have 4 marbles, 
but they're no longer in the cup 
because you turned it upside down! 

They're probably scattered 
around on the floor or counter now!

You still have 4 marbles in the cup, 
even though it is now upside down and
in the freezer