Question
Explain how recycling practices can lead to environmental sustainability.
Answers
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Recycling practices can lead to environmental sustainability in several ways:
1. Reducing the amount of waste in landfills: By recycling materials such as paper, plastics, glass, and metals, there is less waste going to landfills. This helps to reduce the amount of greenhouse gas emissions and other harmful environmental pollutants that are generated by landfills.
2. Conserving natural resources: Recycling conserves natural resources by reducing the need to extract new materials from the earth. For example, recycling aluminum cans can conserve up to 95% of the energy needed to produce new cans.
3. Reducing energy consumption: Recycling materials rather than producing new materials from scratch uses less energy. This is because recycling often requires less processing and transportation than producing new materials.
4. Preserving wildlife and habitats: By reducing the amount of waste going to landfills, recycling helps to preserve habitats and wildlife. It also reduces the amount of pollution in the air and water, which can have a positive impact on wildlife health.
Overall, recycling practices help to create a more sustainable environment by reducing waste, conserving natural resources, and reducing energy consumption and pollution levels.
1. Reducing the amount of waste in landfills: By recycling materials such as paper, plastics, glass, and metals, there is less waste going to landfills. This helps to reduce the amount of greenhouse gas emissions and other harmful environmental pollutants that are generated by landfills.
2. Conserving natural resources: Recycling conserves natural resources by reducing the need to extract new materials from the earth. For example, recycling aluminum cans can conserve up to 95% of the energy needed to produce new cans.
3. Reducing energy consumption: Recycling materials rather than producing new materials from scratch uses less energy. This is because recycling often requires less processing and transportation than producing new materials.
4. Preserving wildlife and habitats: By reducing the amount of waste going to landfills, recycling helps to preserve habitats and wildlife. It also reduces the amount of pollution in the air and water, which can have a positive impact on wildlife health.
Overall, recycling practices help to create a more sustainable environment by reducing waste, conserving natural resources, and reducing energy consumption and pollution levels.
The water level varies from 10 inches at low tide to 40 inches at high tide. Low tide occurs at 9:15 a.m. and high tide occurs at 3:30 p.m. What is a cosine function that models the variation in inches above and below the water level as a function of time in hours since 9:15 a.m.?
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EXPLAIN SHOW WORK
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We can use a cosine function to model the variation in inches above and below the water level as a function of time since 9:15 a.m. The basic form of a cosine function is:
y = A cos(Bx + C) + D
where:
- A is the amplitude of the function (half the distance between the highest and lowest points),
- B is the frequency of the function (the number of full cycles in 2π units of the input variable),
- C is the phase shift of the function (the horizontal displacement of the wave),
- D is the vertical shift of the function (the midpoint or mean value).
In this case, we know that the water level varies from 10 inches (lowest point or D) to 40 inches (highest point or D + A) and completes one cycle (from low tide to high tide and back to low tide) in 6 hours (from 9:15 a.m. to 3:15 p.m.).
So we can set up the cosine function as follows:
y = 15 cos(π/3 x + π/2) + 25
where:
- A = 15 (half the distance between 40 and 10)
- B = π/3 (one cycle in 6 hours or 2π/6 = π/3 radians per hour)
- C = π/2 (the function reaches its maximum value halfway between the low tide and high tide times)
- D = 25 (the midpoint or mean value between 10 and 40)
To see why the phase shift is π/2, we can think of it this way: at 9:15 a.m., the water level is at its lowest point (10 inches), so we want the cosine function to start at the low point on the horizontal axis. The standard cosine function starts at its maximum point (at x = 0), so we need to shift it by π/2 radians to the right to start at the low point. This means that the maximum point of the function will be halfway between the low tide (9:15 a.m.) and high tide (3:30 p.m.) times, which is at 12:22:30 p.m. (halfway between 9:15 a.m. and 3:30 p.m.).
So the cosine function that models the variation in inches above and below the water level as a function of time in hours since 9:15 a.m. is:
y = 15 cos(π/3 x + π/2) + 25
where x is the time in hours since 9:15 a.m. For example, at 10:30 a.m. (1.25 hours since 9:15 a.m.), the water level would be:
y = 15 cos(π/3 x + π/2) + 25
y = 15 cos(π/3 (1.25) + π/2) + 25
y ≈ 27.5 inches (above the 10-inch low tide level)
y = A cos(Bx + C) + D
where:
- A is the amplitude of the function (half the distance between the highest and lowest points),
- B is the frequency of the function (the number of full cycles in 2π units of the input variable),
- C is the phase shift of the function (the horizontal displacement of the wave),
- D is the vertical shift of the function (the midpoint or mean value).
In this case, we know that the water level varies from 10 inches (lowest point or D) to 40 inches (highest point or D + A) and completes one cycle (from low tide to high tide and back to low tide) in 6 hours (from 9:15 a.m. to 3:15 p.m.).
So we can set up the cosine function as follows:
y = 15 cos(π/3 x + π/2) + 25
where:
- A = 15 (half the distance between 40 and 10)
- B = π/3 (one cycle in 6 hours or 2π/6 = π/3 radians per hour)
- C = π/2 (the function reaches its maximum value halfway between the low tide and high tide times)
- D = 25 (the midpoint or mean value between 10 and 40)
To see why the phase shift is π/2, we can think of it this way: at 9:15 a.m., the water level is at its lowest point (10 inches), so we want the cosine function to start at the low point on the horizontal axis. The standard cosine function starts at its maximum point (at x = 0), so we need to shift it by π/2 radians to the right to start at the low point. This means that the maximum point of the function will be halfway between the low tide (9:15 a.m.) and high tide (3:30 p.m.) times, which is at 12:22:30 p.m. (halfway between 9:15 a.m. and 3:30 p.m.).
So the cosine function that models the variation in inches above and below the water level as a function of time in hours since 9:15 a.m. is:
y = 15 cos(π/3 x + π/2) + 25
where x is the time in hours since 9:15 a.m. For example, at 10:30 a.m. (1.25 hours since 9:15 a.m.), the water level would be:
y = 15 cos(π/3 x + π/2) + 25
y = 15 cos(π/3 (1.25) + π/2) + 25
y ≈ 27.5 inches (above the 10-inch low tide level)