# A blue supergiant star has a radius of 7.4 × 10^10 m. The spherical surface behaves as a blackbody radiator. If the surface temperature is 2.4 × 10^4 K, what is the rate at which energy is radiated from the star?

2 × 10^3 J/s
4 × 10^19 J/s
2 × 10^32 J/s
3 × 10^33 J/s

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## Well, this blue supergiant star sure knows how to bring the heat! Let's calculate the rate at which energy is radiated from this stellar firework.

To do that, we can use the Stefan-Boltzmann law, which states that the rate of energy radiated by a blackbody is proportional to the fourth power of its temperature, multiplied by the surface area.

So, the rate at which energy is radiated (P) can be calculated as:

P = σ * A * T^4

Where σ is the Stefan-Boltzmann constant (5.67 × 10^-8 W/m^2K^4), A is the surface area of the star, and T is the temperature.

The surface area of a sphere is given by:

A = 4πr^2

Plugging in the values:

r = 7.4 × 10^10 m
T = 2.4 × 10^4 K

A = 4π(7.4 × 10^10)^2
A ≈ 6.85 × 10^22 m^2

Now let's calculate the rate of energy radiated from the star:

P = 5.67 × 10^-8 * 6.85 × 10^22 * (2.4 × 10^4)^4

P ≈ 2 × 10^32 J/s

So, it looks like the rate at which energy is radiated from this star is approximately 2 × 10^32 J/s! That's a lot of energy! Keep shining, blue supergiant!

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## To calculate the rate at which energy is radiated from the blue supergiant star, we can use the Stefan-Boltzmann Law, which states that the power radiated by a blackbody is proportional to the fourth power of its temperature and its surface area. The equation for the power radiated is:

P = σ * A * T^4

Where:
P = power radiated (in watts)
σ = Stefan-Boltzmann constant (5.67 × 10^-8 W m^-2 K^-4)
A = surface area of the star (4πr^2)
T = temperature of the star (in Kelvin)
r = radius of the star

Given:
Radius of the star (r) = 7.4 × 10^10 m
Temperature of the star (T) = 2.4 × 10^4 K

First, let's calculate the surface area of the star:

A = 4πr^2
A = 4π * (7.4 × 10^10)^2

Next, let's substitute the values into the equation:

P = σ * A * T^4
P = 5.67 × 10^-8 * 4π * (7.4 × 10^10)^2 * (2.4 × 10^4)^4

Calculating the above expression will give us the power radiated from the star in watts. Let's calculate it:

P ≈ 3.6 × 10^33 J/s

Therefore, the rate at which energy is radiated from the blue supergiant star is approximately 3.6 × 10^33 J/s.

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## To calculate the rate at which energy is radiated from the blue supergiant star, you can use the Stefan-Boltzmann Law. This law states that the total power radiated by a blackbody is proportional to the fourth power of its temperature and its surface area.

The formula for the power radiated by a blackbody is given by:

P = σ * A * T^4

Where:
σ = Stefan-Boltzmann constant (5.67 × 10^-8 W/m^2K^4)
A = Surface area
T = Temperature

To solve this, you need to calculate the surface area of the blue supergiant star.

The surface area (A) of a sphere is given by:

A = 4πr^2

Where:
A = Surface area
π = Pi (approximately 3.14159)

Now, let's calculate the surface area of the blue supergiant star:

A = 4πr^2
A = 4 * 3.14159 * (7.4 × 10^10)^2
A ≈ 7.2419 × 10^22 m^2

Now you can substitute the values into the Stefan-Boltzmann law equation:

P = σ * A * T^4
P = (5.67 × 10^-8) * (7.2419 × 10^22) * (2.4 × 10^4)^4

By calculating this expression, you'll find that the power radiated, or the rate at which energy is radiated from the star, is approximately 2 × 10^32 J/s.

Therefore, the correct option is:

2 × 10^32 J/s