Final answer:
The mean monthly rate of growth of the coral, with a change in diameter from 50 cm to 58 cm over one year, is approximately 2391.79 cm³/month. The calculation involves finding the change in volume of the spherical coral and dividing by the number of months in a year.
Step-by-step explanation:
To calculate the mean rate of growth of a coral in cm³ per month, if its diameter changes from 50 cm to 58 cm in one year, we need to find the change in volume and then divide it by the number of months in a year. The formula for the volume V of a sphere is V = 4/3 πr³, where r is the radius of the sphere.
First, calculate the initial volume (V1) with a radius of 25 cm (half of 50 cm):
- V1 = 4/3 π (25 cm)³ = 4/3 π (15625 cm³) ≈ 52359.85 cm³
Then, calculate the final volume (V2) with a radius of 29 cm (half of 58 cm):
- V2 = 4/3 π (29 cm)³ = 4/3 π (24389 cm³) ≈ 81061.35 cm³
The change in volume ΔV is V2 - V1 which is 81061.35 cm³ - 52359.85 cm³ = 28701.5 cm³.
To find the monthly rate of growth, divide the change in volume by the number of months in a year (12):
- Rate = ΔV / 12 months = 28701.5 cm³ / 12 months ≈ 2391.79 cm³/month
The mean monthly rate of growth of the coral is approximately 2391.79 cm³/month.