A circular water stain on your home ceiling had a diameter of about 4 inches across on Monday at 8 A.M. On Tuesday at the same time it was about 5.3 inches across. The unit for time is days.
(1) Calculate the growth rate as a percentage to at least the nearest 10-thousandth over this time period.
(2) Assume that the growth rate can be applied over the next 5 days. Calculate the estimated diameter of the stain in 5 days from the initial measurement using discrete exponential growth. inches across
(3) Calculate the estimated diameter of the stain in 5 days from the initial measurement using a linear growth. inches across
(1) To calculate the growth rate as a percentage, we can use the formula:
Growth Rate = (Final Size - Initial Size) / Initial Size * 100
Given that the initial size was 4 inches and the final size was 5.3 inches, the growth rate can be calculated as follows:
Growth Rate = (5.3 - 4) / 4 * 100
= 1.3 / 4 * 100
= 0.325 * 100
= 32.5%
Therefore, the growth rate as a percentage is 32.5%.
(2) To calculate the estimated diameter of the stain in 5 days using discrete exponential growth, we can use the formula:
Final Size = Initial Size * (1 + Growth Rate)^Time
Given that the initial size was 4 inches, the growth rate is 32.5%, and the time is 5 days, the estimated diameter of the stain can be calculated as follows:
Final Size = 4 * (1 + 0.325)^5
= 4 * (1.325)^5
≈ 4 * 2.069
≈ 8.276 inches
Therefore, the estimated diameter of the stain in 5 days using discrete exponential growth is approximately 8.276 inches.
(3) To calculate the estimated diameter of the stain in 5 days using linear growth, we can use the formula:
Final Size = Initial Size + (Growth Rate * Time)
Given that the initial size was 4 inches, the growth rate is 32.5%, and the time is 5 days, the estimated diameter of the stain can be calculated as follows:
Final Size = 4 + (0.325 * 5)
= 4 + 1.625
= 5.625 inches
Therefore, the estimated diameter of the stain in 5 days using linear growth is 5.625 inches.
To find the growth rate as a percentage, we can use the formula:
Growth rate = ((Final diameter - Initial diameter) / Initial diameter) * 100
Let's calculate it:
(1) Growth rate = ((5.3 - 4) / 4) * 100 ≈ 32.5%
Now, let's calculate the estimated diameter of the stain in 5 days using discrete exponential growth. Discrete exponential growth assumes that the growth rate remains constant over the given time period.
(2) Estimated diameter = Initial diameter * (1 + growth rate)^(number of time periods)
Here, the initial diameter is 4 inches, the growth rate is 0.325 (32.5% as a decimal), and the number of time periods is 5.
Estimated diameter = 4 * (1 + 0.325)^5 ≈ 7.844 inches
Now, let's calculate the estimated diameter of the stain in 5 days using linear growth. Linear growth assumes a constant increase in diameter over each time period.
(3) Estimated diameter = Initial diameter + (growth rate * number of time periods)
Here, the initial diameter is 4 inches, the growth rate is 0.325 (32.5% as a decimal), and the number of time periods is 5.
Estimated diameter = 4 + (0.325 * 5) = 4 + 1.625 ≈ 5.625 inches
Therefore, the estimated diameter of the stain in 5 days using discrete exponential growth is approximately 7.844 inches, and using linear growth is approximately 5.625 inches.