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
To solve this problem, we first need to calculate the growth rate as a percentage. Then, we can use this growth rate to estimate the diameter of the stain in the future, both using discrete exponential growth and linear growth.
(1) To calculate the growth rate, we need to find the difference in diameter over the time period. The diameter increased from 4 inches to 5.3 inches, which is a difference of 5.3 - 4 = 1.3 inches.
Next, we divide this difference by the initial diameter (4 inches) and multiply by 100 to convert it into a percentage:
Growth rate = (1.3/4) * 100
Calculating this expression, we find that the growth rate is approximately 32.5%. Therefore, the growth rate over this time period is 32.5%.
(2) To estimate the diameter of the stain in 5 days using discrete exponential growth, we'll start with the initial diameter (4 inches) and apply the growth rate of 32.5% for each day. Since we need to calculate the diameter after 5 days, we raise (1 + growth rate) to the power of 5:
Estimated diameter = 4 * (1 + 0.325)^5
Calculating this expression, we find that the estimated diameter of the stain after 5 days using discrete exponential growth is approximately 8.19 inches.
(3) To estimate the diameter of the stain in 5 days using linear growth, we'll find the average growth rate over the given time period. Linear growth assumes a constant increase over time. To find the average growth rate, we divide the difference in diameter (1.3 inches) by the number of days (1 day):
Average growth rate = 1.3/1 = 1.3 inches per day
Now, we can estimate the diameter after 5 days using this average growth rate:
Estimated diameter = 4 + (1.3 * 5)
Calculating this expression, we find that the estimated diameter of the stain after 5 days using linear growth is approximately 10.5 inches.