An overweight man makes lifestyle changes in order to lose weight. His starting weight is 290 pounds and he has set a target weight of 200 pounds. Let t be the time (in months), and let D be the difference (in pounds) between his weight at time t and his target weight. Each month the difference D decreases by 15%.
(a) Make an exponential model of D versus the time t in months since the diet began.
D(t) =
(b) How long will it take for his weight to reach 220 pounds? (Round your answer to two decimal places.)
months
D(t) = 290 * 0.85^t
(a) To create an exponential model of D versus the time t in months, we need to understand how the weight loss progresses each month. We are told that the difference D decreases by 15% each month. This means that each month, the difference D is multiplied by (1 - 0.15) or 0.85.
Given that the initial weight is 290 pounds and the target weight is 200 pounds, the initial difference D is 290 - 200 = 90 pounds.
Using this information, we can create the exponential model:
D(t) = 90 * (0.85)^t
(b) To find how long it will take for his weight to reach 220 pounds, we need to solve for t in the equation D(t) = 90 * (0.85)^t = 220.
We can rearrange the equation as follows:
(0.85)^t = 220/90
(0.85)^t = 2.44
Taking the logarithm of both sides, we have:
t * log(0.85) = log(2.44)
Now, we can solve for t:
t = log(2.44) / log(0.85)
Using a calculator, we find:
t ≈ 5.91
Therefore, it will take approximately 5.91 months for his weight to reach 220 pounds. Round the answer to two decimal places, the answer is 5.91 months.