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Chapter 0: Problem 7

The percentage of obese children aged \(12-19\) years in the United States is approximately $$P(t)=\left\\{\begin{array}{ll}0.04 t+4.6 & \text { if } 0 \leq t<10 \\ -0.01005 t^{2}+0.945 t-3.4 & \text { if } 10 \leq t \leq 30 \end{array}\right.$$ where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1970 . What was the percentage of obese children aged \(12-19\) years at the beginning of \(1970 ?\) At the beginning of \(1985 ?\) At the beginning of 2000 ?

Short Answer

Expert verified
The percentages of obese children aged 12-19 at the beginning of 1970, 1985, and 2000 were 4.6%, approximately 8.66%, and approximately 17.65%, respectively.

Step by step solution

01

Determine the appropriate function for the given year

First, we need to find the corresponding value of t for each year: - Beginning of 1970: t = 0 - Beginning of 1985: t = 1985 - 1970 = 15 - Beginning of 2000: t = 2000 - 1970 = 30 Now, we will determine which function applies to each year. For t=0, since 0 鈮� t < 10, we will use the first function (linear). For t=15 and t=30, since 10 鈮� t 鈮� 30, we will use the second function (quadratic).
02

Calculate the percentage at the beginning of 1970

Use the first function with t = 0: P(0) = 0.04 * 0 + 4.6 P(0) = 4.6 So, the percentage of obese children aged 12-19 at the beginning of 1970 was 4.6%.
03

Calculate the percentage at the beginning of 1985

Use the second function with t = 15: P(15) = -0.01005 * (15)^2 + 0.945 * 15 - 3.4 P(15) 鈮� 8.6625 So, the percentage of obese children aged 12-19 at the beginning of 1985 was approximately 8.66%.
04

Calculate the percentage at the beginning of 2000

Use the second function with t = 30: P(30) = -0.01005 * (30)^2 + 0.945 * 30 - 3.4 P(30) 鈮� 17.65 So, the percentage of obese children aged 12-19 at the beginning of 2000 was approximately 17.65%. In conclusion, the percentages of obese children aged 12-19 at the beginning of 1970, 1985, and 2000 were 4.6%, approximately 8.66%, and approximately 17.65%, respectively.

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