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(1 point) The table below gives U.S. lead emissions, L, into the environment in millions of metric tons for 1970−1992. Let t=0 in 1970. (a) Find an exponential model for these data. Lead emission, L(t)= (b) Find a fourth-degree polynomial model for these data. Lead emission, L(t)= (c) Which of these curves gives a better model for the data? A. fourth-degree polynomial B. exponential model (d) Estimate the lead emission in 1981 according to the polynomial model in part (b).

User Mad
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Final answer:

Creating an exponential or polynomial model requires the specific data, which is missing here. However, an exponential model often follows L(t) = a · e^{kt}, while a fourth-degree polynomial model generally looks like L(t) = at^4 + bt^3 + ct^2 + dt + e. To choose the better model, compare the R-squared values; to estimate emissions in 1981 with the polynomial, substitute t = 11 into the equation.

Step-by-step explanation:

The student has presented a scenario that requires the development of both an exponential model and a fourth-degree polynomial model to represent U.S. lead emissions over a period of time. Without the specific data on lead emissions, creating exact models is not possible.

However, an exponential model typically has the form L(t) = a · ekt, where a is the initial amount of lead emissions, e is the base of the natural logarithm, k is the growth/decay rate, and t is the time in years since 1970. A fourth-degree polynomial model could have the form L(t) = at4 + bt3 + ct2 + dt + e, with constants a, b, c, d, and e to be determined by fitting the model to the data points.

To determine which model gives a better fit, one would typically calculate the coefficient of determination (R-squared value) for each model and compare these values. The model with the higher R-squared value fits the data better.

For part (d), without the actual polynomial equation, we are unable to directly compute the lead emission for the year 1981. However, if the polynomial model were known, one would simply substitute t = 11 into the model (since 1981 is 11 years after 1970) and calculate the lead emission.

User Saru
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