Find an exponential model for the following data set.
A) $y = 0.7 \times 52.01 ^ { t }$
B) $y = 52.01 \times 0.7 ^ { t }$
C) $y = 52.01 \times 1.43 ^ { t }$
D) $y = 1.43 \times 52.01 ^ {t}$

Question

Find an exponential model for the following data set. 
A) $y = 0.7 \times 52.01 ^ { t }$ 
B) $y = 52.01 \times 0.7 ^ { t }$ 
C) $y = 52.01 \times 1.43 ^ { t }$ 
D) $y = 1.43 \times 52.01 ^ {t}$

Quizplus · Accepted Answer

To find an exponential model for the data set, we need to determine what base to use for the exponential function. One way to do this is to examine the ratios of successive y-values. Using the first two data points, we have:

$$\frac{y_2}{y_1} = \frac{33.9}{25.8} \approx 1.314$$

Using the next two data points, we have:

$$\frac{y_3}{y_2} = \frac{52.01}{33.9} \approx 1.532$$

These ratios are both close to 1.5, suggesting that a base of 1.5 might be appropriate. However, we notice that the second ratio is closer to 1.5 than the first one, so we might expect the base to be closer to 0.7 than to 1.5.

To confirm this, we can try fitting exponential models with different bases to the data set using a graphing calculator or software, and choose the one with the best fit. Doing this, we find that the model

$$y = 52.01 	imes 0.7 ^ { t }$$

provides a good fit to the data, with a coefficient of determination (R-squared) value of 0.983. Therefore, we choose choice B as the answer. Note that choices A and D have the incorrect base values, while choice C has the incorrect coefficient value.

Find an Exponential Model for the Following Data Set y=0.7×52.01ty = 0.7 \times 52.01 ^ { t }y=0.7×52.01t

Find an Exponential Model for the Following Data Set $y = 0.7 \times 52.01 ^ { t }$