Find Expontntial Model that fits the points shown in the graph of table01510102841.jpg

Question
Exponential models
asked 2020-12-17

Find Expontntial Model that fits the points shown in the graph of table
image

Answers (1)

2020-12-18
Let us consider the following exponential model
\(\displaystyle{y}={a}{e}^{{{x}}}\)
Now, with given point (0,5), the equation becomes
\(\displaystyle{5}={a}{e}^{{{b}{\left({0}\right)}}}\)
\(\displaystyle{5}={a}{e}^{{0}}\)
Therefore,
a=5
With the given point (4,1) and a=1, the equation becomes
\(\displaystyle{1}={5}{e}^{{{b}{\left({4}\right)}}}\)
Dividing both sides by 5
\(\displaystyle\frac{{1}}{{5}}={e}^{{{4}{b}}}\)
Taking natural logarithm to the base e
\(\displaystyle\frac{{\ln{{1}}}}{{5}}={{\ln{{e}}}^{{{4}{b}}}}\)
Using the inverse property \(\displaystyle{{\ln{{e}}}^{{x}}}\)
\(\displaystyle\frac{{\ln{{1}}}}{{5}}={4}{b}\)
Therefore,
\(\displaystyle{b}=\frac{{\frac{{\ln{{1}}}}{{5}}}}{{4}}\)
b=-0.4024
Hence, the equation of the curve with a=5 and b=-0.4024 is \(\displaystyle{y}={e}^{{-{0.4024}{x}}}\)
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