Question # A share of ABC stock was worth $60 in 2005 and only worth$45 in 2010. a. Find the multiplier and the percent decrease. b. Write an exponential functi

Exponential models
ANSWERED A share of ABC stock was worth $60 in 2005 and only worth$45 in 2010. a. Find the multiplier and the percent decrease. b. Write an exponential function that models the value of the stock starting from 2005. c. Assuming that the decline in value continues at the same rate , use your answer to (b) to predict the value in 2020. 2021-05-08

Deternmine the multiplier. Since 2005 is the start, this will be 0. The point will be (0,60).
$$\displaystyle{y}={a}{b}^{{x}}$$ Write the equation
$$\displaystyle{60}={a}{b}^{{0}}$$ Substitute the values
$$60=a(1)$$ Use zero rule of exponent
$$60=a$$ Simplify
In order to determine b, use point (5,45) for 2010.
$$\displaystyle{y}={a}{b}^{{x}}$$ Write the equation
$$\displaystyle{45}={60}{\left({b}\right)}^{{5}}$$ Substitute the values
$$\displaystyle\frac{{45}}{{65}}={60}\frac{{\left({b}\right)}^{{5}}}{{60}}$$ Divide both sides by 60
$$0.75=b^5$$ Simplify
$$\displaystyle{5}\sqrt{{0.75}}={5}\sqrt{{b}}^{{5}}$$ Use radical to remove exponent
0.944=b Simplify
The multiplier is 0.944. Subtract the multiplier from 1 to determine the percent decrease. 1-0.944=0.056= 5.6%
The exponential function for the given situation is: $$\displaystyle{f{{\left({x}\right)}}}={60}{\left({0.944}\right)}^{{x}}$$
Using the exponential function, $$\displaystyle{f{{\left({x}\right)}}}={60}{\left({0.944}\right)}^{{x}}$$, determine the value in 2020, which is 15 years from 2005.
$$\displaystyle{f{{\left({x}{0}\right)}}}={60}{\left({0.944}\right)}^{{x}}$$ Write the function
$$\displaystyle{f{{\left({15}\right)}}}={60}{\left({0.944}\right)}^{{15}}$$ Substitute the values
$$f(15)=25.28$$ Perform operation