Depreciation of equipment A large die-casting machine used to make automobile engine blocks is purchased for $2.5 million. For tax purposes, the value of the machine can be depreciated by 6.8% of its current value each year.


b. After how many years is the value of the machine 10% of its original value?

27–30. Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.

Crime rate The homicide rate decreases at a rate of 3%/yr in a city that had 800 homicides/yr in 2018. At this rate, when will the homicide rate reach 600 homicides/yr?

27–30. Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time.

Valium metabolism The drug Valium is eliminated from the bloodstream with a half-life of 36 hr. Suppose a patient receives an initial dose of 20 mg of Valium at midnight. How much Valium is in the patient’s blood at noon the next day? When will the Valium concentration reach 10% of its initial level?

After the introduction of foxes on an island, the number of rabbits on the island decreases by 4.5% per month. If y(t) equals the number of rabbits on the island t months after foxes were introduced, find the rate constant k for the exponential decay function y(t) = y₀eᵏᵗ.

Depreciation of equipment A large die-casting machine used to make automobile engine blocks is purchased for $2.5 million. For tax purposes, the value of the machine can be depreciated by 6.8% of its current value each year.

a. What is the value of the machine after 10 years?

13–14. Absolute and relative growth rates Two functions f and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant.

f(t) = 100 + 10.5t, g(t) = 100e^(t/10)

Determine if the function is an exponential function.  

If so, identify the power & base, then evaluate for $x=4$.

$f\left(x\right)=\left(-2\right)^{x}$

Determine if the function is an exponential function.  

If so, identify the power & base, then evaluate for $x=4$ .

$f\left(x\right)=3\left(1.5\right)^{x}$

Determine if the function is an exponential function.  

If so, identify the power & base, then evaluate for $x=4$ .

$f\left(x\right)=\left(\frac12\right)^{x}$