27–30. Designing exponential decay functions Devise an exponen...

Question

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?

Accepted Answer

Welcome back, everyone. A town had 950 car thefts per year in 2015. If the number of car thefts decreases by 2.5% per year, after how many years will the annual car thefts reach 700? Round your answer to the nearest whole number A 5 years, B 8 years, C 12 years, and D 14 years. For this problem because we have a constant. Rate of growth, we're going to use an exponential model, right? Let's recall that N of T is equal to. And subscript 0 multiplied by 1 + R raise to the power of T. So, what we want to do is simply identify N subscript zero, which is the initial number of car thefts, and subscript 0 is 950. R is the growth rate, and we have to understand that it is going to be negative because it says decreases. So, we're going to add a negative sign. And that'll be 2.5%. We want to use it in its decimal form, which would be 0.025. So now we can show that our model is N of T equals 950 multiplied by 1 minus 0.025, raised the power of T. Which is 950 multiplied by 0. 975 to the power of T. We know that at some point. The number of annual car thefts will reach 700, right, we're going to set NFT. Equals 700. On the left hand side we get 700, and on the right hand side of our equation we have 950. Multiplied by 0.975 to the power of T, and this is the equation that we want to solve for time T. What we can do is simply divide both sides by 950, so we get 700. Divided by 950. Is equal to 0.975 to the power of T. From here we can take a land of both sides. And if we do that, according to the properties of logarithms, we can rewrite our right hand side as T multiplied by LN of 0.975, right? We can basically bring down the exponent. And now solving for time, we can show that T is equal to our left hand side, LN of 700 divided by 950. Divided by the right hand side, LN of 0.975, right? We're dividing by that factor LN of 0.975 to solve for C. Now, what we're going to do is approximate the results. This is approximately equal to 12 years, which corresponds to the answer to C. Thank you for watching.

Watch next