# Question Video: Writing and Evaluating Exponential Functions to Model Exponential Decay in a Real-World Context Mathematics • 9th Grade

A carβs value depreciates by π% every year. A new car costs π dollars. Write a function that can be used to calculate π, the carβs value in dollars, after π‘ years. What is the value of π for which the carβs value will be halved in 3 years? Give your answer to the nearest whole number.

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### Video Transcript

A carβs value depreciates by π percent every year. A new car costs π dollars. Write a function that can be used to calculate π, the carβs value in dollars, after π‘ years. What is the value of π for which the carβs value will be halved in three years? Give your answer to the nearest whole number.

For the first part of this question, weβre asked to find a function π, which is the value of a car, after π‘ years. Weβre told that the carβs value depreciates by π percent every year and that a new car costs π dollars. We can model depreciation of this type with an exponential decay function, recalling that an exponential decay function can take the form π of π‘ is equal to π multiplied by π raised to the power π‘, where π is between zero and one. π is the initial value. Thatβs π at time π‘ is equal to zero. And π is equal to uppercase π plus one. Thatβs where π is less than zero and represents the constant rate of change of π. And so, with this substitution, we have π of π‘ equal to π multiplied by one plus π raised to the power π‘.

Now, since the carβs value π depreciates by lowercase π percent every year, we can represent the decay rate β thatβs uppercase π β as a percentage in terms of lowercase π, that is, negative lowercase π over 100. Weβre told that a new car costs π dollars. So the initial value π in our exponential decay function is equal to π. And so our function for the value of a car after π‘ years is π of π‘ is equal to π multiplied by one minus lowercase π over 100 all raised to the power π‘.

Now, for the second part of the question, we want to find the value of π for which the carβs value will be halved in three years. That is the yearly percentage decrease for which the carβs value will be halved in three years. Now, if the initial value of the car is π, then half of this is π over two. And if the carβs value is halved in three years, then we have π of three is equal to π over two. But now if we substitute π‘ equal to three into our function π of π‘, we have that π of three is equal to π multiplied by one minus π over 100 raised to the power three.

And now equating our two expressions, we have an equation we can solve for π. First, dividing through by π, weβre left with one-half on the right-hand side and one minus π over 100 to the power three on the left-hand side. Next, taking the cubed root on both sides, we have one minus π over 100 is the cubed root of one-half. Now, multiplying both sides by negative one and adding one, we have π over 100 is equal to one minus the cubed root of one-half. And multiplying both sides by 100 on our left-hand side, we isolate π. And so we have π is equal to 100 multiplied by one minus the cubed root of one-half. And keying the right-hand side into our calculator gives us π is equal 20.6299 and so on. To the nearest whole number then, we have π is equal to 21 percent.

If a carβs value depreciates by π percent every year and a new car costs π dollars, we can calculate π, the carβs value, using an exponential decay function π of π‘ is equal to π multiplied by one minus π over 100 raised to the power π‘. And the value of π for which the carβs value will be halved in three years is 21 percent.