The price of a computer component is decreasing at a rate of 12​% per year. State whether this decrease is linear or exponential. If the component costs ​$50 ​today, what will it cost in three​ years?

The decrease in price at a constant percentage rate of 12% per year suggests that the change is **exponential** rather than linear. This is because a percentage-based decrease affects the current value, not the initial value, and hence compounds over time. 

To calculate the price in three years, we use the formula for exponential decay:

\[
P = P_0 \times (1 - r)^t
\]

where:
- $ P_0 = 50 $ (initial price today),
- $ r = 0.12 $ (the rate of decrease),
- $ t = 3 $ (the number of years).

Substitute the values:

\[
P = 50 \times (1 - 0.12)^3
\]
\[
P = 50 \times (0.88)^3
\]

Calculating $ (0.88)^3 $:

\[
(0.88)^3 \approx 0.6815
\]

Thus:

\[
P \approx 50 \times 0.6815 = 34.08
\]

**Therefore, the cost of the component in three years will be approximately $34.08.**

Would you like more details or have any questions?

Here are 5 related questions to explore this topic further:

1. How would the calculation change if the decrease rate was compounded monthly instead of annually?
2. How much would the component cost in 5 years at this rate?
3. If the rate of decrease changes to 15%, how does the final cost after 3 years compare?
4. Can you calculate the time it would take for the component to reach half of its original price?
5. What would the cost be if the decrease were linear rather than exponential?

**Tip:** In exponential decay, the percentage decrease applies to the remaining value each period, leading to a gradually slowing decrease over time.

The price of a computer component is decreasing at a rate of 12% per year. State whether this decrease is linear or exponential. If the component costs $50 today, what will it cost in three years?

Understand exponential decay with this example, where the price of a computer component decreases by 12% per year. Learn how to calculate the component's price in three years if it costs $50 today. Explore key mathematical concepts like exponential decay and percentage decrease, suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation