Jason wants to buy a new HD television but he thinks
that if he waits, the quality of HD televisions will improve. The television he wants
to buy costs $2500 now, and based on pricing trends, Jason thinks that the price
will increase by 4% each year. Use the equation to predict when a new HD television will
cost $3000.

To determine when the cost of the television will reach $3000, we can use an exponential growth equation. The general formula for exponential growth is:

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

where:
- $ P $ is the future price of the television,
- $ P_0 $ is the initial price of the television ($2500 in this case),
- $ r $ is the rate of increase per year (4% or 0.04),
- $ t $ is the number of years from now.

Given:
- $ P = 3000 $,
- $ P_0 = 2500 $,
- $ r = 0.04 $.

Now we substitute and solve for $ t $:

\[
3000 = 2500 \times (1 + 0.04)^t
\]

1. First, divide both sides by 2500:

\[
1.2 = (1.04)^t
\]

2. To solve for $ t $, take the natural logarithm of both sides:

\[
\ln(1.2) = t \cdot \ln(1.04)
\]

3. Now, solve for $ t $:

\[
t = \frac{\ln(1.2)}{\ln(1.04)}
\]

Calculating the values:

\[
t \approx \frac{0.1823}{0.0392} \approx 4.65
\]

Thus, it will take approximately 4.65 years for the price of the television to reach $3000. Jason can expect the price to reach this amount in about **4 years and 8 months**.

Would you like further details on the steps or have additional questions?

Here are some related questions you might explore:
1. What would the price be after 5 years if the rate remains the same?
2. How would the waiting time change if the growth rate was only 3% instead of 4%?
3. What is the formula to find the price in any future year given the rate and initial cost?
4. How does exponential growth compare with linear growth in pricing trends?
5. How would the price change if it decreases by 4% each year?

**Tip:** When solving for time in exponential growth, logarithms are very useful for isolating the exponent.

Jason wants to buy a new HD television but he thinks that if he waits, the quality of HD televisions will improve. The television he wants to buy costs $2500 now, and based on pricing trends, Jason thinks that the price will increase by 4% each year. Use the equation to predict when a new HD television will cost $3000.

Learn how to calculate when the price of an HD television will reach $3000, given a 4% annual price increase starting from $2500. This solution uses exponential growth equations and logarithms, suitable for students in Grades 9-12.

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