Write a function in terms of t that represents the situation. A smartphone costs $850, and its resell value decreases by 11.2% each year.

Understand the Problem

The question is asking us to formulate a mathematical function that describes the resale value of a smartphone over time, given its initial cost and the rate at which its value decreases annually.

Answer

The resale value function is $V(t) = 850 \cdot (0.888)^t$.

Steps to Solve

Identify the initial value and depreciation rate

The initial cost of the smartphone is $850. The resale value decreases by 11.2% each year. To express this mathematically, we note that after one year, the value will be reduced to 88.8% of its original value (since $100% - 11.2% = 88.8%$).

Determine the decay factor

The decay factor can be represented as a decimal. To find this, convert the percentage decrease into a decimal: $$ \text{Decay factor} = 1 - 0.112 = 0.888 $$

Write the function in terms of time

After $t$ years, the value of the smartphone can be represented using the formula for exponential decay: $$ V(t) = V_0 \cdot (decay , factor)^{t} $$ Where:

$V_0$ is the initial value ($850$)
The decay factor is $0.888$

Thus, the function becomes: $$ V(t) = 850 \cdot (0.888)^t $$

The function representing the resale value of the smartphone is given by: $$ V(t) = 850 \cdot (0.888)^t $$

More Information

This function models the depreciation of the smartphone's value over time. Each year, its resale value decreases to approximately 88.8% of what it was the previous year due to the 11.2% annual depreciation.

Tips

Ignoring the conversion of percentage rates: Always convert percentage decreases properly into a decimal format.
Using incorrect formula: Make sure to apply the exponential decay formula correctly, as it specifically accounts for continuous depreciation over time.

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