Question

revisiting exponents & their functions quick check
two years ago, marc bought a new cell phone. his purchase price was $750. the value of the phone for the next 2 years was $637.50 and $541.88, respectively. find the exponential function that represents the value of marc’s phone t years after it was purchased. (1 point)

• $f(t) = 750 cdot 0.85^t$
• $f(t) = 750 cdot 0.85t$
• $f(t) = 637.50 cdot 0.85^t$
• $f(t) = 750 cdot 1.15^t$

Explanation:

Step1: Recall exponential decay formula

The general form of an exponential function for decay is \( f(t) = a \cdot r^t \), where \( a \) is the initial value and \( r \) is the common ratio (decay factor, \( 0 < r < 1 \) for decay). Here, the initial purchase price (at \( t = 0 \)) is \( \$750 \), so \( a = 750 \).

Step2: Find the decay factor \( r \)

We know at \( t = 1 \) (1 year after purchase), the value is \( \$637.50 \). Using the formula \( f(1)=750\cdot r^1 \), substitute \( f(1) = 637.50 \):
\[
637.50 = 750r
\]
Solve for \( r \): \( r=\frac{637.50}{750}=0.85 \).

Step3: Check the function form

Now, check the options. The exponential function should have initial value \( 750 \) and decay factor \( 0.85 \), so the function is \( f(t)=750\cdot 0.85^t \). Let's verify with \( t = 2 \): \( f(2)=750\cdot 0.85^2 = 750\cdot 0.7225 = 541.875\approx541.88 \), which matches.

Answer:

A. \( f(t) = 750 \cdot 0.85^t \)