Question

f(t) = 8,000(0.65)^t
the given function f models the number of coupons a company sent to their customers at
the end of each year, where t represents the number of years since the end of 1998, and
0 ≤ t ≤ 5. if y = f(t) is graphed in the ty-plane, which of the following is the best
interpretation of the y-intercept of the graph in this context?
a the minimum estimated number of coupons the company sent to their
customers during the 5 years was 1,428.

b the minimum estimated number of coupons the company sent to their
customers during the 5 years was 8,000.

c the estimated number of coupons the company sent to their customers at
the end of 1998 was 1,428.

d the estimated number of coupons the company sent to their customers at
the end of 1998 was 8,000.

Explanation:

Step1: Recall y-intercept definition

The y - intercept of a function \(y = f(t)\) occurs when \(t = 0\).

Step2: Substitute \(t = 0\) into the function

Given \(f(t)=8000(0.65)^{t}\), when \(t = 0\), we use the property \(a^{0}=1\) (for \(a
eq0\)). So \(f(0)=8000(0.65)^{0}=8000\times1 = 8000\).

Step3: Interpret the context

\(t\) represents the number of years since the end of 1998. When \(t = 0\), it is the end of 1998. So \(f(0) = 8000\) means the estimated number of coupons sent at the end of 1998 is 8000. Now let's analyze the options:

• Option A: The minimum value of \(f(t)\) for \(0\leq t\leq5\) occurs at \(t = 5\) (since \(0.65\lt1\), the function is decreasing). \(f(5)=8000(0.65)^{5}\approx8000\times0.1785 = 1428\), but this is not the y - intercept, so A is wrong.
• Option B: As we saw, the minimum is at \(t = 5\) (1428), not 8000, so B is wrong.
• Option C: When \(t = 0\), \(f(0)=8000

eq1428\), so C is wrong.

• Option D: When \(t = 0\) (end of 1998), \(f(0) = 8000\), which matches the interpretation.

Answer:

D. The estimated number of coupons the company sent to their customers at the end of 1998 was 8,000.