Question

a cell phone manufacturer recorded the number of phones sold in a year. they expect the sales volume to increase every quarter. the approximate number of phones sold by the manufacturer, t years, after the data was recorded is shown by the following expression. 55,000(1.08)^{4t} which statement below best describes the coefficient, 55,000? ○a. the number of times the number of phones sold has compounded since it was recorded ○b. the increase in the number of phones sold every quarter ○c. the rate at which the number of phones sold is increasing ○d. the recorded number of phones sold in the initial year

Explanation:

The general form of an exponential growth function is \(y = a(1 + r)^{nt}\), where \(a\) is the initial amount. In the given expression \(55000(1.08)^{4t}\), when \(t = 0\) (the initial year, right after the data was recorded), the value of the expression is \(55000(1.08)^{0}=55000\times1 = 55000\).

• Option A: The number of times compounded is related to \(4t\) (since compounding is quarterly, \(n = 4\) and \(t\) is the number of years).
• Option B: The increase per quarter is related to the growth factor \((1.08)\) (where the growth rate per quarter \(r=0.08\) or \(8\%\)).
• Option C: The rate of increase is \(8\%\) per quarter (from the factor \((1 + 0.08)\)).
• Option D: When \(t = 0\) (initial year), the value of the function \(55000(1.08)^{4t}\) is \(55000\), which represents the initial - recorded number of phones sold.

Answer:

D. the recorded number of phones sold in the initial year