Question

Find the exponential function that satisfies the given conditions: initial value = 34, increasing at a rate of 7% per year.

Option 1: f(t) = 34 ⋅ 1.07ᵗ
Option 2: f(t) = 7 ⋅ 1.07ᵗ
Option 3: f(t) = 34 ⋅ 7ᵗ
Option 4: f(t) = 34 ⋅ 0.07ᵗ

Answer 1

The correct exponential function which starts with an initial value of 34 and increases at a rate of 7% per year is expressed as f(t) = 34 · 1.07^t, where t is the number of years.

Step-by-step explanation:

The question is asking us to find the exponential function that represents a situation where we have an initial value and a consistent percent growth rate per year. The initial value is given as 34, and the rate at which the function is increasing is 7% per year. These conditions suggest that the correct form of the exponential function would incorporate the initial value as the coefficient (the number in front of the base), and the growth rate as the base raised to the power of t, where t represents time in years.

Option 1, f(t) = 34 · 1.07t, correctly represents the initial value of 34 multiplying the growth factor of 1.07 raised to the power of t. Each year (each increment of t), the 1.07 factor represents a 7% increase over the previous year's value. Therefore, considering the conditions given, Option 1 is the correct exponential function representing this situation.