1. A local bakery opened with new equipment worth $80,000. Ten years later, the owners’ accountants told them that their equipment had depreciated at a rate of 6 percent per year. Which equation would you use to create a sequence modeling this situation?

Question

1. A local bakery opened with new equipment worth $80,000. Ten years later, the owners’ accountants told them that their equipment had depreciated at a rate of 6 percent per year. Which equation would you use to create a sequence modeling this situation?
A) an=80,000(0.94)^n−1
B) an=80,000(0.06)^n
C) an=80,000(0.06)^n−1
D) an=80,000(0.94)^n

2. In a certain geographic location, a herd of elephants is declining at a rate of four percent every year. If there are currently 62 elephants in the herd, create an exponential decay function to model the problem. Let y represent the number of elephants after t years.
A) y = 62(0.04)^t
B) y = 62(1.04)^t
C) y = 62(0.6)^t
D) y = 62(0.96)^t

4. Use the graph of the exponential decay function that models an exponential decay sequence to discuss the properties and determine the equation for the horizontal asymptote of the graph.
A) The horizontal asymptote is at y = −7
B) The horizontal asymptote is at y = 0.3
C) The horizontal asymptote is at y = 5
D) The horizontal asymptote is at y = 0

GPT-4o mini · Answer

1. To determine the correct equation for the depreciation of the equipment, we can use the formula for exponential decay. The equipment is worth $80,000 and depreciates at a rate of 6% per year. This means that after each year, the value is multiplied by (1 - 0.06) or 0.94. The correct formula for the value $ a_n $ after $ n $ years would be: $$ a_n = 80,000(0.94)^n $$ However, since the problem specifies $ n-1 $, the correct answer must be: **D) $ a_n = 80,000(0.94)^{n} $** 2. To create an exponential decay function for the herd of elephants, we start with the current population of 62 elephants and apply the decay rate, which is 4% per year. The decay factor will be (1 - 0.04) or 0.96. Therefore, the equation representing the number of elephants after $ t $ years is: $$ y = 62(0.96)^t $$ So, the correct answer is: **D) $ y = 62(0.96)^t $** 4. The horizontal asymptote of an exponential decay function represents the value that the function approaches as the input (time, in this case) goes to infinity. As the population of elephants decays, it will approach 0 but never actually reach it. Hence, the horizontal asymptote is: **D) The horizontal asymptote is at $ y = 0 $**.