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

Responses

an=80,000(0.06)n

an=80,000(0.94)n

an=80,000(0.94)n−1

an=80,000(0.06)n−1

Answer 1

To model the depreciation of the equipment, you would use the equation that accounts for the remaining value of the equipment after each year, which decreases by 6% annually.

This means that each year, the equipment retains 94% (which is 100% - 6%) of its value. Therefore, the sequence modeling this situation can be represented by the equation:

an = 80,000(0.94)^n

Where:

• \( a_n \) is the value of the equipment after \( n \) years.
• \( 80,000 \) is the initial value of the equipment.
• \( 0.94 \) represents the value remaining after 6% depreciation each year.
• \( n \) is the number of years since the equipment was purchased.

So, the correct response is:

an=80,000(0.94)^n

Answer 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.
Responses

y=62(1.04)t

y=62(0.04)t

y=62(0.6)t

y=62(0.96)t

Answer 3

To model the exponential decay of the elephant herd, we need to account for the information that the population is declining at a rate of 4% each year. This means that each year the population retains 96% of its previous population (100% - 4% = 96%).

The general form for an exponential decay function is:

\[ y = y_0 \times (1 - r)^t \]

Where:

• \( y_0 \) is the initial quantity (62 elephants in this case),
• \( r \) is the rate of decline (0.04 for a 4% decline),
• \( t \) is the number of years.

Thus, the equation would be:

\[ y = 62 \times (0.96)^t \]

Hence, the correct response from your options is:

y = 62(0.96)^t