Negative Exponents in a Declining Balance
A financial planner compares two ways of writing a declining balance. The original model is A(t) = 5000(1 − 0.04)^t, where t is years and the balance is in dollars. A colleague rewrites it as B(t) = 5000(1 + 0.04)^(−t), arguing that a negative exponent turns the growth factor into a decay factor. For t = 5, the planner's spreadsheet gives A(5) ≈ 4076.86, while the colleague's formula gives B(5) ≈ 4109.64. The planner needs to know which rewriting is algebraically equivalent to A(t), whether either expression can be simplified to 5000(0.96)^t, and how to check the discrepancy without relying on the spreadsheet. The account has a 4% annual decline, and t is measured in whole years. A second question is whether B(t) can be made equal to A(t) by changing the base to 1/(1.04) instead of 1.04.
1. Rewrite A(t) = 5000(1 − 0.04)^t in the form 5000b^t and evaluate A(5) to two decimal places.
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Since 1 − 0.04 = 0.96, A(t) = 5000(0.96)^t. At t = 5, compute 0.96^5. One check: 0.96² = 0.9216; 0.96⁴ = 0.9216² = 0.84934656; multiply by 0.96 again to get 0.96⁵ ≈ 0.8153726976. Then A(5) = 5000(0.8153726976) ≈ 4076.863488, which rounds to $4076.86.
2. Determine whether B(t) = 5000(1 + 0.04)^(−t) is equivalent to A(t). Use exponent laws in your explanation.
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Use the negative-exponent rule: (1.04)^(−t) = 1/(1.04)^t = (1/1.04)^t. Therefore B(t) = 5000(1/1.04)^t ≈ 5000(0.961538)^t. This is not equivalent to 5000(0.96)^t because the bases differ: 1/1.04 ≈ 0.961538, not 0.96. The negative exponent correctly reciprocates 1.04, but it does not turn a 4% increase factor into a 4% decrease factor. Numerically, B(5) ≈ 4109.64, which differs from A(5) ≈ 4076.86.
3. Does replacing the base 1.04 with 1/(1.04) make B(t) equal to A(t)? Explain.
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Replacing 1.04 by 1/(1.04) gives 5000(1/(1.04))^t = 5000(1.04)^(−t), which is exactly B(t), not A(t). The base is still about 0.961538, not 0.96. To match A(t), the base must be 0.96 = 1 − 0.04. Equivalently, one could write A(t) = 5000(1/0.96)^(−t), using the reciprocal of 0.96. Changing to 1/(1.04) does not make the two models equal.