To calculate the after-tax salvage value from the sale of the machinery, we need to follow these steps:

1. Calculate the total depreciation over the first 3 years using the MACRS rates.
2. Determine the book value of the machinery after 3 years.
3. Compute the gain or loss on the sale.
4. Determine the tax impact of the sale.
5. Calculate the after-tax salvage value.

Step 1: Total depreciation over the first 3 years

The depreciation rates for the first three years are:

• Year 1: 20%
• Year 2: 32%
• Year 3: 19.2%

So, the total depreciation is:

$\text{Total Depreciation} = 20\% + 32\% + 19.2\% = 71.2\%$

Now, apply this to the initial cost of the machinery, which is $27,776:

$\text{Depreciation Amount} = 71.2\% \times 27,776 = 19,769.47$

Step 2: Book value after 3 years

The book value of the machinery after 3 years is the original cost minus the total depreciation:

$\text{Book Value} = 27,776 - 19,769.47 = 8,006.53$

Step 3: Gain or loss on the sale

The company sells the machinery for $14,939, which is greater than the book value of $8,006.53. This results in a gain:

$\text{Gain} = 14,939 - 8,006.53 = 6,932.47$

Step 4: Tax impact

The tax on the gain is calculated using the tax rate of 40%:

$\text{Tax on Gain} = 40\% \times 6,932.47 = 2,772.99$

Step 5: After-tax salvage value

The after-tax salvage value is the sale price minus the tax on the gain:

$\text{After-tax Salvage Value} = 14,939 - 2,772.99 = 12,166$

So, the after-tax salvage value is $12,166.

Would you like more details on any part of the calculation?

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Here are some related questions to expand on this:

1. How does MACRS depreciation differ from straight-line depreciation?
2. What impact does a higher or lower tax rate have on the after-tax salvage value?
3. How would the after-tax salvage value change if the machine was sold for a loss?
4. What are the key differences between book value and market value in asset sales?
5. How does the sale of a fully depreciated asset affect financial statements?

Tip: When calculating depreciation, always ensure you apply the correct depreciation method and rates for the specific asset class according to tax regulations.