Let's break down the problem step by step.

(a) Expression for the book value in the $t$th year

The machine is depreciated linearly over 10 years, with an initial cost of $250,000 and a scrap value of $10,000. Linear depreciation means the machine loses the same amount of value every year.

1. Initial Cost (C): $250,000
2. Scrap Value (S): $10,000
3. Depreciation Period (n): 10 years

The annual depreciation $D$ is calculated as:

$D = \frac{\text{Initial Cost} - \text{Scrap Value}}{\text{Number of Years}} = \frac{250,000 - 10,000}{10} = \frac{240,000}{10} = 24,000 \text{ dollars per year}$

The book value $V(t)$ at the end of year $t$ is given by:

$V(t) = \text{Initial Cost} - t \times D$

Substituting the values:

$V(t) = 250,000 - 24,000t$

This is the expression for the book value in the $t$th year.

(b) Finding the machine's book value in 2021

The machine was installed in 2016, so the year 2021 is the 5th year of operation (since $2021 - 2016 = 5$).

Substitute $t = 5$ into the expression for $V(t)$:

$V(5) = 250,000 - 24,000 \times 5 = 250,000 - 120,000 = 130,000$

So, the machine's book value in 2021 is $130,000.

(c) Rate of depreciation

The rate of depreciation is simply the annual depreciation $D$ calculated earlier:

$D = 24,000 \text{ dollars per year}$

So, the machine is depreciated at a rate of $24,000 per year.

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Would you like more details or have any other questions? Here are some related questions:

1. How would the book value change if the machine had a longer depreciation period?
2. What would the book value be after 8 years?
3. How does scrap value affect the annual depreciation?
4. What happens to the book value after the 10th year?
5. How does linear depreciation differ from other methods like declining balance?

Tip: When dealing with depreciation, always start by determining the annual depreciation rate, as it simplifies calculating the book value for any year.