Question

Samuel bought a cement mixer for $54,205. The value of the cement mixer depreciated at a constant rate per year. The table shows the value of the cement mixer after the first and second years: Year 1 2 Value (in dollars) 47,158. 35 41,027. 76 Which function best represents the value of the cement mixer after t years? f(t) = 47,158. 35(0. 87)t f(t) = 54,205(0. 13)t f(t) = 47,158. 35(0. 13)t f(t) = 54,205(0. 87)t.

Answer 1

The value of cement mixer after t year is $f(t) = 54,205(0. 87)^{t}$

Given to us

The value of cement mixer when bought, $P$= $ 54,205

the value of cement mixer after 1 year, $P_1$=   $ 47,158. 35

the value of cement mixer after 2 year,$P_2$=  $ 41,027. 76

To find out depreciation we can use the formula for depreciation,

$P_n= P(1-r)^n\\where,\\P_n= Value\ of\ asset\ after\ n\ year\\P= Value\ of\ asset\ when\ bought\\r= rate\ of\ depreciation\\n= number\ of\ years$

By putting the value, in the formula we get,

$P_1= P(1-r)^n\\\\47,158.35= 54,205\times (1-r)^1\\\\\dfrac{47,158.35}{54,205} = (1-r)\\\\0.87=1-r\\\\r=0.13$

Therefore, putting the value of $r$ and $P$ in depreciation formula for $t$ years we get,

$P_n= P(1-r)^n$

$f(t) = 54,205(0. 87)^{t}$

Hence, the value of cement mixer after t year is $f(t) = 54,205(0. 87)^{t}$.

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