Question

part a
manufacturing a textile company bought a piece of weaving equipment for $60,000. it is expected to depreciate at an average rate of 10% per year.
a. write an equation for the value of the piece of equipment $ after t years.

part b
b. find the value of the piece of equipment after 6 years. round the nearest dollar.
about $

Explanation:

Step1: Define depreciation formula

This is exponential depreciation, where the remaining value each year is \(1 - 0.10 = 0.90\) of the previous year's value. The initial value \(V_0 = 60000\).
The general formula for exponential depreciation is \(V(t) = V_0(1 - r)^t\), so substituting values gives:
\(V(t) = 60000(0.90)^t\)

Step2: Substitute \(t=6\) into the formula

Calculate the value after 6 years by plugging \(t=6\) into the equation from Step1.
\(V(6) = 60000(0.90)^6\)

Step3: Calculate \(0.90^6\)

Compute the exponential term first:
\(0.90^6 = 0.531441\)

Step4: Multiply by initial value

Multiply the result from Step3 by the initial equipment cost:
\(V(6) = 60000 \times 0.531441 = 31886.46\)

Step5: Round to nearest dollar

Round the calculated value to the nearest whole number:
\(\text{Round}(31886.46) = 31886\)

Answer:

a. \(V(t) = 60000(0.90)^t\)
b. 31886