8. green giant daycare center purchased new furniture for its toddler room. the estimated useful life of the…
8. green giant daycare center purchased new furniture for its toddler room. the estimated useful life of the furniture is 10 years. the total cost was $3650 and the value at the end of 10 years is expected to be $250. (a) what is the rate of change for this function? (b) write a function that can be used to estimate the value of the furniture, v, at any time during the 10 - year period. (c) calculate v(8) and explain its meaning.
Answer
Explanation:
Step1: Calculate rate of change
The rate of change (slope) $m$ for depreciation is given by the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $x_1 = 0$ (start - time), $y_1=3650$ (initial value), $x_2 = 10$ (end - time), $y_2 = 250$ (end - value). So, $m=\frac{250 - 3650}{10-0}=\frac{- 3400}{10}=-340$.
Step2: Write the linear function
The linear function has the form $V(t)=mt + b$, where $m$ is the slope and $b$ is the initial value. We know $m=-340$ and $b = 3650$. So the function is $V(t)=-340t + 3650$, where $t$ is the number of years since the purchase and $0\leq t\leq10$.
Step3: Calculate $V(8)$
Substitute $t = 8$ into the function $V(t)=-340t + 3650$. So $V(8)=-340\times8 + 3650=-2720+3650 = 930$. The meaning of $V(8)$ is that the value of the furniture 8 years after it was purchased is $930$.
Answer:
(a) - 340 (b) $V(t)=-340t + 3650$, $0\leq t\leq10$ (c) $V(8)=930$. It means the value of the furniture 8 years after purchase is $930$.