Question

1. 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.

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 $930$.