An evergreen nursery sells a certain shrub after 4 years. The growth rate of the shrub is given by $d h / d t = 2.5 t + 6$ , where t is the time in years and h is the height in centimeters. The seedlings are 10 centimeters tall when planted (t = 0). How tall are the shrubs when they are sold?
A)36 centimeters
B)40 centimeters
C)44 centimeters
D)54 centimeters
E)70 centimeters

Question

Quizplus · Accepted Answer

To find the height when the shrubs are sold, integrate the growth rate function $dh/dt = 2.5t + 6$ from $t = 0$ to $t = 4$. The integral is $h(t) = \int (2.5t + 6) dt = 1.25t^2 + 6t + C$. Given the initial height $h(0) = 10$, solve for $C$: $10 = 1.25(0)^2 + 6(0) + C$, so $C = 10$. Therefore, $h(t) = 1.25t^2 + 6t + 10$. Evaluate at $t = 4$: $h(4) = 1.25(4)^2 + 6(4) + 10 = 20 + 24 + 10 = 54$ centimeters.

An Evergreen Nursery Sells a Certain Shrub After 4 Years dh/dt=2.5t+6d h / d t = 2.5 t + 6dh/dt=2.5t+6

An Evergreen Nursery Sells a Certain Shrub After 4 Years $d h / d t = 2.5 t + 6$