Question

An evergreen nursery usually sells a certain shrub after 7 years of growth and shaping. The growth rate during those 7 years is approximated by dh/dt = 1.3t + 2, where t is the time in years and h is the height in centimeters. The seedlings are 17 centimeters tall when planted (t = 0). (a) Find the height after t years. h(t) = (b) How tall are the shrubs when they are sold? cm

Answer 1

Answer:

(a)$h(t)=\frac{1.3t^2}{2} + 2t +17$

(b)62.85cm

Step-by-step explanation:

The growth rate of the shrub is given as:

$\frac{dh}{dt} = 1.3t + 2$

Where t=time in years, h=height in centimeters

(a)First, we solve for the height h(t) by integrating.

$\int \frac{dh}{dt} dt= \int (1.3t + 2)dt\\h(t)=\frac{1.3t^2}{2} + 2t +C, C a constant of Integration \\ We sunstitute the initial value to find the value of C \\ When t=0, h=17cm \\17=C\\Therefore:\\h(t)=\frac{1.3t^2}{2} + 2t +17$

(b)The shrub are sold after 7 years of growth. Therefore, we determine the value of h(t) when t=7 years.

$h(t)=\frac{1.3t^2}{2} + 2t +17\\h(7)=\frac{1.3*7^2}{2} + 2(7) +17\\=31.85+14+17\\h(7)=62.85cm$

The Shrubs are 62.85cm tall when they are sold.