Question

An evergreen nursery usually sells a certain shrub after 5 years of growth and shaping.
The growth rate during those 5 years is approximated by

\frac{dh}{dt}=1.4t+8

where t is the time in years and h is the height in centimeters. The seedlings are 14
centimeters tall when planted. How tall are the shrubs when they are sold?

Answer 1

Answer:

The height of shrubs is 71.5 cm

Step-by-step explanation:

An evergreen nursery usually sells a certain shrub after 5 years of growth and shaping.

The growth rate during those 5 years is approximated by

$\dfrac{dh}{dt}=1.4t+8$

where t is the time in years and h is the height in centimeters.

Multiply both sides by dt.

$dh=(1.4t+8)dt$

Integrate both sides.

$\int dh=\int (1.4t+8)dt$

$h=1.4(\dfrac{t^2}{2})+8t+C$

$h=0.7t^2+8t+C$                 .... (1)

The seedlings are 14 centimeters tall when planted. it means h=14 at t=0.

Substitute h=14 and t=0 in equation (1).

$14=0.7(0)^2+8(0)+C$

$14=C$

Substitute C=14 in equation (1).

$h=0.7t^2+8t+14$

We need to find the height of shrubs when they are sold.

Substitute t=5 in the above equation.

$h=0.7(5)^2+8(5)+14$

$h=0.7(25)+40+14$

$h=71.5$

Therefore, the height of shrubs is 71.5 cm when they are sold.