An evergreen nursery usually sells a certain type of shrub after 6 yea

Judith McQueen 2021-12-15 Answered
An evergreen nursery usually sells a certain type of shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by dhdt=1.5t+5, where t is the time in years and h is the height in centimeters. The seedlings are 12 centimeters tall when planted (t=0).
(a) Find the height after t years.
(b) How tall are the shrubs when they are sold?
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Expert Answer

hysgubwyri3
Answered 2021-12-16 Author has 43 answers
a) dhdt1.5t+5
Since we want to know the height of the trees as a function of time, we can know that height by integrating the function dh/dt because h(t) is the anti-derivative of dh/dt as follows
h(t)=dhdtdt
=1.5tdt+5dt (sum and constant multiple rule)
=1.5t22+5l+C (Clndt=ln+1n+1 and kdt=kl)
=34l2+5l+C(1) (simplify)
we can know C by using the condition h(0)=12cm as follows
h(0)=3402+50+C=12
C=12(2)
h(t)=34t2+5t+12 (from (1) and (2))
b) To know its height on selling use the function we have obtained to get h(6) as follows
h(6)=3462+56+12=69cm

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Maria Lopez
Answered 2021-12-17 Author has 32 answers
We are given the following data:
- The time duration is t=6y
- The growth rate function is dhdt=1.5t+5
- The length of the seeding is h=12cm
a) The expression for the growth rate is,
dhdt=1.5t+5
dh=(1.5t+5)dt
Integrate the above function within the given limits.
dh=0t(1.5t+5)dt
h(t)=(1.5(t1+11+1)+5(t0+10+1))+C
h=0.75t2+5t+C(1)
Put initial condition in Eq (1) to calculate the constant of integration.
h(0)=12
12=0.75(0)2+5(0)+C
C=12
Substitute 12 for C in Eq (1).
h=0.75t2+5t+12
Thus, the height after t years is 0.75t2+5t+12.
b) The expression for the height of the shrubs after t years is,
h=0.75t2+5t+12
Substituting the given values in the above expression, we will get
h=0.75(6)2+5(6)+12
=27+30+12
=69cm
Thus, the height of the shrubs when they sold is 69cm.

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nick1337
Answered 2021-12-27 Author has 575 answers

Step 1
Rewrite the relation for growth and time. Integrate it.
dhdt=1.5t+5
dh=(1.5t+5)dt
dh=(1.5t+5)dt
h=1.5t22+5t+C eqn.(1)
Step 2
Find the value of constant C, by substituting h=12 and t=0 in eqn.(1)
(12)=1.5(0)22+5(0)+C
12=C
Therefore, the height “h” after “t” years can be given as:
h=1.5t22+5t+12
Step 3
h=1.5t22+5t+12
The shrubs sold after 6 years, so use to calculate the height of shrubs when they are sold by substituting t=6
h=1.5(6)22+5(6)+12
=1.5·362+30+12
h=69cm

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