Find the height after t years. h(t) = (b) How tall are the shrubs when they are sold?

Question

An evergreen nursery usually sells a certain shrub after 9 years of growth and shaping. The growth rate during those 9 years is approximated by dh

dt

 = 1.7t + 3,

 where t is the time (in years) and h is the height (in centimeters). The seedlings are 18 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

Since you've listed this question under precalc, I'm going to answer accordingly. Consider the graph of height vs time. To draw this graph, we would start with the point (0, 18) (at time t = 0 the height is 18 cm). The growth rate will be dh/dt = 1.7t + 3 and since t = 0, it will be 3 cm/year. So after 1 year, the height will be 21 cm so we can put another point on the graph of (1, 21). Now, the growth rate is 4.7 cm/year (t = 1 makes 1.7t + 3 equal to 4.7) so after a year the height will be 25.7 cm making another point on the graph (2, 25.7).

We would continue this getting the following set of points:

(0, 18)

(1, 21)

(2, 25.7)

(3, 32.1)

(4, 40.2)

(5, 50)

(6, 61.5)

(7, 74.7)

(8, 89.6)

(9, 106.2)

So after 9 years, an approximation of the height is 106.2 cm. In reality, this is really an approximation because we are only incrementing the height every year instead of continually. The growth rate is continually changing as you move through time.

An equation in terms of "t" could be:

Answer 2

Hi Latreyll!

This question is listed under precalculus, but it actually involved calculus concepts!

For this question we're given the rate of change (derivative) of the height for any given time and we're asked to find the height, so we're going to need to do an integral.

Part A

The easiest way to do this in my opinion is like this:

Take the equation

(dh/dt) = 1.7t + 3

and take the integral of both sides with respect to t

that will give you ∫ dh = ∫(1.7t + 3)dt

The bounds of integration for the left hand side will be from H(0) to H(t), and the bounds for the right hand side will be from 0 to t.

Taking the antiderivative of both sides and then evaluating along the bounds respectively, we get that

H(t) - H(0) = (0.85t² + 3t)

so

H(t) = (0.85t² + 3t) + H(0)

And H(0) is just the height of the plant when it's planted, which is given to us as 18cm, so H(0) = 18

That gives our final equation as

H(t) = 0.85t² + 3t + 18

Part B

We know that the shrubs are sold after 9 years, so we just need to find what H(9) is.

H(9) = 0.85(9)² + 3(9) + 18

= 113.85 cm

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Hope this helped!