Question

Read the following passage and answer the questions given below:

The relation between the height of the plant ('y' in cm) with respect to its exposure to the sunlight is governed by the following equation y = `4x - 1/2 x^2`, where 'x' is the number of days exposed to the sunlight, for x ≤ 3.

1. Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
2. Does the rate of growth of the plant increase or decrease in the first three days? What will be the height of the plant after 2 days?

Answer 1

y = `4x - 1/2 x^2`

i. The rate of growth of the plant with respect to the number of days exposed to sunlight is given by `dy/dx` = 4 – x.

ii. Let rate of growth be represented by the function g(x) = `dy/dx`.

Now, g'(x) = `d/dx(dy/dx)` = –1 < 0.

`\implies` g(x) decreases.

So the rate of growth of the plant decreases for the first three days.

Height of the plant after 2 days is y = `4 xx 2 - 1/2(2)^2` = 6 cm.