Question

How quickly a leaf grows is proportional how big [ie the surface area] the leaf is. If the area of the leaf grows from 2cm2 to 3cm2 in 3 days, how long will it take for the leaf's area to increase to 5 cm2

Answer 1

Answer: 9 days

Explanation:

• Step 1

Let the rate of Leaf growth r be defined as, $\frac{Increase in area}{time taken}$ = $\frac{A1 - A}{t}$

where A is initial area of the leaf, A1 is the final area of the leaf and t is the time taken for the increase in Area.

• Express the proportional relationship in equation.

Given that rate of leaf growth, r is proportional to the surface area of the leaf A. we have r ∝ A.

r = kA, where k is the rate constant.

therefore, k = $\frac{r}{A}$

when A = 2$cm^{2}$, A1 = 3

so k = $\frac{\frac{3 - 2}{3}}{2}$

= $\frac{1}{3}$ ÷ 2

= 0.33 ÷ 2

k = 0.167

• After calculating the rate constant k, we then find the time t when A1 is 5$cm^{2}$

• we have r = k × A1 = $\frac{A1 - A}{t}$

so, 0.167 × 2 = $\frac{5 - 2}{t}$

0.33 = $\frac{3}{t}$.

t = 3/0.33

Therefore, t = 9 days.