The height, h metres, of a tree, t years after being planted, is modelled by the equation $$h^2 = at + b \\ 0 \leq t < 25$$ where a and b are constants - Edexcel - A-Level Maths Pure - Question 7 - 2022 - Paper 1

To find the constants a and b, we can create a system of equations using the height data provided.

Using the given information:

1. For t = 2 years, the height of the tree is 2.60 m:

   $h^2 = a(2) + b \\ 2.60^2 = 2a + b \\ 6.76 = 2a + b \qquad (1)$

2. For t = 10 years, the height of the tree is 5.10 m:

   $h^2 = a(10) + b \\ 5.10^2 = 10a + b \\ 26.01 = 10a + b \qquad (2)$

Now, we can solve this system of equations:

From equation (1):

$b = 6.76 - 2a$

Substituting for b in equation (2):

$26.01 = 10a + (6.76 - 2a) \\ 26.01 = 10a + 6.76 - 2a \\ 26.01 - 6.76 = 8a \\ 19.25 = 8a \\ a = \frac{19.25}{8} = 2.40625 \approx 2.41$

Now substituting back to find b:

$b = 6.76 - 2(2.41) \\ b = 6.76 - 4.82 = 1.94$

Thus, rounding to 3 significant figures, we find:

• a = 2.41
• b = 1.94

The complete equation is:

$h^2 = 2.41t + 1.94$