Question

A diver is on the 10m platform, preparing to perform a dive. The diver’s height above the water, in metres,
at time t can be modelled using the equation h(t) = 14 + 3t – 4.9t 2 .
a) Determine when the diver will enter the water.

b) Estimate the rate at which the diver’s height above the water is changing as the diver enters the water.

Answer 1

Given: A diver is on the 10m platform, preparing to perform a dive. The diver’s height above the water, in metres, at time t can be modelled using the equation h(t) = 14 + 3t – 4.9t 2 .

To find:

a) The time when the diver will enter the water.

b) The rate at which the diver’s height above the water is changing as the diver enters the water.

Solution:

a)

According to the question, the diver's height can be calculated using the following formula.

$h (t) = 14 + 3t - 4.9t^{2}$

Here, h(t) is the height of the diver above the water which is 10 m and t is the time taken which is to be calculated. Now, the formula can be written as follows.

$10 = 14 + 3t - 4.9t^{2}$

$4.9t^{2} - 3t - 4 = 0$

On solving the quadratic equation, the values of t are found to be

$t = 1.26 s$ or $t = - 0.65 s$

The value of t cannot be negative so the time taken is 1.26 s.

b)

In order to calculate the rate at which the diver’s height above the water is changing, the given equation must be divided by the time taken (t).

$\frac{h (t)}{t} = \frac{14 + 3t - 4.9t^{2}}{t}$

      $= \frac{14}{t} + 3 + 4.9t$

Therefore,

a) the time when the diver will enter the water is after 1.26 s.

b) the rate at which the diver’s height above the water is changing as the diver enters the water is (14/t + 3 + 4.9t).