Ciarán is preparing food for his baby and must use cooled boiled water - Leaving Cert Mathematics - Question 9 - 2014

After five minutes, the temperature of the water relative to room temperature is:

Using the equation:

Taking the natural logarithm:

Calculating,

Setting $y = 50 - 23 = 27$:

We rearrange and take the natural logarithm:

This results in approximately:

Using $A = 77$ and $k = -0.0339$, we can calculate points to plot:

• At $t = 0$: $f(0) = 77$
• At $t = 10$: $f(10) = 54.9$
• At $t = 20$: $f(20) = 39.1$
• At $t = 30$: $f(30) = 27.9$
• At $t = 40$: $f(40) = 19.8$, and so on.

These points can be plotted to create an exponentially decaying curve.

To sketch $g(t)$, we can choose a value for $m$ which is less than $k$; for instance, $m = -0.05$ results in:

• At $t = 0$: $g(0) = 77$
• At $t = 10$: $g(10) = 45.116$

This will clearly show that $g(t)$ decays faster than $f(t)$. Label both curves appropriately.

One possible value for $m$ could be $m = -0.05$. This value is chosen as it is less than $k$, ensuring the cooling rate is faster than that determined previously while still remaining a valid constant.

To find the rates of change, we calculate:

For $t = 1$:

For $t = 10$:

To show that the rate of change is increasing, we calculate the second derivative:

Since $A$ is positive, $k$ is negative, making $k^2$ positive. Therefore, the expression: