11 A hot drink when first made has a temperature which is \(65 ^ { \circ } \mathrm { C }\) higher than room temperature. The temperature difference, \(d ^ { \circ } \mathrm { C }\), between the drink and its surroundings decreases by \(1.7 \%\) each minute.
- Show that 3 minutes after the drink is made, \(d = 61.7\) to 3 significant figures.
- Write down an expression for the value of \(d\) at time \(n\) minutes after the drink is made, where \(n\) is an integer.
- Show that when \(d < 3 , n\) must satisfy the inequality
$$n > \frac { \log _ { 10 } 3 - \log _ { 10 } 65 } { \log _ { 10 } 0.983 }$$
Hence find the least integer value of \(n\) for which \(d < 3\).
- The temperature difference at any time \(t\) minutes after the drink is made can also be expressed as \(d = 65 \times 10 ^ { - k t }\), for some constant \(k\). Use the value of \(d\) for 1 minute after the drink is made to calculate the value of \(k\). Hence find the temperature difference 25.3 minutes after the drink is made.