Exponential inequality

1 Find the set of values of \(x\) for which \(2 \left( 3 ^ { 1 - 2 x } \right) < 5 ^ { x }\). Give your answer in a simplified exact form. [4]

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.

1. Show that 3 minutes after the drink is made, \(d = 61.7\) to 3 significant figures.
2. Write down an expression for the value of \(d\) at time \(n\) minutes after the drink is made, where \(n\) is an integer.
3. 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 \(d < 3\).
4. 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.