A bottle of water is put into a refrigerator - Edexcel - A-Level Maths Pure - Question 11 - 2012 - Paper 1

Given that the initial temperature is 16 °C, we substitute:

$θ(0) = 16 = A + 3$ This results in:

$A = 16 - 3 = 13.$ Thus, we express θ as:

$θ = 13e^{-0.008t} + 3.$ To find the time when θ = 10 °C:

$10 = 13e^{-0.008t} + 3.$ Rearranging gives:

$10 - 3 = 13e^{-0.008t}$ which simplifies to:

$7 = 13e^{-0.008t}.$ Now, dividing both sides by 13:

$e^{-0.008t} = \frac{7}{13}.$ Taking natural logarithms gives:

$-0.008t = \ln \frac{7}{13}.$ Rearranging results in:

$t = \frac{-\ln \frac{7}{13}}{0.008}.$ Calculating this yields:

t ≈ 77.3799 minutes, rounded to the nearest minute gives:

t = 77 minutes.