Question

Jane wishes to bake an apple pie for dessert. The baking instructions say that she should bake the pie in an oven at a constant temperature of 230◦C. Being a mathematician, she knows that the temperature T of the pie after t minutes of baking will be given by T(t) = 230 − Ae−kt , where A and k are constants. After 18 minutes of baking she notices that the temperature of the pie is 138◦C, while after 36 minutes it is 184◦C. Determine the constants A and k.

Answer 1

Therefore
`k= (ln2 )/(18)`, A=184

Explanation:

Given function is

`T(t)=230 -e^(-kt)`

where T(t) is the temperature in °C and t is time in minute and A and k are constants.

She noticed that after 18 minutes the temperature of the pie is 138°C

Putting T(t) =138°C and t= 18 minutes

`138=230 -Ae^(-k* 18)`

`\Rightarrow -Ae^(-18k)=138-230`

`\Rightarrow Ae^(-18k)=92` .....(1)

Again after 36 minutes it is 184°C

Putting T(t) =184°C and t= 36 minutes

`184=230-Ae^(-k* 36)`

`\Rightarrow Ae^(-36k)=230-184`

`\Rightarrow Ae^(-36k)=46`.......(2)

Dividing (2) by (1)

`(Ae^(-36k))/(Ae^(-18k))=(46)/(92)`

`\Rightarrow e^(-18k)=(46)/(92)`

Taking ln both sides

`ln e^(-18k)=ln(46)/(92)`

`\Rightarrow -18k =ln (\frac12)`

`\Rightarrow -18k= ln1-ln2`

`\Rightarrow k= (ln2 )/(18)`

Putting the value k in equation (1)

`Ae^{-18(ln2)/(18)}=92`

`\Rightarrow A e^{ln2^(-1)}=92`

`\Rightarrow A.2^(-1)=92`

`\Rightarrow (A)/(2)=92`

`\Rightarrow A= 92 * 2`

⇒A= 184.

Therefore
`k= (ln2 )/(18)`, A=184