Question

Hello!

I have an equation where I can't really find To (Initial temperature). The equation is from the formula Newton's law of cooling. To is the initial temperature of the object and ln is the natural log. The -0.00124 is the constant k multiplied by the time.
I just need to find the initial temp.

Answer 1

`T_0=80695.17162...`

Step-by-step explanation:

Given equation:

`\ln \left((T_0-100)/(T_0)\right)=-0.00124`

To solve the given equation:

`\textsf{Apply log rules}: \quad e^(\ln (x))=x`

`\implies (T_0-100)/(T_0)=e^(-0.00124)`

Multiply both sides by T₀:

`\implies T_0-100=T_0e^(-0.00124)`

Add 100 to both sides:

`\implies T_0=T_0e^(-0.00124)+100`

Subtract
`T_0e^(-0.00124)` from both sides:

`\implies T_0-T_0e^(-0.00124)=100`

Factor out the common term T₀:

`\implies T_0(1-e^(-0.00124))=100`

Divide both sides by
`(1-e^(-0.00124))`

`\implies T_0=(100)/(1-e^(-0.00124))`

Carry out the calculation:

`\implies T_0=(100)/(1-0.99876...)`

`\implies T_0=(100)/(0.001239231...)`

`\implies T_0=80695.17162...`