Question

PLS PLS HELP W MATH!! ITS CALCULUS

Answer 1

3. First solve the differential equation:

`(\mathrm dT)/(\mathrm dt)=-k(T-A)`

This is separable, as

`(\mathrm dT)/(T-A)=-k\,\mathrm dt`

Integrating both sides gives

`\ln|T-A|=-kt+C`

`\implies T-A=e^(-kt+C)=Ce^(-kt)`

`\implies T(t)=A+Ce^(-kt)`

Now use the given known temperatures at the specified times to solve for the unknown constants. The egg starts with a temperature of
`T(0)=100`; it cools to 90º C after 1 minute so that
`T(1)=90`; and the ambient temperature is
`A=30`.

`T(0)=100\implies100=30+C\implies C=70`

`T(1)=90\implies90=30+70e^(-k)\implies\frac67=e^(-k)\implies-k=\ln\frac67`

So the temperature of the egg at time
`t` is

`T(t)=30+70e^(t\ln\frac67)`

and after 4 minutes it will have cooled to about

`T(4)=30+70e^(4\ln\frac67)\approx68`

but unfortunately none of the provided answers are correct...

4. Vertical tangents occur wherever
`(\mathrm dy)/(\mathrm dx)` is undefined, and horizontal tangents occur wherever the derivative is 0. This means the slope field will show vertical tangents when
`y=3`, so (I) is true, but (II) is not.

(III) is also true because, for instance, we can pick infinitely many values of
`y` so long as
`x=-1`, for which we get horizontal tangent lines.