Question

PLS PLS HELP W MATH!! ITS CALCULUS

Answer 1

3. First solve the differential equation:

$\dfrac{\mathrm dT}{\mathrm dt}=-k(T-A)$

This is separable, as

$\dfrac{\mathrm dT}{T-A}=-k\,\mathrm dt$

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 $A=30$.

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

$T(1)=90\implies90=30+70e^{-k}\implies\dfrac67=e^{-k}\implies-k=\ln\dfrac67$

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 $\frac{\mathrm dy}{\mathrm dx}$ 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.