Question

A) If you get a slice of pizza with perimeter 90cm,what will be the diameter of the pizza for you to have gotten the largest pizza?

B)A curve passes through the point (0,5) and has the property that the slope of the curve at every point P is twice the y -coordinate of P. What is the equation of the curve?

Answer 1

$A:\\\alpha -the\,\,angle\,\,of\,\,a\,\,slice\\R-radius\\Perimeter:\\90=2R+\alpha R\Rightarrow \alpha =\frac{90}{R}-2\\Area:\\S=\frac{\alpha}{2}R^2=\left( \frac{45}{R}-1 \right) R^2=45R-R^2\\45R-R^2\rightarrow \max \\R_{\max}=\frac{-45}{2\cdot \left( -1 \right)}=22.5\\Diameter:\\d=2R=45\\B:\\The\,\,slope\,\,is\,\,y'\left( x \right) \\y'\left( x \right) =2y\left( x \right) \Rightarrow \frac{dy}{y}=2dx\Rightarrow \ln \left| y \right|=2x+C'\Rightarrow y=Ce^{2x}\\y\left( 0 \right) =5\Rightarrow C=5\Rightarrow y=5e^{2x}$