A right triangle has one vertex at the origin and one vertex on the curve y = e-for 1 <* < 5. One of the two perpendicular sides is along the r-axis; the other is parallel to the y-axis.

Question

A right triangle has one vertex at the origin and one vertex on the curve y = e-for 1 <* < 5. One of the two perpendicular sides is along the r-axis; the other is parallel to the y-axis. Find the maximum and minimum areas for such a triangle.

Round your answers to three decimal places.

Answer 1

I think you mean vertex on y=e^-r and 1≤r≤5

and vertex at origin and vertex on r axis at (r,0)

area A=(1/2)*r*e^-r, r≥0 to keep A posiitve

there is not much calculus, A'=(1/2)*[e^-r-(r²)*e^-r]

A'=0 when r=1 which is a local maximimum, and A' is negatvie for r>1 (A decreasing)

so maximum area is when r=1 and minimum area is when r=5