Answers edited by Barry H.

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• The sum of two positive number is 16. Use optimization to find the smallest possible value of the sum of their squares?
  
• How to find the general solution for #x(x^2 + y^2)dy/dx = y(y^2-x^2)# using homogeneous ?
  
• A gutter is to be made using a 5m long rectangular piece of metal that has to be bent to form the open topper gutter. the cross section of the gutter is an isosceles trapezoid with sides making angles 120 degrees with the base?
  
• If an isosceles triangle has perimeter P, how long must the legs of the triangle be to maximize its area? (Your answer may depend on P).
  
• Find the rectangle with the maximum area, which can be turned in the corner. ?
  
• What is the derivative of tan(sin x)?
  
• What is the surface area produced by rotating #f(x)=x/pi^2, x in [-3,3]# around the x-axis?
  
• How do you express the concept of the rate of decay of a radioactive substance as a differential equation?
  
• A large vase has a square base of side length 6 cm, and flat sides sloping outwards at an angle of 120◦ with the base. How to find the rate when height is rising?
  
• An open-top rectangular box is constructed from a 10-in.- by-16-in. piece of cardboard by cutting squares of equal side length from the corners and folding up the sides. Find analytically the dimensions of the box of largest volume and the maximum volume?
  
• Find the volume using disk/washer method of region bound by curves y=e^-x/2, y=ln(9)? The solid is generated when R is revolves around x-axis. The boundaries are ln(9) and 0. I just need help setting up the integral. Thank you.
  
• The volume of a cube is increasing at a rate of 10 cm^3/min. How fast is the surface area increasing when the length of an edge 90 cm?
  
• Pressure (p) and volume (v) given in eqn #pv^1.4=k# where k is constant. At certain instant pressure is #(25gm)/(cm^3# and volume is #32cm^3#. If volume is increasing at rate of #(5cm^3)/s#, how do you find the rate at which pressure is changing?
  
• Help regarding this optimization problem to find dimensions?
  
• Please help me with this calc question?