Please show step by step of working out the value of r for which is A is aminimum and calculate the minimum surface area of the container. The total surface area, of each container is modelled by function A= πr^2+1000/r.

(remember to use the derivative to show you have found the minimum)

The minimum surface area of the container is 276.791 square units.

Step-by-step explanation:

Let be , , . The first and second derivatives of such function are, respectively:

First derivative

Second derivative

The critical values of are determined by equalizing first derivative to zero and solving it: (First Derivative Test)

(since radius is a positive variable)

To determine if critical value leads to an absolute minimum, this input must be checked in the second derivative expression: ()

The critical value leads to an absolute minimum, since value of the second derivative is positive.

Finally, the minimum surface area of the container is:

The minimum surface area of the container is 276.791 square units.

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4