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# Tag Archives: calculus optimization problems and solutions

## Inscribed Rectangle Optimization example question

Find the area of the largest rectangle that can be inscribed in a circle of radius 4.

Solution to this Calculus Optimization practice problem is given in the video below!

## Rain Gutter Optimization example problem

A rain gutter is to be constructed from a metal sheet of width 30 centimeters by bending up one-third of the sheet on each side through an angle θ. How should angle θ be chosen so that the gutter will carry the maximum amount of water?

Solution to this Calculus Optimization practice problem is given in the video below!

## Distance Optimization example

Find the point on the curve y = cos(x) closest to the point (0,0)

Solution to this Calculus Optimization practice problem is given in the video below!

## Running Track Optimization example question

A running track is to be built around a rectangular field, with two straightaways and two semicircular curves at the ends, as indicated in the figure. The length of the track is to be 400 meters. Find the dimensions that will maximize the area of the enclosed rectangle.

Solution to this Calculus Optimization practice problem is given in the video below!

## Ticket Cost Optimization example problem

Suppose that group tickets to a concert are priced at \$40 per ticket if 20 tickets are ordered, up to a maximum of 50 tickets. (For example, if 22 tickets are ordered, the price is \$38 per ticket). Find the number of tickets that maximizes the total cost of the tickets.

Solution to this Calculus Optimization practice problem is given in the video below!

An advertisement consists of a rectangular printed region plus 1-inch margins on the sides and 2-inch margins at the top and bottom. If the area of the printed region is to be 92 square inches, find the dimensions of the printed region and overall advertisement that minimize the total area.

Solution to this Calculus Optimization practice problem is given in the video below!

## Soda Can Optimization example question

A soda can is to hold 12 fluid ounces. Find the dimensions that will minimize the amount of material used in its construction, assuming that the thickness of the material is uniform.

Solution to this Calculus Optimization practice problem is given in the video below!

## Graph Optimization example problem

Suppose that

For each x in the interval (0,1), consider the rectangle formed in the following manner:

a) Its right side is the line segment connecting the points (x, f(x)) and (x, g(x))

b) Its left side lies along the y-axis

Which value of x in the interval (0,1) results in the rectangle of largest area?

Solution to this Calculus Optimization practice problem is given in the video below!

## Optimization Proof example

Show that the Rectangle of Maximum Area for a given Perimeter is always a Square.

Solution to this Calculus Optimization practice problem is given in the video below!

## Optimization Proof example question

Show that the Rectangle of Minimum Perimeter for a given Area is always a Square.

Solution to this Calculus Optimization practice problem is given in the video below!