







The maximum area of a rectangle with a given perimeter 
20170602 

From Bob: How would I go about finding the maximum area of a rectangle given its perimeter (20m, for example)? Answered by Penny Nom. 





A calculus optimization problem 
20150514 

From Ali: Given an elliptical piece of cardboard defined by (x^2)/4 + (y^2)/4 = 1. How much of the cardboard is wasted after the largest rectangle (that can be inscribed inside the ellipse) is cut out? Answered by Robert Dawson. 





A max/min problem 
20121214 

From bailey: A right angled triangle OPQ is drawn as shown where O is at (0,0).
P is a point on the parabola y = ax – x^2
and Q is on the xaxis.
Show that the maximum possible area for the triangle OPQ is (2a^3)/(27) Answered by Penny Nom. 





Maximum area of a rectangle 
20111004 

From Lyndsay: A rectangle is to be constructed having the greatest possible area and a perimeter of 50 cm.
(a) If one of the sides of the rectangle measures 'x' cm, find a formula for calculating the area of the rectangle as a function of 'x'.
(b) Determine the dimensions of the rectangle for which it has the greatest area possible. What is the maximum area? Answered by Penny Nom. 





Maximize the floor area 
20100707 

From shirlyn: A rectangular building will be constructed on a lot in the form of a right triangle with legs
of 60 ft. and 80 ft. If the building has one side along the hypotenuse,
find its dimensions for maximum floor area. Answered by Penny Nom. 





A maxmin problem 
20090420 

From Charlene: A fixed circle lies in the plane. A triangle is drawn
inside the circle with all three vertices on the circle and two of the vertices at the
ends of a diameter. Where should the third vertex lie to maximize the perimeter
of the triangle? Answered by Penny Nom. 





A maxmin problem 
20090324 

From Jay: Determine the area of the largest rectangle that can be inscribed between the xaxis and the curve defined by y = 26  x^2. Answered by Harley Weston. 





What point on the graph y = e^x is closest to the origin? 
20080303 

From elvina: What point on the graph y = e^x is closest to the origin? Justify your answer. Answered by Stephen La Rocque. 





A ball bearing is placed on an inclined plane 
20080215 

From Leah: A ball bearing is placed on an inclined plane and begins to roll.
The angle of elevation of the plane is x.
The distance (in meters) that the ball bearing rolls in t seconds is s(t) = 4.9(sin x)t^2.
What is the speed of the ball bearing,
and what value of x will produce the maximum speed at a particular time? Answered by Penny Nom. 





Minimum cost for a fixed volume 
20070418 

From James: My question goes: A silo is to be constructed and surmounted by a hemisphere. The material of the hemisphere cost twice as much as the walls of the silo. Determine the dimensions to be used of cost is to be kept to a minimum and the volume is fixed. Answered by Penny Nom. 





A maxmin problem 
20051216 

From Julie: A car travels west at 24 km/h. at the instant it passes a tree, a horse and buggy heading north at 7 km/h is 25 km south of the tree. Calculate the positions of the vessels when there is a minimum distance between them. Answered by Penny Nom. 





Mrs. Faria lives on an island 
20051215 

From Julie: Mrs. Faria lives on an island 1 km from the mainland. She paddles her canoe at 3 km/h and jogs at 5 km/h. the nearest drug store is 3 km along the shore from the point on the shore closest to the island. Where should she land to reach the drug store in minimum time? Answered by Penny Nom. 





Getting to B in the shortest time 
20011219 

From Nancy: A motorist in a desert 5 mi. from point A, which is the nearest point on a long, straight road, wishes to get to point B on the road. If the car can travel 15 mi/hr on the desert and 39 mi/hr on the road to get to B, in the shortest possible time if...... A.) B is 5 mi. from A B.) B is 10 mi. from A C.) B is 1 mi. from A Answered by Penny Nom. 





An emergency response station 
20010329 

From Tara: Three cities lying on a straight line want to jointly build an emergency response station. The distance between each town and the station should be as short as possible, so it cannot be built on the line itself, but somewhere east or west. Also, the larger the population of a city, the greater the need to place the station closer to that city. You are to minimize the overall sum of the products of the populations of each city and the square of the distance between that city and the facility. City A is 6 miles from the road's origin, City B is 19 miles away from the origin, and City C is 47 miles from the origin. The populations are 18,000 for City A, 13,000 for City B, and 11,000 for City C. Where should the station be located? Answered by Claude Tardif and Penny Nom. 





An integer maxmin problem 
20000313 

From Paul Servic: Maximize Q = xy^{ 2} where x and y are positive integers such that x + y^{ 2} = 4 Answered by Penny Nom. 





Maximize 
20000312 

From Tara Doucet: My question is Maximize Q=xy^2 (y is to the exponent 2) where x and y are positive integers such that x + y^2 ( y is to the exponent 2)=4 Answered by Harley Weston. 





Slant height of a cone 
20000224 

From Jocelyn Wozney: I need help with this problem for my high school calculus class. Any help you can give me will be greatly appreciatedI am pretty stumped. "Express the volume of a cone in terms of the slant height 'e' and the semivertical angle 'x' and find the value of 'x' for which the volume is a maximum if 'e' is constant. Answered by Harley Weston. 





Area of a circle and an inequality 
19991030 

From Adam Anderson: I have two problems. The first: prove that the area of a cirlce is pi times radius squared without using calculus. The second: show that ln(x) < x  1 for all x > 0. Answered by Harley Weston.


