







Maximizing the area of a two lot region 
20160403 

From yousef: A man wishes to enclose two separate lots with 300m of fencing. One lot is a square and the other a rectangle whose length is twice its width. Find the dimensions of each lot if the total area is to be a minimum. Answered by Penny Nom. 





Maximizing the ticket revenue 
20141007 

From Allen: An airplane whose capacity is 100 passengers is to be chartered for a flight to Europe. The fare is to be $150 per person, if 60 people buy tickets. However, the airline agrees to reduce the fare for every passenger by $1 for each additional ticket sold. How many tickets should be sold to maximize the ticket revenue for this flight? Answered by Chris Fisher. 





The popcorn box problem 
20131107 

From Dave: We know that calculus can be used to maximise the volume of the tray created when cutting squares from 4corners of a sheet of card and then folding up.
What I want is to find the sizes of card that lead to integer solutions for the size of the cutout, the paper size must also be integer. EG 14,32 cutout 3 maximises volume as does 13,48 cutout 3.
I have done this in Excel but would like a general solution and one that does not involve multiples of the first occurence, as 16, 10 cutout 2 is a multiple of 8,5 cutout 1. Answered by Walter Whiteley. 





Maximize the volume of a cone 
20131009 

From Conlan: Hi I am dong calculus at school and I'm stumped by this question:
A cone has a slant length of 30cm. Calculate the height, h, of the cone
if the volume is to be a maximum.
If anyone can help me it would be greatly appreciated.
thanks. Answered by Penny Nom. 





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. 





A maximization problem 
20120409 

From Nancy: After an injection, the concentration of drug in a muscle varies according to a function of time, f(t). Suppose that t is measured in hours and f(t)=e^0.02t  e^0.42t. Determine the time when the maximum concentration of drug occurs. Answered by Penny Nom. 





A spherical ball in a conical wine glass 
20111026 

From Jules: A heavy spherical ball is lowered carefully into a full conical wine
glass whose depth is h and whose generating angle (between the axis
and a generator) is w. Show that the greatest overflow occurs when the
radius of the ball is (h*sin(w))/(sin(w)+cos(2w)). Answered by Claude Tardif. 





What is the maximum weekly profit? 
20101010 

From Joe: A local artist sells her portraits at the Eaton Mall.
Each portrait sells for $20 and she sells an average of 30 per week.
In order to increase her revenue, she wants to raise her price.
But she will lose one sale for every dollar increase in price.
If expenses are $10 per portrait, what price should be set to maximize the weekly profits?
What is the maximum weekly profit? Answered by Stephen La Rocque and Penny Nom. 





Maximizing the volume of a cylinder 
20100831 

From Haris: question: the cylinder below is to be made with 3000cm^2 of sheet metal. the aim of this assignment is to determine the dimensions (r and h) that would give the maximum volume.
how do i do this?
i have no idea. can you please send me a steptostep guide on how t do this?
thank you very much. Answered by Penny Nom. 





A max min problem 
20100819 

From Mark: a rectangular field is to be enclosed and divided into four equal lots by fences parallel to one of the side. A total of 10000 meters of fence are available .Find the area of the largest field that can be enclosed. 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 max min problem 
20100406 

From Terry: The vertex of a right circular cone and the circular edge of its base lie on the surface of a sphere with a radius of 2m. Find the dimensions of the cone of maximum volume that can be inscribed in the sphere. Answered by Harley Weston. 





Maximizing the area of a rectangle 
20091217 

From rachel: A rectangular field is to be enclosed by 400m of fence. What dimensions will give a maximum area? Answered by Penny Nom. 





Maximize profit 
20091114 

From Willie: Profit is the difference between Total Revenue and Total Cost.
Therefore, to MAXIMIZE PROFIT you must maximize Total Revenue.
True or False? Explain answer. Answered by Penny Nom. 





A rectangular pen 
20090813 

From Kari: A rectangular pen is to be built using a total of 800 ft of fencing. Part of this fencing will be used
to build a fence across the middle of the rectangle (the rectangle is 2 squares fused together so if you can
please picture it).
Find the length and width that will give a rectangle with maximum total area. Answered by Stephen La Rocque. 





Maximum profit 
20090511 

From Sally: a manufacturer of dresses charges $90 per dress up to 100 units and the average production cost is $60 per dress. to encourage larger orders the company will drop the price per dress by .10 for orders in excess of 100. I need to find the largest order the company should allow with the special discount to realize maximum profit. Answered by Harley Weston. 





A discount on a charter plane 
20090506 

From karen: a charter plane company advertises that it will provide a plane for a fare of $60. if your party is twenty or less and all passengers will receive a discount of $2 per person if the party is greater than 20. what number of passengers will maximize revenue for the company Answered by Stephen La Rocque. 





A maximum area problem 
20090113 

From Kylie: Help me please! I don't know how or where to start and how to finish.
The problem is: A window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 15 ft., find the dimensions that will allow the maximum amount of light to enter. Answered by Harley Weston. 





A sphere in a can of water 
20081212 

From Meghan: A cylindrical can open at the top has (inside) base radius equal to 1.
The height of the can is greater than 2.
Imagine placing a steel sphere of radius less than 1 into the can, then pouring water into the can until the top of the sphere is just covered.
What should be the radius of the sphere so the volume of water used is as large as possible? Answered by Harley Weston. 





Largest Inscribed Rectangle 
20080903 

From astrogirl: find the shape and area of the largest rectangle that can be inscribed in a circle of a diameter a=2 Answered by Janice Cotcher. 





How many presses should be used? 
20080504 

From Sarah: Hi! I am in Calculus and this problem is on my study guide and i just cant figure it out!?
A printing company had eight presses, each of which can print 300 copies per hour. It costs $5.00 to set up each press for a run and 12.5+6n dollars to run n presses for an hour. How many presses should be used to print 6000 copies most profitably? Let h equal the number of hours used to print the 6000 copies. Answered by Harley Weston. 





The maximum area of a pizza slice 
20080412 

From charles: A slice of pizza in the form of a sector of a circle has a perimeter of 24 inches. what value for the radius of the pizza makes the slice largest[when o is the central angle in radians, the area of the sector is given by A= r^20/2and the length on the circle is given by s=r0 Answered by Harley Weston. 





Maximize income 
20080118 

From Chris: Lemon Motors have been selling an average of 60 new cars per month at
$800 over the factory price. They are considering an increase in this
markup. A marketing survey indicates that for every $20 increase, they
will sell 1 less car per month. What should their new markup be in order
to maximize income? Answered by Stephen La Rocque and Harley Weston. 





Maximum volume of a box 
20080115 

From Rajesh: A square piece of a cardboard of sides ten inches has four equal peices are removed at the corners, then the sides are turned up to form an open box. What is the maximum volume such a box can have? Answered by Stephen La Rocque. 





Maximize the product 
20071125 

From David: Hi i have this site call calcchat.com, but i dont understand how they explained this can you take a look? The question is:
Direction: Find two positive numbers that satisfy the given requirements.
The sum is S and the product is a maximum
this is what they did
1) Let x and y be two positive numbers such that x + y = S
2)P = xy
3) = x (S  x)
4) =Sx  x^2
5)...etc. the thing i dont get is how did they go from step 2 to step 3
and also i know this sound dumb but how did they get step 2? =) Answered by Harley Weston. 





A rectangular plot of farmland 
20071125 

From Christy: A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a singlestrand electric fence. With 800m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions? Answered by Harley Weston. 





Maximize his profit 
20071112 

From apoorva: During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $10 each and his sales averaged 20 per day. When he increased the price by $1, he found that he lost two sales per day.
a. Find the demand function, assuming it is linear.
b. If the material for each necklace costs Terry $6, what should the selling price be to maximize his profit? Answered by Penny Nom. 





Maximize profit 
20071022 

From Dina: A meat market purchases steak from a local meat packinghouse. The meat is purchased on Monday at a price of $2 per pound, and the meat market sells the steak for $3 per pound. Any steak left over at the end of the week is sold to a local Zoo for $0.50 per pound. The demand for steak and the probabilities of occurrence are as follows:
Demand Probability
20 10%
21 10%
22 15%
23 20%
24 20%
25 15%
26 10%
Determine the amount of stock to maximize the profit. Draw the graph and explain. Answered by Penny Nom. 





Maximizing profits II 
20071005 

From a student: Suppose there are three firms with the same demand function. The function is Q=100040P. Each firm also a a cost function.
Firm 1: 4000+5Q,
Firm 2: 3000+5Q,
Firm 3: 3000+7Q.
What price should each firm charge if it wants to maximize profits. Answered by Harley Weston. 





Maximizing profit 
20071005 

From a student: Use the following equation to demonstrate how a firm that produces at MR=MC can also maximize its total profit. The equations to use are
P=1705Q
TC=40+50Q+5Q^2 Answered by Harley Weston. 





Find the dimensions of the rectangle that will contain the greatest area 
20070806 

From Julirose: The perimeter of a rectangle is 38 meters. Find the dimensions of the rectangle that will contain the greatest area. Answered by Penny Nom. 





The isosceles triangle of largest area with perimeter 12cm 
20070716 

From sharul: find the dimension of isosceles triangle of largest area with perimeter 12cm Answered by Harley Weston. 





Using Heron's Formula to help maximize the area of a triangle 
20070627 

From Claire: Given one side of a triangle is 4 cm and the ratio 1:3 for the other 2 sides. What is the largest area of the triangle? Answered by Stephen La Rocque and Harley Weston. 





Maximizing the volume of a cone given the slant length 
20070514 

From Christina: A coffee filter for a new coffee maker is to be designed using a conical filter. The filter is to be made from a circle of radius 10cm with a sector cut from it such that the volume of coffee held in the filter is maximised. Determine the dimensions of the filter such that the volume is maximised. Answered by Stephen La Rocque and Kerstin Voigt. 





Maximize revenue 
20070308 

From San: A movie theatre sells tickets for $8.50 each. The manager is considering raising the prices but knows that for every 50 cents the price is raised, 20 fewer people go to the movies. The equation R= 40c^2+84c describes the relationship between the cost of the tickets, c dollars, and the amount of revenue, R dollars, that the theatre makes. What price should the theatre charge to maximize revenue? This question comes from my gr.11 corresponding study homework and I not yet solve it. Please help! Thank you, I will appreciate your help. Answered by Stephen La Rocque. 





Maximize the area of the yard 
20070208 

From Andy: I have 60 m to construct a fence adjacent to my house. What are the values of x and y that maximize the area of the yard? Answered by Penny Nom. 





Maximizing profit 
20070123 

From Denise: Total Profit= Total RevenueTotal Cost P(x)=R(x)C(x) Where x is the number of units sold. Find the maximum profit and the number of units that must be sold in order to get that profit. R(x)=5x C(x)=.001x^2+1.2x+60 Answered by Stephen La Rocque. 





A field with the largest possible area 
20050925 

From Louise: A FARMER HAS FENCING OF 1000M AND WANTS A FIELD WITH THE BIGGEST POSSIBLE AREA HOW DO I GO ABOUT DOING THIS Answered by Penny Nom. 





Maximizing revenue 
20050513 

From Jackie: 1.The manager of a 100unit apartment complex knows from experience that all units will be occupied if the rent is $400 per month. A market survey suggests that, on the average, one additional unit will remain vacant for each $5 increase in rent. What rent should the manager charge to maximize revenue?
2.During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $10 each and his sales averaged 20 per day. When he increased the price by $1, he found that he lost two sales per day.
a. Find the demand function, assuming it is linear.
b. If the material for each necklace costs Terry $6, what should the selling price be to maximize his profit?
Answered by Penny Nom. 





Largest square inside a circle 
20041025 

From Bob: my granddaughter asked
what is the largest size square in inches
would fit in a 60 inch circle?
I believe it to be around 42.3 inches but
would like to teach her how to do it mathematically. Answered by Penny Nom. 





Maximize income 
20041024 

From Connie: A company that sells x units of a product generates an income (I, in dollars) which is a function of x. The income generated is described by the equation
I = (1/2)x^2 + 100x.
Discuss how to determine the number of units that must be sold so that the company can maximize its income. What is the maximum income? Answered by Penny Nom. 





Maximizing the angle to the goal mouth 
20040515 

From Yogendra: You are running down the boundary line dribbling the ball in soccer or hockey. Investigate where in your run the angle the goal mouth makes with your position is at a maximum. Answered by Penny Nom. 





Maximizing the area 
20040327 

From Petey: Please could you tell me why for my coursework (where I have to find the largest area that a fence 1000m long can cover) why I should only test equilateral and isoceles triangles? We were told NOT to do right angled triangles but I was wondering why not?
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. 





A lighthouse problem 
20011102 

From A student: A lighthouse at apoint P is 3 miles offshore from the nearest point O of a straight beach. A store is located 5 miles down the beach from O. The lighthouse keeper can row at 4 mph and walk at 3.25 mph.
a)How far doen the beach from O should the lighthouse keeper land in order to minimize the time from the lighthouse to the store?
b)What is the minimum rowing speed the makes it faster to row all the way? Answered by Harley Weston. 





Maximize the area 
20011013 

From Mike:
I have no clue how to do this problem. Here is what the professor gave to us: A=LW
C=E(2L+2W) + I(PL) Where P = # of partitions E= cost of exterior of fence I = cost of interior of fence C = total cost of fence . . . Answered by Harley Weston. 





Maximize profit 
20010509 

From Brian: The marginal cost for a certain product is given by MC = 6x+60 and the fixed costs are $100. The marginal revenue is given by MR = 1802x. Find the level of production that will maximize profit and find the profit or loss at that level. Answered by Harley Weston. 





Pillows and Cushions 
20000927 

From Fiona:
The following problem was given to grade eleven algebra students as a homework assignment. To manufacture cushions and pillows, a firm uses two machines A and B. The time required on each machine is shown. Machine A is available for one full shift of 9.6 hours. Machine B is available for parts of two shifts for a total of 10.5 hours each day. Answered by Harley Weston. 





Minimizing the metal in a can 
20000502 

From May Thin Zar Han: A can is to be made to hold 1 L of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can. Answered by Harley Weston. 





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. 

