3.6 Optimization Problems
Chapter 3
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Solutions 26 Videos

The height, h, of a ball t seconds after being thrown into the air is given by the function h(t)=-4.9t^2+19.6t+2. Find the maximum height of the ball.

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Find the two integers that have a sum of 20 and a maximum product}

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At G&W Industries, it has been shown that the number of gizmos an employee can produce each day can be represented by the equation N(t)=-0.05t^2+3t+5, where t is the number of years of experience the employee has, and 0\leq t \leq 40. How many years of experience does it take for an employee to achieve maximum productivity?

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A rectangular pen is to be built with 1200m of fencing. The pen is to be divided into three parts using two parallel partitions.

(a) Find the maximum possible area of the pen.

(b) Explain how the maximum area would change if each side of the pen had to be at least 180m long.

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Two pens with one common side are to be built with 60m of fencing. one pen is to be square, and the other rectangular, as shown. Find the dimensions that maximize the total area.

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A showroom for a car dealership is to be built in the shape of a rectangle with brick on the back and sides, and glass on the front. The floor of the showroom is to have an area of 500 m^2.

a) If a brick wall costs $1200/m while a glass wall costs $600/m, what dimensions would minimize the cost of the showroom?

b) Should the cost of the roof be considered when deciding on the dimensions? Explain your reasoning. Assume the roof is flat.

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A Norman window has the shape of a rectangle with a semicircular top, as shown.

a) For a given perimeter, find the ratio of height to radius that will maximize the window’s area.

b) Assume that the semicircular portion of the perimeter is three times as costly to build per metre as the straight edges. For a given area, what ratio of height to radius would minimize cost?

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A cylindrical can is to have a volume of 1 L.

a) Find the height and radius of the can that will minimize the surface area.

b) What is the ratio of the height to the diameter?

c) Do pop cans have a similar ratio? If not, suggest why.

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A soup can of volume 500 cm^3 is to be constructed. The material for the top costs 0.4¢/cm^2 while the material for the bottom and sides costs 0.2¢/cm^2. Find the dimensions that will minimize the cost of producing the can.

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A cylindrical drum with an open top is to be constructed using 1 m^2 of aluminum.

a) Write an equation for the volume of the drum in terms of the radius.

b) What radius gives the maximum volume?

c) Use a graphing calculator to graph the equation from part a).

d) What is the maximum volume if the radius can be a maximum of 0.2 m? Refer to the graph as you explain your reasoning.

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A rectangular piece of paper with perimeter 100 cm is to be rolled to form a cylindrical tube. Find the dimensions of the paper that will produce a tube with maximum volume.

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There are 50 apple trees in an orchard, and each tree produces an average of 200 apples each year. For each additional tree planted within the orchard, the average number of apples produced drops by 5. What is the optimal number of trees to plant in the orchard?

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For an outdoor concert, a ticket price of $30 typically attracts 5000 people. For each $1 increase in the ticket price, 100 fewer people will attend. The revenue, R, is the product of the number of people attending and the price per ticket.

a) Let x represent the number of $1 price increases. Find an equation expressing the total revenue in terms of x.

b) State any restrictions on x. Can x be a negative number? Explain.

c) Find the ticket price that maximizes revenue.

d) Will your answer to part c) change if the concert area holds a maximum of 1200 people? Explain.

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Find the area of the largest rectangle that can be inscribed between the x-axis and the graph defined by y = 9-x^2.

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The cost of fuel per kilometre for a truck travelling v kilometres per hour is given by the eqwuation \displaystyle C(v) = \frac{v}{100} + \frac{25}{v}

a) What speed will result in the lowest fuel cost per kilometre?

b) Assume the driver is paid $40/h. What speed would give the lowest cost, including fuel and wages, for a 1000-km trip?

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Find a positive number such that the sum of the square of the number and its reciprocal is a minimum.

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Brenda drives an 18-wheeler. She plans to buy her own truck. Her research indicates that the expected running costs, C, in dollars, per 100 km, are given by C(v) = 0.9 + 0.0016v^2, where v is the speed, in kilometres per hour. Brenda’s first trip will be 1500 km, round trip. She plans to pay herself $30/h. Determine the speed that will minimize Brenda’s costs for the trip.

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A piece of plexiglass is in the shape of a semicircle with a radius 2 m. Determine the dimensions of the rectangle with the greatest area that can be cut from the piece of plexiglass.

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Recall that 20 m of edging is to form the two sides of the square garden and the curved edge of the garden in the shape of a quarter circle. Let x represent the length of edging used for the quarter circle.

a) Find an expression for the length of edging used for each side of the square garden.

b) Find an expression for the radius of the quarter circle.

c) Use your answers to parts a) and b) to find an equation for the combined area of the two gardens.

d) Show that your equation from part c) is equivalent to \displaystyle A(x) = (\frac{4+ \pi}{4\pi})x^2 - 10x + 100 .

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A 60-cm by 40-cm piece of tin has a square cut out of each corner, and then the sides are folded up to form an open-top box.

a) Let x represent the side length of the squares that are cut out. Draw a diagram showing all dimensions.

b) Find an equation to represent the volume of the box.

c) State the domain of the function. Explain your reasoning.

d) Find the dimensions that will maximize the volume of the box.

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Oil is shipped to a remote island in cylindrical containers made of steel. The height of each container equals the diameter. Once the containers are emptied on the island, the steel is sold. Shipping costs are \$10/m^3 of oil, and the steel is sold for \$7/m^2.

a) Determine the radius of the container that maximizes the profit per container. Ignore any costs (other than shipping) or profits associated with the oil in the barrel.

b) Determine the maximum profit per container.

c) Check your answers to parts a) and b) by graphing.

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A box with a square base and closed top is to be constructed out of cardboard.

a) Without finding derivatives, what shape do you expect would give the minimum surface area for a given volume? Explain.

b) Does the volume affect the shape of the box? Explain.

c) If the box has no top, will the shape change? Verify your answer by choosing a volume and solving the problem using calculus.

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A series of rectangular fenced pens is to be built, each using 1000 m of fencing.

a) a) Find the dimensions that will maximize the total area in each situation.

  • i) a rectangular pen with no restrictions
  • ii) a rectangular pen, divided into two as shown #9345
  • iii) a rectangular pen, placed against a barn so it only requires three sides to be fenced

b) Consider the three situations in part a). Do you see a pattern? Explain.

c) Explain how you could find the dimensions if the pens were similar to those in part a)ii), but divided into four equal parts.

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a) For a rectangular prism of a given volume, what shape has the least surface area?

b) What three-dimensional shape has the least surface area for a given volume?

c) Think of the shapes of various packages you see at the grocery store. Why do you suppose they are not necessarily the shapes you described in parts a) and b)? What other factors might be considered that would affect the shape of containers and packaging?

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A piece of wire of length L is cut into two pieces. One piece is bent into a square and the other into a circle. How should the wire be cut so that the total area enclosed is a minimum?

A Use all of the wire for the circle.

B Use all of the wire for the square.

C Use \displaystyle \frac{L}{2} for the circle.

D Use \displaystyle \frac{4L}{\pi + 4} for the circle.

E Use \displaystyle \frac{\pi L}{\pi + 4} for the circle.

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Consider the function \displaystyle f(x) = \frac{x^n}{x - 1} , where n is a positive integer. Let G represent the greatest possible number of local local extrema for f(x), and let L represent the least possible number of local extrema. Which statement is true?

A G = n and L = n -1

B G =n and L = 1

C G = n -1 and L = 0

D G = 2 and L = 1

E G = 2 and L = 0

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Lectures 7 Videos

ex A farmer has 2400 m of fencing and wants to fence off a rectangular field that borders a straight river. He needs to fence along the river. What are the dimensions of the field that has the largest area?

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2 Introduction to Optimization with example with boundaries

If 800 cm^2 of material is available to make a box with a square base, find the largest volume of the box.

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3 ex2 Optimizing Volume of a Box

A cylindrical 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.

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4 ex3 surface area of cydlinder