3.4 Optimization in Economics and Science
Chapter
Chapter 3
Section
3.4
Solutions 25 Videos

The cost, in dollars, to produce x litres of maple syrup festival is C(x) = 75(\sqrt{x} -10), where x \geq 400.

• What is the average cost of producing 625 L?
0.38mins
Q1a

The cost, in dollars, to produce x litres of maple syrup festival is C(x) = 75(\sqrt{x} -10), where x \geq 400.

• The marginal cost at C'(x), and the marital revenue is R'(x). Marginal cost at x litres is the expected change in cost if we were to produce one additional litre of syrup. Similarly for marginal revenue. What is the marginal cost at 1225 L?
0.50mins
Q1b

The cost, in dollars, to produce x litres of maple syrup festival is C(x) = 75(\sqrt{x} -10), where x \geq 400.

• How much production is needed to achieve a marginal cost of \$0.50/L? Buy to View 0.37mins Q1c A sociologist determines that a foreign-language student has learned N(t) = 20t - t^2 vocabulary terms after t hours of uninterrupted study. • How many terms are learned between times t = 2 and t =3? Buy to View 1.06mins Q2a A sociologist determines that a foreign-language student has learned N(t) = 20t - t^2 vocabulary terms after t hours of uninterrupted study. • What is the rate, in terms per hour, at which the student is learning at time t = 2? Buy to View 0.28mins Q2b A sociologist determines that a foreign-language student has learned N(t) = 20t - t^2 vocabulary terms after t hours of uninterrupted study. • What is the maximum rate, in terms per hour, at which the student is learning? Buy to View 0.43mins Q2c A researcher found that the level of antacid in a person's stomach, t minutes after a certain brand of antacid tablet is taken, is L(t) = \displaystyle{\frac{6t}{t^2 + 2t + 1}}. • Determine the value of t for which L'(t) = 0. Buy to View 1.42mins Q3a A researcher found that the level of antacid in a person's stomach, t minutes after a certain brand of antacid tablet is taken, is L(t) = \displaystyle{\frac{6t}{t^2 + 2t + 1}}. Determine L(1) Buy to View 0.30mins Q3b A researcher found that the level of antacid in a person's stomach, t minutes after a certain brand of antacid tablet is taken, is L(t) = \displaystyle{\frac{6t}{t^2 + 2t + 1}}. i) Using your graphing calculator, graph L(t). ii) From the graph, what can you predict about the level of antacid in a person's stomach after 1 min? iii) What is happening to the level of antacid in person's stomach from 2\leq t \leq 8. Buy to View 0.26mins Q3cde The operating cost, C, in dollars per hour, for an airplane cursing at a height of h metres and an air speed of 200 km/h is given by C = 4000 + \displaystyle{\frac{h}{15}} + \displaystyle{\frac{15 000 000}{h}} for the domain 1000 \leq h \leq 20 000. Determine the height at which the operating cost is at a minimum, and find the operating cost per hour at this height. Buy to View 1.56mins Q4 A rectangular piece of land is to be fenced using two kinds of fencing. Two opposite sides will be fenced using standard fencing that costs$6/m, while the other two sides will require heavy-duty fencing that costs $9/m. What are the dimensions of the rectangular lot of greatest area that can be fenced for a cost of$9000?

3.12mins
Q5

A real estate office manages 50 apartments in a downtown building. When the rent is $900 per month, all the units are occupied. for every$25 increase in rent, one unit becomes vacant. On average, all units require $75 in maintenance and repairs each month . How much rent should the real testate office charge to maximize profits? Buy to View 2.46mins Q6 A bus service carries 10 000 people daily between Ajax and Union Station, and the company has space to serve up to 15 000 people per day. The cost to ride the bus is \$20. Market research shows that if the fare increases by \$0.50, 200 fewer people will ride the bus. What fare should be charged to get the maximum revenue, given that the bus company must have at least \$130 000 in fares a day to cover operating costs?

3.12mins
Q7

The fuel cost per hour for running a ship is approximately one half the cube of the speed (measured in knots) plus additional fixed costs of $216 per hour. Find the most economical speed to run the ship for a 500 M (nautical mile) trip. Note: Assume that there are no major disturbances, such as heavy tides or stormy seas. Buy to View 3.44mins Q8 The cost of producing an ordinary cylindrical tin can is determined by the materials used for the wall and the end pieces. If the end pieces are twice as expensive per square centimetre as the wall, find the dimensions (to the nearest millimetre) to make a 1000 cm^3 can at minimal cost. Buy to View 4.31mins Q10 Sandy is making a closed rectangular jewellery box with a square base from two different woods. The wood for the top and bottom costs 20/m^2. The wood for the sides costs 30/m^2 . Find the dimensions that will minimize the cost of the wood for a volume of 4000$ cm^3.

3.47mins
Q12

An electronics store is selling personal CD players. The regular price for each CD player is $90. During a typical two weeks, the store sells 50 units. Past sales indicate that for every \$1  decrease in price, the store sells five more units during two weeks. Calculate the price that will maximize revenue.

A professional basketball team plays in an arena that holds 20 000 spectators. Average attendance at each game has been 14 000. The average ticket price is \$75. Market research shows that for each \$5 reduction in the ticket price, attendance increases by 800. Find the price that will maximize revenue.