Exercises


  1. Find the maximum and minimum values of the given functions over each of the given intervals or explain why they do not exist.
    1. \( f(x) = 2x^3 - 3x^2 - 12x + 5 \) on the intervals
      1. \( [-3, 4] \)
      2. \( [-1, 3] \)
      3. \( [-2, 4] \)
    2. \( f(x) = x^3 - 3x + 1 \) on the intervals
      1. \( [0, 2] \)
      2. \( [-3, 2] \)
    3. \( f(x) = x^{\frac{2}{3}} \) on the intervals
      1. \( [1, 8] \)
      2. \( [-8, 8] \)
    4. \( f(x) = \dfrac{3x^2}{x - 3} \) on the intervals
      1. \( [-2, 2] \)
      2. \( [2, 8] \)
  2. The number of salmon swimming upstream to spawn, \( N \), is approximated by the function\[ N(t) = -t^3 + 3t^2 + 864t + 5000\] \(6 \leq t \leq 20 \), where \( t \) represents the temperature of the water in degrees Celsius. Determine the optimal temperature to maximize the size of the salmon population.
  3. A three-sided fence is to be built next to a straight section of river, which forms the fourth side of a rectangular region, as shown in the diagram below. The enclosed area is to equal \( 1800 \) m\(^2 \) and the fence running parallel to the river must be set back at least \( 20 \) m from the river. Determine the minimum perimeter of such an enclosure and the dimensions of the corresponding enclosure.
    Rectangle with 3 sides of fence and a fourth side of the river.
  4. A two-pen corral is to be built. The outline of the corral forms two identical adjoining rectangles, as shown in the diagram below. If there is \( 120 \) m of fencing available and the fence width cannot be less than \( 6 \) m, what dimensions of the corral will maximize the enclosed area?
    A rectangle divided into two equal rectangles with the width going the same way as the dividing line
  5. An open rectangular box is to be made by cutting four equal squares from each corner of a \( 12 \) cm by \( 12 \) cm piece of metal and then folding up the sides (sample diagram shown below). The finished box must be at least \( 1.5 \) cm deep, but not deeper than \( 3 \) cm. What are the dimensions of the finished box if the volume is to be maximized?
    Diagram of situation described in question
  6. A rectangular picnic area of \( 8000 \) m\(^2 \) is being constructed by the Parks Commission along the edge of a river. It will be fenced on three sides, but not along the river. Ornamental fencing costing \( $12 \)/m will be used on the side opposite the river and chain-link fencing costing \($3 \)/m will be used on the other two sides.
    1. If the Parks Commission wants to have at least \( 40 \) m of riverfront, what are the dimensions of the picnic area that will minimize the cost of the project?
    2. If the Parks Commission wants to have at least \( 80 \) m of riverfront, what are the dimensions of the picnic area that will minimize the cost of the project?
  7. A closed box with a square base is to be made to have a volume of \( 16~000 \) cm\(^3\). The material for the top and bottom of the box costs \( 3 \) cents per square centimetre, while the material for the sides costs \( 1.5 \) cents per square centimetre. If no dimension can be less than \( 10 \) cm, determine the dimensions of the box that will minimize cost.
  8. A natural gas line runs east-west. A town wants to connect two new housing developments to the line by running lines from a single point on the existing line to the two developments. One development, D1, is \( 3 \) km due south of the existing line; the other development, D2, is \( 4 \) km due south of the existing line and \( 5 \) km east of the first development. Find the place on the existing line to make the connection to minimize the total length of new line.
    Diagram of situation described in question
  9. A carpenter is building an open box with a square base and with no dimension less than \( 1 \) m. The material for the sides of the box will cost \( $3 \) per square metre, and the material for the base will cost \( $4 \) per square metre. What are the dimensions of the box of greatest volume that can be constructed for \( $72 \).
  10. Determine the coordinates of the point \( P \) on the parabola \( y = 9 - x^2 \) that is closest to the point \( A\,(3, 9) \).
  11. A rectangle is inscribed in a right triangle with side lengths \( 5 \) m, \( 12 \) m, and \( 13 \) m. What are the dimensions of the inscribed rectangle of greatest area?
  12. A company is running a utility cable from point \( A \) on the shore to an installation on an island, at point \( B \). The island is \( 6 \) km from the shore (at point \( C \)) and point \( A \) is \( 9 \) km from point \( C \) (see diagram below). It costs \( $400 \) per kilometer to run the cable on land and \( $500 \) per kilometer to run the cable under water. At what distance along the shore from \( A \) should the company start running the cable under the water if they wish to minimize the cost of the project?
    Point C is 9 km due west from point A and 6 km due south of point B
  13. Two industrial plants, \( A \) and \( B \), are located \( 15 \) km apart and emit \( 75 \) parts per million (ppm) and \( 300 \) ppm of particulate matter, respectively. Each plant is surrounded by a restricted area of radius \( 1 \) km in which no housing is allowed and the concentration of pollutant arriving at any other point \( H \) from each plant decreases with the reciprocal of the distance between that plant and \( H \). Where should a house be located on a road joining the two plants to minimize the total pollution arriving from both plants?