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?
A piece of wire, 100 cm long, needs to be bent to form a rectangle. Determine the dimensions of a rectangle with the maximum area.
Discuss the result of maximizing the area of a rectangle, given a fixed perimeter.
A farmer has 600 m of fence and wants to enclose a rectangular field beside a river. Determine the dimensions of the fenced field in which the maximum area is enclosed. (Fencing is required on only three sides: those that aren't next to the river.)
A rectangular piece of cardboard, 100 cm by 40 cm, is going to be used to make a rectangular box with an open top by cutting congruent squares from the corners. Calculate the dimensions (to one decimal place) for a box with the largest volume.
A rectangle has a perimeter of 440 cm. What dimensions will maximize the area of the rectangle?
What are the dimensions of a rectangle with an area of 64 m^2 and the smallest possible perimeter?
A rancher has 1000 m of fencing to enclose two rectangular corrals. The corrals have the same dimensions and one side in common. What dimensions will maximize the enclosed area?
A net enclosure for practising golf shots is open at one end, as shown. Find the dimensions that will minimize the amount of netting needed and give a volume of 144 m^2. (Netting is required only on the sides, the top, and the far end.)
The volume of a square-based rectangular cardboard box needs to be 1000 cm^3. Determine the dimensions that require the minimum amount of material to manufacture all six faces. Assume that there will be no waste material. The machinery available cannot fabricate material smaller than 2 cm in length.
Determine the area of the largest rectangle that can be inscribed inside a semicircle with a radius of 10 units. Place the length of the rectangle along the diameter.
A cylindrical-shaped tin can must have a capacity of 1000 cm^3.
a) Determine the dimensions that require the minimum amount of tin for the can. (Assume no waste material.) According to the marketing department, the smallest can that the market will accept has a diameter of 6 cm and a height of 4 cm.
b) Express your answer for part a. as a ratio of height to diameter. Does this ratio meet the requirements outlined by the marketing department?
Determine the area of the largest rectangle that can be inscribed in a right triangle if the legs adjacent to the right angle are 5 cm and 12 cm long. The two sides of the rectangle lie along the legs.
An isosceles trapezoidal drainage gutter is to be made so that the angles at A and B in the cross-section ABCD are each 120^{\circ}. If the 5 m long sheet of metal that has to be bent to form the open-topped gutter and the width of the sheet of metal is 60 cm, then determine the dimensions so that the cross-sectional area will be a maximum.
Calculate the maximum volume of water that can be held by this gutter.
The 6 segments of the window frame shown in the diagram are to be constructed from a piece of window framing material 6 m in length. A carpenter wants to build a frame for a rural gothic style window, where \triangle ABC is equilateral.
- Determine the dimensions that should be used to the six pieces so that the maximum amount of light will be admitted. Assume no waste material for corner cuts and so on.
The 6 segments of the window frame shown in the diagram are to be constructed from a piece of window framing material 6 m in length. A carpenter wants to build a frame for a rural gothic style window, where \triangle ABC is equilateral.
- Would the carpenter get more light if the window was built in the shape of an equilateral triangle only? Explain.
A train leaves the station at 10:00 am and travels due south at a speed of 60 km/h. Another train has been heading due west at 45 km/h and reaches the same station at 11:00am. At what time were the two trains closest together?
A north-south highway intersect an east-west highway at point P. A vehicle crosses P at 1:00 pm, travelling east at a constant speed of 60 km/h. At the same instant, another vehicle is 5 km north of P, travelling south at 80 km/h. Find the time when the two vehicles are closest to each other and the distance between them at this time.
You looked at two specific right triangles and observed that a rectangle with the maximum area that can be inscribed inside the triangle had dimensions equal to half the lengths of the sides adjacent to the rectangle. Prove that this is true for any right triangle.
Prove that any cylindrical can of volume k cubic units that is to be made using a minimum amount of material must have the height equal to the diameter.
A piece of wire, 100 cm long, is cut into two pieces. One piece is bent to form a square, and the other piece is bent to form a circle. Determine how the wire should be cut so that the total area enclosed is
a) a maximum
b) a minimum
Determine the minimal distance from point (-3, 3) to the curve given by y = (x -3)^2.
A chord joins any two points A and B on the parabola whose equation is y^2 = 4x. If C is the midpoint of AB, and CD is drawn parallel to the x-axis to meet the parabola at D, prove that the tangent at D is parallel to chord AB.
A rectangle lies in the first quadrant, with one vertex at the origin and two of the sides along the coordinate axes. If the fourth vertex lies on the line defined by x + 2y -10 = 0, find the rectangle with the maximum area.
The base of a rectangle lies along the x-axis, and the upper two vertices are on the curve defined by y = k^2 - x^2.Determine the dimensions of the rectangle with the maximum area.