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Worksheet: Optimization Using Derivatives

Q1:

A ladder leans against a building and also touches the top of a fence. If the fence is 6 m high and 6.25 m away from the building, what is the shortest ladder that will do? Give your answer correct to the nearest thousandth.

Q2:

A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

  • A75 trees per acre
  • B82 trees per acre
  • C12 trees per acre
  • D41 trees per acre
  • E560 trees per acre

Q3:

A rectangular box with a square base is going to be built and it needs to have a volume of 20 cubic feet. The material for the base costs 30 cents per square foot. The material for the sides costs 10 cents per square foot. The material for the top costs 20 cents per square foot. Determine the dimensions that would yield the minimum cost.

  • A 2 ft by 5 ft by 5 ft
  • B 2 ft by 2 ft by 10 ft
  • C 2 ft by 2 ft by 2 ft
  • D 2 ft by 2 ft by 5 ft
  • E 5 ft by 5 ft by 5 ft

Q4:

A farmer wants to create a rectangular field on his land using an existing wall to bound one side. Determine, to the nearest thousandth, the maximum area he can obtain if he has 177-meters of fence to surround the other three sides.

  • A 11 748.375 m2
  • B 5 874.1875 m2
  • C 7 832.25 m2
  • D 3 916.125 m2

Q5:

A right circular cylinder is to have a volume of 40 cubic inches. It costs 4 cents/square inch to construct the top and bottom and 1 cent/square inch to construct the rest of the cylinder. Find the radius which would yield the minimum cost.

Q6:

A rectangular prism-shaped box has a square base. If the sum of all its edges equals 792 cm, calculate the dimensions of the box that will maximize its volume.

  • A 33 cm, 33 cm, 132 cm
  • B 198 cm, 198 cm, 99 cm
  • C 99 cm, 99 cm, 66 cm
  • D 66 cm, 66 cm, 66 cm

Q7:

A rectangular-shaped playground ends in two semicircles. Given that the perimeter of the playground is 594 m, determine its maximum area.

  • A 88 209 m2
  • B 1 7 6 4 1 8 𝜋 m2
  • C 176 418 m2
  • D 8 8 2 0 9 𝜋 m2

Q8:

A right circular cylinder without a top has a volume of 50 cubic meters. What is the radius giving the minimum surface area?

Q9:

A wire of length 41 cm is used to make a rectangle. What dimensions give the maximum area?

  • A 4 1 5 cm, 1 2 3 1 0 cm
  • B 4 1 3 cm, 4 1 6 cm
  • C 4 1 3 cm, 8 2 3 cm
  • D 4 1 4 cm, 4 1 4 cm
  • E 4 1 2 cm, 4 1 2 cm

Q10:

The rectangular cross section of a block of wood is cut from a cylindrical log of diameter 67 cm. The resistance of this block is proportional to its width and the square of its length. What dimensions give the maximum resistance?

  • A 6 7 3 6 cm, 6 7 6 6 cm
  • B 6 7 3 cm, 6 7 6 cm
  • C 6 7 6 cm, 6 7 6 cm
  • D 6 7 3 3 cm, 6 7 6 3 cm

Q11:

In the figure, what is the minimum value of 𝑥 + 2 𝑦 ? Give your answer to 3 decimal places.

Q12:

A rectangular parallelepiped has height twice the width of the base. If its volume is 7 375, what dimensions will minimise its surface area?

  • A22.28, 44.56, 7.43
  • B17.69, 35.38, 11.78
  • C14.04, 28.08, 18.71

Q13:

An open-topped box is constructed by removing equal squares from the corners of a square sheet of side 12 cm, then turning up the sides. Find the side length of the removed squares that maximises the volume of the box.

Q14:

In the figure, a semicircle is attached to a rectangle. What is the smallest perimeter if the area enclosed is 100? Give your answer exactly and in terms of 𝜋 .

  • A 2 ( 𝜋 + 4 )
  • B 1 0 𝜋 + 2
  • C 1 0 2 𝜋 + 4
  • D 1 0 2 ( 𝜋 + 4 )
  • E 1 0 2 ( 𝜋 + 8 )

Q15:

If the hypotenuse length of a right triangle equals 33.5 cm, find the lengths of the two sides of the right angle to the nearest thousandth when the area of the triangle is as maximum as possible.

  • A 16.75 cm, 16.75 cm
  • B 23.688 cm, 7.089 cm
  • C 23.688 cm, 41.029 cm
  • D 23.688 cm, 23.688 cm

Q16:

The cost of petrol for a truck is proportional to the square of its speed. The cost is 6 3 L E per hour at a speed of 63 km/h and the other costs amount to 1 6 2 L E per hour regardless of speed. Find the speed that would minimise the total cost of running the truck.

Q17:

In the figure, given that 0 𝑥 1 0 , what is the maximum value of 𝑥 + 2 𝑦 ? Give your answer to 3 decimal places.

Q18:

All the vertices of a certain rectangle lie on an equilateral triangle with sides of length 14 cm; one side of the rectangle lies on the base of the triangle and the vertices of the opposite side of the rectangle lie on the two other sides of the triangle. What is the maximum area this rectangle could have?

  • A 196 cm2
  • B 49 cm2
  • C 1 9 6 2 cm2
  • D 4 9 3 2 cm2

Q19:

A tank with a square bottom has a capacity 268 m3. Coating the bottom, cover, and sides costs 5 1 L E , 1 6 L E , and 2 7 L E per square metre, respectively. Find the dimensions that give the lowest coating cost.

  • A 5 m, 2 6 8 2 5 m
  • B 4 m, 6 7 4 m
  • C 3 m, 2 6 8 9 m
  • D 6 m, 6 7 9 m

Q20:

Given that 𝐴 𝐵 𝐶 𝐷 is a square whose side length is 11 cm, 𝑀 𝐵 𝐶 , where 𝐵 𝑀 = 𝑥 c m , and 𝑁 𝐶 𝐷 , where 𝐶 𝑁 = 9 5 𝑥 , find the value of 𝑥 that makes the area of 𝐴 𝑀 𝑁 as minimum as possible.

  • A 9 9 1 0
  • B 5 5 9
  • C 9 9 5
  • D 5 5 1 8

Q21:

The length of the hypotenuse in a right triangle is 43 cm. Find the length of the other two sides given that the length of the altitude drawn from the right angle to the hypotenuse is as large as possible.

  • A 43 cm, cm
  • B cm, cm
  • C cm, cm
  • D cm, cm
  • E cm, cm

Q22:

In a circuit of an alternating current, the current 𝐼 , measured in amperes, at any moment 𝑡 , measured in seconds, is given by the relation 𝐼 = 1 6 𝑡 + 1 9 𝑡 s i n c o s . What is the maximum value of the current in this circuit rounded to two decimal places?

Q23:

A rectangular enclosure is divided into two identical pens and fenced with 300 feet of fencing. What dimensions would give the maximum area?

  • Alength: 75 ft, width: 75 ft
  • Blength: 50 ft, width: 37.5 ft
  • Clength: 75 ft, width: 37.5 ft
  • Dlength: 75 ft, width: 50 ft
  • Elength: 50 ft, width: 50 ft

Q24:

A rectangle lies inside an equilateral triangle with one of its sides on the base on the triangle and a vertex on each of the other two sides. Given that the triangle has a side of 77 cm, what are the dimensions of the rectangle with maximum area?

  • A 77 cm, 77 cm
  • B 7 7 4 cm, 77 cm
  • C 7 7 2 cm, 77 cm
  • D 7 7 3 4 cm, 7 7 2 cm

Q25:

A factory producing appliances makes a profit of 57 per appliance when it produces 16 per month. Suppose that this profit decreases by 25 piastres for every extra appliance produced. How many appliances produced per month ensures maximum profit?

  • A327 appliances
  • B76 appliances
  • C362 appliances
  • D122 appliances