Lesson Worksheet: Optimization: Applications on Extreme Values Mathematics • Higher Education

In this worksheet, we will practice applying derivatives to real-world problems to find the maximum and the minimum values of a function under certain conditions.

Q1:

Find the two numbers whose sum is 96, and their product is as large as possible.

  • A48, 144
  • B192, 288
  • C192, 96
  • D48, 48

Q2:

Find two numbers whose sum is 156 and the sum of whose squares is the least possible.

  • A452, 608
  • B78, 78
  • C344, 500
  • D148, 8

Q3:

What is the maximum volume of a right circular cylinder with surface area 24𝜋 cm2? Give your answer in terms of 𝜋.

  • A8𝜋 cm3
  • B16 cm3
  • C4𝜋 cm3
  • D16𝜋 cm3

Q4:

The sum of the sides of a rectangular parallelepiped with a square base is 12 cm. Find the dimensions that maximize the volume.

  • A2 cm, 2 cm, and 2 cm
  • B1 cm, 1 cm, and 1 cm
  • C4 cm, 2 cm, and 6 cm
  • D1 cm, 2 cm, and 2 cm

Q5:

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.

Q6:

Given that the volume of a hot air balloon grows according to the relation 𝑓(𝑡)=7,000𝑡𝑡+49+4,000, where the time is measured in hours, determine its maximum volume.

Q7:

A window is made of a semicircle on top of a rectangle, with the semicircle’s diameter equal to the rectangle’s width. Given that the window’s perimeter is 30 m, determine the semicircle’s radius that maximizes the window’s area.

  • A304+𝜋 m
  • B4+𝜋30 m
  • C2+𝜋𝜋 m
  • D302+𝜋 m
  • E14+𝜋 m

Q8:

Find the points on the curve 𝑦=2𝑥+21 that are closest to the point (6,0).

  • A5,31, 5,31
  • B4,13, 4,13
  • C7,7, 7,7
  • D7,35, 7,35

Q9:

A rectangular piece of cardboard paper has two dimensions of 10 cm and 16 cm. If congruent squares of side length 𝑥 cm are cut from its four corners, and the projected parts are folded upward to form a box without a cover, calculate the dimension of the formed box when its volume is as maximum as possible.

  • A2 cm, 6 cm, 12 cm
  • B6 cm, 4 cm, 10 cm
  • C6 cm, 2 cm, 4 cm
  • D2 cm, 8 cm, 14 cm

Q10:

A sector of a circle has area 16 cm2. Find the radius 𝑟 that minimizes its perimeter, and then determine the corresponding angle 𝜃 in radians.

  • A𝑟=4cm, 𝜃=116rad
  • B𝑟=4cm, 𝜃=2rad
  • C𝑟=4cm, 𝜃=12rad
  • D𝑟=16cm, 𝜃=18rad
  • E𝑟=8cm, 𝜃=12rad

Q11:

A vertical cylindrical silo of capacity 384𝜋 m3 is to be built with a hemispherical dome top. If painting one square meter of the dome costs three times as much as painting one square meter of the side, what dimensions will minimize the painting cost?

  • Ar = 6 m, h = 36 m
  • Br = 8 m, h = 24 m
  • Cr = 5 m, h = 8 m
  • Dr = 4 m, h = 24 m
  • Er = 3 m, h = 30 m

Q12:

What is the maximum area of an isosceles triangle inscribed in a circle of radius 47 cm? Give your answer to the nearest hundredth.

Q13:

Given that the sum of the surface areas of a sphere and a right circular cylinder is 1,000𝜋 cm2, and their radii are equal, find the radius of the sphere that makes the sum of their volume at its maximum value.

Q14:

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.

Q15:

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.

  • A99 cm, 99 cm, and 66 cm
  • B198 cm, 198 cm, and 99 cm
  • C33 cm, 33 cm, and 132 cm
  • D66 cm, 66 cm, and 66 cm

Q16:

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

  • A88,209𝜋 m2
  • B176,418 m2
  • C176,418𝜋 m2
  • D88,209 m2

Q17:

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

  • A412 cm, 412 cm
  • B415 cm, 12310 cm
  • C414 cm, 414 cm
  • D413 cm, 823 cm
  • E413 cm, 416 cm

Q18:

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?

  • A6733 cm, 6763 cm
  • B676 cm, 676 cm
  • C673 cm, 676 cm
  • D6736 cm, 6766 cm

Q19:

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

Q20:

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

  • A17.69, 35.38, 11.78
  • B14.04, 28.08, 18.71
  • C22.28, 44.56, 7.43

Q21:

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 maximizes the volume of the box.

Q22:

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.

  • A2(𝜋+4)
  • B102(𝜋+4)
  • C10𝜋+2
  • D102𝜋+4
  • E102(𝜋+8)

Q23:

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

  • A23.688 cm, 41.029 cm
  • B16.750 cm, 16.750 cm
  • C23.688 cm, 7.089 cm
  • D23.688 cm, 23.688 cm

Q24:

In the figure, given that 0𝑥10, what is the maximum value of 𝑥+2𝑦? Give your answer to 3 decimal places.

Q25:

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?

  • A196 cm2
  • B4932 cm2
  • C49 cm2
  • D1962 cm2

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