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Lesson: Using Derivatives in Optimization Problems

Sample Question Videos

Worksheet • 25 Questions • 1 Video

Q1:

Find the two numbers whose sum is 74, and their product is as maximum as possible.

  • A37, 37
  • B 3 7 , 111
  • C148, 7 4
  • D 1 4 8 , 222

Q2:

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

  • A48, 48
  • B 4 8 , 144
  • C192, 9 6
  • D 1 9 2 , 288

Q3:

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

  • A 1 cm, 1 cm, 1 cm.
  • B 2 cm, 2 cm, 2 cm.
  • C 4 cm, 2 cm, 6 cm.
  • D 1 cm, 2 cm, 2 cm.

Q4:

The sum of the sides of a rectangular parallelepiped having a square base is 24 cm. Find the dimensions that maximize the volume.

  • A 2 cm, 2 cm, 2 cm.
  • B 4 cm, 4 cm, 4 cm.
  • C 8 cm, 4 cm, 12 cm.
  • D 2 cm, 4 cm, 4 cm.

Q5:

What is the maximum volume of a right circular cylinder with surface area 2 4 𝜋 cm2? Give your answer in terms of 𝜋 .

  • A 1 6 𝜋 cm3
  • B 8 𝜋 cm3
  • C 16 cm3
  • D 4 𝜋 cm3

Q6:

What is the maximum volume of a right circular cylinder with surface area 2 1 6 𝜋 cm2? Give your answer in terms of 𝜋 .

  • A 4 3 2 𝜋 cm3
  • B 7 2 𝜋 cm3
  • C 432 cm3
  • D 3 6 𝜋 cm3

Q7:

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.

  • A 2 cm, 6 cm, 12 cm
  • B 6 cm, 2 cm, 4 cm
  • C 6 cm, 4 cm, 10 cm
  • D 2 cm, 8 cm, 14 cm

Q8:

A rectangular piece of cardboard paper has two dimensions of 5 cm and 8 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.

  • A 1 cm, 3 cm, 6 cm
  • B 3 cm, 1 cm, 2 cm
  • C 3 cm, 2 cm, 5 cm
  • D 1 cm, 4 cm, 7 cm

Q9:

Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.

  • A 5 0 × 5 0 f t f t
  • B 4 0 0 × 4 0 0 f t f t
  • C 1 0 0 × 1 0 0 f t f t
  • D 2 0 0 × 2 0 0 f t f t
  • E 2 5 × 2 5 f t f t

Q10:

Given that the sum of the surface areas of a sphere and a right circular cylinder is 1 0 0 0 𝜋 cm2, and their radii are equal, find the radius of the sphere that makes the sum of their volume at its maximum value.

Q11:

Given that the sum of the surface areas of a sphere and a right circular cylinder is 8 1 0 𝜋 cm2, and their radii are equal, find the radius of the sphere that makes the sum of their volume at its maximum value.

Q12:

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.

  • A 3 0 4 + 𝜋 m
  • B 4 + 𝜋 3 0 m
  • C 1 4 + 𝜋 m
  • D 3 0 2 + 𝜋 m
  • E 2 + 𝜋 𝜋 m

Q13:

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 58 m, determine the semicircle’s radius that maximizes the window’s area.

  • A 5 8 4 + 𝜋 m
  • B 4 + 𝜋 5 8 m
  • C 1 4 + 𝜋 m
  • D 5 8 2 + 𝜋 m
  • E 2 + 𝜋 𝜋 m

Q14:

A rectangular corral is split into 3 pens with identical dimensions. Given that 500 feet of fencing was used to construct the corral and it was designed to have the maximum area possible, find the dimensions of the corral.

  • Alength = 125 ft, width = 62.5 ft
  • Blength = 62.5 ft, width = 50 ft
  • Clength = 62.5 ft, width = 62.5 ft
  • Dlength = 125 ft, width = 125 ft
  • Elength = 62.5 ft, width = 41.7 ft

Q15:

Given that the volume of a hot air balloon grows according to the relation 𝑓 ( 𝑡 ) = 7 0 0 0 𝑡 𝑡 + 4 9 + 4 0 0 0 2 , where the time is measured in hours, determine its maximum volume.

Q16:

Given that the volume of a hot air balloon grows according to the relation 𝑓 ( 𝑡 ) = 6 0 0 0 𝑡 𝑡 + 9 + 6 0 0 0 2 , where the time is measured in hours, determine its maximum volume.

Q17:

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

Q18:

What is the maximum area of an isosceles triangle inscribed in a circle of radius 28 cm? Give your answer to the nearest hundredth.

Q19:

Find the points on the curve 𝑦 = 2 𝑥 + 2 1 2 that are closest to the point ( 6 , 0 ) .

  • A 7 , 7 , 7 , 7
  • B 5 , 3 1 , 5 , 3 1
  • C 4 , 1 3 , 4 , 1 3
  • D 7 , 3 5 , 7 , 3 5

Q20:

Find the points on the curve 𝑦 = 3 𝑥 + 2 4 2 that are closest to the point ( 3 , 0 ) .

  • A 3 2 , 1 1 4 2 , 3 2 , 1 1 4 2
  • B 9 2 , 4 2 2 , 9 2 , 4 2 2
  • C 0 , 2 6 , 0 , 2 6
  • D 3 2 , 7 8 2 , 3 2 , 7 8 2

Q21:

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

  • A78, 78
  • B 3 4 4 , 500
  • C148, 8
  • D 4 5 2 , 608

Q22:

Find two numbers whose sum is 4 0 and the sum of whose squares is the least possible.

  • A 2 0 , 2 0
  • B 7 3 , 33
  • C 4 2 4 , 384
  • D 4 6 8 , 428

Q23:

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

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q24:

A vertical cylindrical silo of capacity 3 8 4 𝜋 m3 is to be built with a hemispherical domed top. If painting the dome costs three times as much as painting the sides, what dimensions will minimise the painting cost?

  • A radius = 4 m, height = 24 m
  • B radius = 3 m, height = 30 m
  • C radius = 8 m, height = 24 m
  • D radius = 6 m, height = 36 m
  • E radius = 5 m, height = 8 m

Q25:

A vertical cylindrical silo of capacity 1 2 8 𝜋 m3 is to be built with a hemispherical domed top. If painting the dome costs four times as much as painting the sides, what dimensions will minimise the painting cost?

  • A radius = 4 m, height = 8 m
  • B radius = 3 m, height = 10 m
  • C radius = 8 m, height = 8 m
  • D radius = 6 m, height = 12 m
  • E radius = 5 m, height = 8 m
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