Compiled by: GA Mac Tavish Grade 12 Revision Book Calculus II
MATHEMATICS
Matric Intervention Programme
2018
CALCULUS II
Optimisation Mixed
Exam Questions
GRADE 12
Common Exam Type Questions
Question 1: A chapel window consists of four equal rectangles and a semi-circle. The
length of the metal that is being used for the frame is 36 metres.
Page 1 of 27
,Compiled by: GA Mac Tavish Grade 12 Revision Book Calculus II
2
1.1
x
Prove that the area for the frame is given by: A 24x 4x2
6
SOLUTION:
1.2
Determine the length of the base PQ for a maximum area of the window.
SOLUTION:
Question 2 A semi-circle of diameter x is cut from a rectangle with a length of 80 cm.
Calculate the value of x which gives the maximum possible (shaded) area.
Give your answer in terms of .
Page 2 of 27
, Compiled by: GA Mac Tavish Grade 12 Revision Book Calculus II
SOLUTION:
Question 3 A cannon fires a projectile onto the top of a hill, such that the height (in metres)
reached by the projectile seconds after it is fired from the cannon is given by the
parabola equation, s 200t 5t2 .
If the velocity of the projectile, when it hits the hill on the way down, is 170 metres per
second, what is the height of the hill where the projectile hits?
SOLUTION: 𝑠 = 200𝑡 − 5𝑡2
𝑠′ = 200 − 10𝑡 = 170 t = 3 , using symmetry (AOS: x = 20), t = 37
or 𝑠′ = 200 − 10𝑡 = −170 t = 37 𝑠 = 200(37) −
5(37)2 = 555𝑚
Question 4 The perimeter of a shape is given by:
Calculate the value of that will make the perimeter ( ) a minimum.
SOLUTION:
𝜋𝑥2 + 4𝑥2 = 16
𝑥2(𝜋 + 4) = 16
𝑥2 = 2,24
𝑥 = 1,50
Page 3 of 27
MATHEMATICS
Matric Intervention Programme
2018
CALCULUS II
Optimisation Mixed
Exam Questions
GRADE 12
Common Exam Type Questions
Question 1: A chapel window consists of four equal rectangles and a semi-circle. The
length of the metal that is being used for the frame is 36 metres.
Page 1 of 27
,Compiled by: GA Mac Tavish Grade 12 Revision Book Calculus II
2
1.1
x
Prove that the area for the frame is given by: A 24x 4x2
6
SOLUTION:
1.2
Determine the length of the base PQ for a maximum area of the window.
SOLUTION:
Question 2 A semi-circle of diameter x is cut from a rectangle with a length of 80 cm.
Calculate the value of x which gives the maximum possible (shaded) area.
Give your answer in terms of .
Page 2 of 27
, Compiled by: GA Mac Tavish Grade 12 Revision Book Calculus II
SOLUTION:
Question 3 A cannon fires a projectile onto the top of a hill, such that the height (in metres)
reached by the projectile seconds after it is fired from the cannon is given by the
parabola equation, s 200t 5t2 .
If the velocity of the projectile, when it hits the hill on the way down, is 170 metres per
second, what is the height of the hill where the projectile hits?
SOLUTION: 𝑠 = 200𝑡 − 5𝑡2
𝑠′ = 200 − 10𝑡 = 170 t = 3 , using symmetry (AOS: x = 20), t = 37
or 𝑠′ = 200 − 10𝑡 = −170 t = 37 𝑠 = 200(37) −
5(37)2 = 555𝑚
Question 4 The perimeter of a shape is given by:
Calculate the value of that will make the perimeter ( ) a minimum.
SOLUTION:
𝜋𝑥2 + 4𝑥2 = 16
𝑥2(𝜋 + 4) = 16
𝑥2 = 2,24
𝑥 = 1,50
Page 3 of 27
