Name: ___________________________
Maximizing and Rates
A manufacturer of electric kettles performs a cost control study. They discover that to produce x kettles
30−𝑥 2
per day, the cost per kettle is given by 𝐶(𝑥) = 4 ln 𝑥 + ( 10 ) dollars with a minimum production
capacity of 10 kettles per day. How many kettles should be manufactured to keep the cost per kettle to a
minimum?
2
Infinitely many rectangles which sit on the x–axis can be inscribed under the curve 𝑦 = 𝑒 −𝑥 .
a) Draw a picture of the situation.
b) Describe the coordinates such that rectangle ABCD has maximum area.
Consider the manufacture of cylindrical tin cans of 1 L capacity where the cost of the metal used is to be
minimized. This means that the surface area must be as small as possible.
1000
a) Explain why the height h is given by ℎ = 𝜋𝑟 2 cm.
b) Show that the total surface area A is given by 𝐴 = 2𝜋𝑟 2 +
c) Find the value for r that makes A as small as possible.
2000
𝑟
cm2.
Sonya approaches a painting which has its bottom edge 2 m above eye level and its top edge 3 m
above eye level.
a) Given α and 𝜃 as shown in the diagram, find tan 𝛼 and tan(𝛼 + 𝜃)
b) Find 𝜃 in terms of x only. (Hint: 𝜃 = (𝛼 + 𝜃) − 𝛼
𝑑𝜃
2
3
𝑑𝜃
c) Show that 𝑑𝑥 = 𝑥 2 +4 − 𝑥 2 +9 and hence find x when 𝑑𝑥 = 0.
d) What does your answer mean?
The back end of a guided long-range torpedo is to be conical with slant edge s cm, and will be filled with
fuel. Find the ratio of s : r such that the fuel carrying capacity is maximized.
In a practical sense what does the derivative tell us?
𝑑
When you found 𝑑𝑥 𝐶 from the first problem mean in real world terms?
𝑑
What did 𝑑𝑟 𝐴 represent from the question on the bottom of page 1?
Thus far we have only worked with finding derivatives with respect to variables in our equation (for
example we found the derivative of the eye angle with respect to the distance from the wall on page 2).
We now are going to be look at related rates where we will be taking the derivative with respect to time
or some other outside variable.
The classic introduction to related rates involves a ladder leaning against a wall as seen in the diagram at
right. The ladder in this situation is 5 m long and leaning against a
vertical wall y m tall and stands on horizontal ground x m long. The
ladder is gradually slipping and beginning to slide down the wall.
a) Create an equation for the relationship between x, y, and the
length of the ladder.
b) To this point we would have differentiated with respect to x or y, so let’s do so now. Find the
derivative with respect to x, find the derivative with respect to y (should have 2 separate eqn.)
c) Provided a real world interpretation for your two equations.
d) The reasons this is a related rates problem is because we aren’t concerned with how x changes as y
changes, but rather we care how both are changing over time. Take the derivative of your equation
from a) with respect to time (t). (Hint: the process is akin to how we did our implicit differentiation)
𝑑𝑥
𝑑𝑦
e) Explain what 𝑑𝑡 and 𝑑𝑡 mean in terms of the situation.
f) The feet of the ladder on the ground are moving at 10 m/s when they are 3 m from the wall. Use your
equation from d) to find the speed of the of the other end of the ladder at that instant.
A cube is expanding so its volume increases at a constant rate of 10 cm3/s. Find the rate of change in its
total surface area, at the instant when its sides are 20 cm long.
a) How is this problem different from the previous volume and surface area questions we’ve done?
b) Draw a diagram of the situation.
c) What are the variables in the situation and what are the constants? List them
d) Create an equation for the volume and a separate equation for the surface area.
e) Take the derivative of both equations with respect to t to obtain a differential equation.
f) Plug in what you know and solve for the rate of change in surface area at the desired instant.
Mr. Gilmartin’s suggested process for solving related rates problems:
1) Draw a diagram of the situation
2) Identify the variables and constants involved in the problem and label your diagram –
Identify what’s been given and what’s being asked for.
3) Create any and all necessary equations – you may need more than 1 equation
4) Use implicit differentiation to find derivative of your equations – with respect to t
5) Plug in any remaining info and solve
Air is being pumped into a spherical balloon at 10 cm3/min. Calculate the rate at which the radius of the
balloon is increasing when the diameter is 15 cm.
Triangle ABC is right angled at A, and AB = 20 cm. <ABC increases at a constant rate of 1˚ per minute. At
what rate is BC changing at the instant when <ABC measures 30˚?
Two wizards Arthur and Beatrix leave Harry Potter’s house at 130˚ to one another, with constant speeds
of 12 m/s and 16 m s-1 respectively. Find the rate at which the distance between them is changing after 2
minutes.
Knowledge Check-in
The function f is defined by f (x) = x e2x. It can be shown that f (n) (x) = (2n x + n 2n−1) e2x for all n +, where f (n) (x)
represents the nth derivative of f (x).
(a) By considering f (n) (x) for n =1 and n = 2, show that there is one minimum point P on the graph of f, and find
the coordinates of P.
(7)
(b)
Show that f has a point of inflexion Q at x = −1.
(5)
(c)
Determine the intervals on the domain of f where f is
(i) concave up;
(ii) concave down.
(2)
(d)
Sketch f, clearly showing any intercepts, asymptotes and the points P and Q.
(4)
(e)
Use mathematical induction to prove that f (n) (x) = (2nx + n2n−1) e2x for all n
represents the nth derivative of f (x).
+
, where f (n) (x)
(9)
(Total 27 marks)
Challenge Problem
A packaging company makes boxes for chocolates. An example of a box is
shown below. This box is closed and the top and bottom of the box are
identical regular hexagons of side x cm.
diagram not to scale
(a)
3 3x 2
Show that the area of each hexagon is
cm2.
2
(1)
(b)
Given that the volume of the box is 90 cm3, show that when x =
is a minimum, justifying that this value gives a minimum.
3
20 the total surface area of the box
(7)
(Total 8 marks)