LemonmathSelf-study sheet

Grade 12 · Lesson 1.5

Derivatives in real-world problems

Name: Date: Time spent: min

Goals

Turn a real-life problem into finding the maximum or minimum of a function on a domain.
Solve optimisation problems about area, volume, material, cost, profit and paths.
Use derivatives to compute velocity, acceleration and the rate of change of a quantity.
Give a complete answer with units, and check that the result is sensible.

Remember

Four steps: choose the variable and its constraint → set up the function → find the maximum or minimum on the domain → conclude with units.
The domain is an open interval: use a table of variations. A closed interval: compare the values at the critical points and at both endpoints.
Motion: $v(t)=s'(t)$ , $a(t)=v'(t)=s''(t)$ ; the greatest velocity is usually reached when $a(t)=0$ .
$f'(t_0)$ is the instantaneous rate of change of $f$ at $t_0$ ; a minus sign means the quantity is decreasing.
Always check the result: does it satisfy the constraint, and does it make sense in real life?

Exercises

Exercise 1

An object moves along a straight line and has travelled $s(t)=t^3-6t^2+9t$ (metres) after $t$ seconds. Find the times when the velocity is $0$ , and the acceleration at those times.

Exercise 2

Mr Hùng fences a rectangular garden of area $200$ m², with one side against an existing wall (which needs no fence). Find the dimensions that make the total length of fence as small as possible.

Exercise 3

From a square sheet of card with side $60$ cm, four equal squares of side $x$ cm are cut from the corners and the sides are folded up to make an open box. Find $x$ to make the volume of the box as large as possible.

Exercise 4

The population of a town $t$ years after 2025 is estimated as $N(t)=\dfrac{26t+10}{t+5}$ (thousand people). Find the rate of population growth at $t=5$ . In the long run, what does the population approach?

Exercise 5

A hotel has $50$ rooms. At $400$ thousand đồng a night it is full; each increase of $20$ thousand đồng in the price leaves one more room empty. Let $x$ be the number of price increases ( $0\le x\le 50$ ) and $R(x)$ the revenue per night (thousand đồng). Is each claim true or false?

a) $R(x)=(400+20x)(50-x)$ .

b) $R$ is increasing on the interval $(0, 15)$ .

c) The revenue is largest when the room price is $700$ thousand đồng.

d) The largest revenue per night is $24\,000\,000$ đồng.

Exercise 6

A cylindrical can with a lid has volume $1000$ cm³. Find the base radius $r$ (cm) that uses the least material (round to two decimal places).

Exercise 7

During a flu outbreak, the number of people who have caught the illness by day $t$ is estimated as $f(t)=30t^2-t^3$ ( $0\le t\le 20$ ). The rate of spread is $f'(t)$ (people per day). On which day is the rate of spread greatest, and what is it?

Exercise 8

A lorry travels $300$ km at a constant speed $v$ (km/h), $40\le v\le 90$ . Each hour of driving costs $500$ thousand đồng (wages, wear) plus $\dfrac{v^2}{5}$ thousand đồng of fuel. Choose $v$ to make the total cost of the trip as small as possible.

Exercise 9 Review

A metal sheet $30$ cm wide is folded into a gutter: the bottom is $10$ cm wide and each of the two sides is $10$ cm wide, each side tilted outwards at an angle $\theta$ to the vertical ( $0\le\theta<\dfrac{\pi}{2}$ ). Find $\theta$ that makes the cross-section area of the gutter as large as possible.

Exercise 10 Challenge

A rectangular window is set inside a semicircular frame of radius $10$ dm, with one side of the window on the diameter. Find the dimensions of the window with the largest area.

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. An object moves along a straight line and has travelled

s(t)=2t^3-3t^2+t

(metres) after

t

seconds. Its velocity at

t=2

seconds is

6

m/s.B.

13

m/s.C.

18

m/s.D.

37

m/s.

Question 2. A particle moves along a straight line with distance

s(t)=-t^3+6t^2

(metres),

0\le t\le 5

(seconds). Its velocity is greatest at

t=5

.B.

t=4

.C.

t=0

.D.

t=2

Question 3. The cost of making

x

products is

C(x)=0.01x^2+20x+500

(thousand đồng). The instantaneous rate of change of the cost with respect to the number of products at

x=100

21

thousand đồng per product.B.

2600

thousand đồng per product.C.

22

thousand đồng per product.D.

26

thousand đồng per product.

Question 4.

40

m of fencing goes all the way round a rectangular garden of width

x

(m). The area of the garden in terms of

x

S(x)=x(20-x)

.B.

S(x)=x(40-x)

.C.

S(x)=x(40-2x)

.D.

S(x)=2x+2(20-x)

Part II. True or false

Question 1. A ball is thrown straight up; its height after

t

seconds is

h(t)=2+20t-5t^2

(metres).

a) The velocity of the ball is $v(t)=20-10t$ (m/s).	T / F
b) The ball reaches its greatest height at $t=2$ seconds.	T / F
c) The greatest height of the ball is $20$ m.	T / F
d) The speed of the ball when it hits the ground is more than $20$ m/s.	T / F

Part III. Short answer

Question 1. A piece of wire

2

m long is cut into two parts: the first is bent into a square and the second into an equilateral triangle. To make the total area of the two shapes as small as possible, how long (in metres) should the part for the square be? Give your answer to 2 decimal places.

Question 2. A beam with a rectangular cross-section of width

b

and height

h

(cm) is cut from a cylindrical log of diameter

40

cm, with the rectangle inscribed in the circle. The strength of the beam is proportional to

bh^2

. For the strongest beam, what is the width

b

in centimetres? Give your answer to 1 decimal place.

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $v(t)=s'(t)$ , $a(t)=s''(t)$ .

Steps:

Compute $v(t)$ and solve $v(t)=0$ .
Compute $a(t)=6t-12$ at each root.

Exercise 2.

What you need: the steps for an optimisation problem; find the minimum on $(0, +\infty)$ with a variation table.

Steps:

Let $x$ be the side perpendicular to the wall; the side parallel to the wall is $\dfrac{200}{x}$ .
The fence has length $L(x)=2x+\dfrac{200}{x}$ ; find its minimum.

Exercise 3.

What you need: the box has a square base of side $60-2x$ and height $x$ , with $0<x<30$ .

Steps:

Set up $V(x)=x(60-2x)^2$ .
Compute $V'(x)$ , factorise it, and keep only the roots in $(0, 30)$ .

Exercise 4.

What you need: the instantaneous rate of change is $N'(t)$ ; the long-run value is $\lim\limits_{t\to+\infty}N(t)$ (the horizontal asymptote).

Steps:

Compute $N'(t)$ with the quotient rule.
Substitute $t=5$ and compute the limit.

Exercise 5.

What you need: revenue = price of one room $\times$ number of rooms let.

Steps:

Write the price and the number of rooms in terms of $x$ .
Compute $R'(x)$ and find where $R'(x)=0$ .
Compute the room price and the revenue at that point.

Exercise 6.

What you need: the volume $V=\pi r^2h$ ; the total surface area $S=2\pi r^2+2\pi rh$ .

Steps:

From $\pi r^2h=1000$ , write $h$ in terms of $r$ and set up $S(r)=2\pi r^2+\dfrac{2000}{r}$ .
Solve $S'(r)=0$ and check the sign.

Exercise 7.

What you need: find the maximum of $g(t)=f'(t)$ on $[0, 20]$ .

Steps:

Compute $g(t)=f'(t)=60t-3t^2$ .
Solve $g'(t)=0$ and compare with both endpoints.

Exercise 8.

What you need: the driving time is $\dfrac{300}{v}$ hours; the total cost is the cost per hour times the time.

Steps:

Set up $C(v)=\dfrac{300}{v}\left(500+\dfrac{v^2}{5}\right)=\dfrac{150\,000}{v}+60v$ .
Find its minimum on $[40, 90]$ .

Exercise 9.

What you need: the cross-section is a trapezium with height $10\cos\theta$ , shorter base $10$ and longer base $10+20\sin\theta$ ; the derivatives of $\sin$ and $\cos$ and the formula $\cos2\theta=1-2\sin^2\theta$ (grade 11).

Steps:

Set up $A(\theta)=100\cos\theta(1+\sin\theta)$ .
Compute $A'(\theta)$ and turn it into an equation in $\sin\theta$ .
Compare with the value at $\theta=0$ .

Exercise 10.

What you need: if half the width of the window is $x$ , its top corners lie on the semicircle, so its height is $\sqrt{100-x^2}$ (Pythagoras' theorem).

Steps:

Set up $S(x)=2x\sqrt{100-x^2}$ for $0<x<10$ .
Compute $S'(x)$ and solve $S'(x)=0$ .

Self-check

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

Redo the ones you missed after: ☐ 1 day ☐ 3 days ☐ 7 days

Scan for step-by-step solutions

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐

#	Did it alone	Needed a hint	Not yet
1	☐	☐	☐
2	☐	☐	☐
3	☐	☐	☐
4	☐	☐	☐
5	☐	☐	☐
6	☐	☐	☐
7	☐	☐	☐
8	☐	☐	☐
9	☐	☐	☐
10	☐	☐	☐