LemonmathSelf-study sheet

Grade 12 · Lesson 7.3

Revision: optimisation and real-life problems

Name: Date: Time spent: min

Goals

Turn a real-life problem into finding the maximum or minimum of a function on the correct domain.
Use the derivative as a rate of change: velocity, acceleration, growth rate.
Solve cost, revenue and profit problems, and route or installation problems with the least cost.
Give answers with the right units and the requested rounding.

Exercises

Exercise 1 Multiple choice

A rectangular garden has perimeter $40$ m. Its greatest possible area is A. $100$ m². B. $400$ m². C. $96$ m². D. $200$ m².

Exercise 2 Short answer

An object moves with $s(t)=t^3-3t^2+5t$ (m), $t$ in seconds. Find its velocity (m/s) at the moment when its acceleration is $0$ .

Exercise 3 Multiple choice

The total cost of making $x$ items is $C(x)=x^2+20x+900$ (thousand đồng), $x>0$ . The least average cost per item is A. $80$ thousand đồng. B. $30$ thousand đồng. C. $60$ thousand đồng. D. $50$ thousand đồng.

Exercise 4 Written answer

An open tank with a square base has volume $32$ dm³. Find its dimensions so that it uses the least material (the area of the base plus the four sides is least).

Exercise 5 True/False

A shop sells shirts at $200$ thousand đồng each and sells $50$ a month; for every $1$ thousand đồng reduction in price it sells $1$ more shirt. Each shirt costs the shop $120$ thousand đồng. Let $x$ (thousand đồng) be the price reduction, $0\le x\le80$ .

a) At a price of $190$ thousand đồng, the shop sells $60$ shirts a month.

b) The monthly profit is $P(x)=(80-x)(50+x)$ (thousand đồng).

c) The profit is greatest when the price is $170$ thousand đồng.

d) The greatest profit is $4225$ thousand đồng.

Exercise 6 Short answer

A ball is thrown straight up; its height after $t$ seconds is $h(t)=1.5+19.6t-4.9t^2$ (m). Find the greatest height of the ball, in metres.

Exercise 7 Multiple choice

A cylindrical can has volume $16\pi$ cm³. To use the least material (least total surface area), the radius of its base must be A. $2$ cm. B. $4$ cm. C. $8$ cm. D. $1$ cm.

Exercise 8 Short answer

A person at point $A$ on a lake is $4$ km from a straight shore ( $B$ is the point of the shore nearest to $A$ ). They want to reach point $C$ on the shore with $BC=10$ km: they row at $3$ km/h to a point $M$ on $BC$ , then walk at $5$ km/h to $C$ . How far from $B$ should $M$ be to make the time least, in km?

Exercise 9 True/False

Four squares of side $x$ cm are cut from the corners of a $16$ cm × $10$ cm rectangular sheet of card, which is then folded into an open box.

a) The constraint on $x$ is $0<x<5$ .

b) The volume of the box is $V(x)=x(16-x)(10-x)$ .

c) The volume is greatest when $x=\dfrac{20}{3}$ .

d) The greatest volume of the box is $144$ cm³.

Exercise 10 Challenge

A metal sheet $30$ cm wide is folded into a gutter: a base $10$ cm wide and two sides $10$ cm wide, each making an angle $\theta$ with the horizontal, $0<\theta\le\dfrac{\pi}{2}$ (the cross-section is an isosceles trapezium). Find $\theta$ so that the cross-sectional area is greatest, and find that area to 1 decimal place.

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. Two positive numbers add up to

12

. The greatest possible product is

36

.B.

6

.C.

144

.D.

72

Question 2. An object moves with

s(t)=t^3-3t^2+4t

(m). Its acceleration at

t=2

seconds is

6

m/s².B.

8

m/s².C.

4

m/s².D.

0

m/s².

Question 3. The total cost of making

x

items is

C(x)=2x^2+50x+800

(thousand đồng). How many items should be made to minimise the average cost per item?

20

.B.

400

.C.

10

.D.

40

Question 4.

N(t)

is the number of users of an app (thousands) after

t

months, and

N'(5)=12

. Which statement is correct?

A. At

5

months, the number of users is increasing at about

12

thousand per month.B. After

5

months there are

12

thousand users.C. Over the first

5

months, users increased by

12

thousand per month on average.D. After

5

months, the number of users has increased by

12

thousand in total.

Part II. True or false

Question 1. Mr Ba uses

120

m of fencing for three sides of a rectangular plot; the fourth side is a river bank and needs no fence. Let

x

(m) be the length of each side perpendicular to the river.

a) The constraint on $x$ is $0<x<60$ .	T / F
b) The area of the plot is $S(x)=x(120-2x)$ .	T / F
c) The area is greatest when $x=30$ .	T / F
d) The greatest area is $3600$ m².	T / F

Part III. Short answer

Question 1. A ball is thrown upwards with height

h(t)=2+14.7t-4.9t^2

(m) after

t

seconds. After how many seconds does the ball reach its greatest height?

Question 2. Four squares of side

x

cm are cut from the corners of a square sheet of side

12

cm, which is then folded into an open box. What is the greatest volume of the box, in cm³?

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

Hints (only when stuck)

Exercise 1.

What you need: if one side is $x$ , the other is $20-x$ .

Steps:

Write $S(x)=x(20-x)$ , $0<x<20$ .
Solve $S'(x)=0$ .

Exercise 2.

What you need: $v=s'$ , $a=v'$ .

Steps:

Find $v(t)$ and $a(t)$ .
Solve $a(t)=0$ and substitute into $v$ .

Exercise 3.

What you need: the average cost is $\overline C(x)=x+20+\dfrac{900}{x}$ .

Steps:

Solve $\overline C'(x)=1-\dfrac{900}{x^2}=0$ .
Evaluate $\overline C$ at the positive root.

Exercise 4.

What you need: base side $x$ , height $h=\dfrac{32}{x^2}$ .

Steps:

Write $S(x)=x^2+4xh=x^2+\dfrac{128}{x}$ , $x>0$ .
Draw the table of variations on $(0, +\infty)$ .

Exercise 5.

What you need: profit = (selling price − cost price) × quantity.

Steps:

Price $200-x$ , profit per shirt $80-x$ , quantity $50+x$ .
Find the maximum of $P$ on $[0, 80]$ .

Exercise 6.

What you need: the height is greatest when the velocity $h'(t)=0$ .

Steps:

Solve $h'(t)=19.6-9.8t=0$ .
Substitute into $h(t)$ .

Exercise 7.

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

Steps:

Substitute $h=\dfrac{16}{r^2}$ to get $S(r)=2\pi r^2+\dfrac{32\pi}{r}$ .
Solve $S'(r)=0$ .

Exercise 8.

What you need: time = distance ÷ speed; $AM=\sqrt{x^2+16}$ with $x=BM$ .

Steps:

Write $T(x)=\dfrac{\sqrt{x^2+16}}{3}+\dfrac{10-x}{5}$ , $0\le x\le10$ .
Solve $T'(x)=0$ and compare with the two endpoints.

Exercise 9.

What you need: the base of the box measures $(16-2x)$ by $(10-2x)$ .

Steps:

Write $V(x)$ and find $V'(x)$ .
Keep only the roots inside the domain of $x$ .

Exercise 10.

What you need: area of a trapezium $=\dfrac{(\text{sum of the parallel sides})\cdot\text{height}}{2}$ ; $\cos2\theta=2\cos^2\theta-1$ .

Steps:

The height is $10\sin\theta$ and the top is $10+20\cos\theta$ , so $A(\theta)=100\sin\theta(1+\cos\theta)$ .
Find $A'(\theta)$ and factorise it in terms of $\cos\theta$ .

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