Ah, you have finally reached the Huan question.

bii) is easy.

If x ≥ 10,

2x ≥ 20
0 ≥ 20 - 2x

Your box will have 0 or negative width. (Not possible)

If x = 0, the height of the box is 0 (you won't even have a box)

For c), you are not provided with any data points or coordinates.

So you use the equation y = x(20 - 2x)(30 - 2x) and plot for 0 < x < 10.

i.e choose some values of x and find the corresponding values of y. Then plot them.

Next, draw the line y = 900

This line will intersect the curve at 2 points for 0 < x < 10

They are at x = 2.302 and x = 5.777 (4s.f)

This part of the curve is upward sloping so between these two x-values, lies the maximum point. The volume will be greater than 900cm³ for 2.302 < x < 5.777


But of course, you're using graph paper so it won't be possible to get an accuracy of 4s.f.

You'll probably get an accuracy of 0.1cm.or 0.05cm depending on the scale you use.

Since we want the mass of the box to be as small as possible, and we are given that 1 square metre of the material weighs 210 grams,

the smaller the area of the cardboard used for the box, the lighter the box will be.


Area of the cardboard used
= Area of rectangular piece - area of 4 squares that were cut out

20 cm × 30 cm - 4(x cm × x cm)

= (600 - 4x²) cm²


So, we should use a value of x that is as large as possible, such that we can get the smallest area.

This value of x also needs to satisfy the condition that the volume of the box ≥ 900 cm³.

So we can use x = 5.777 cm

thx :) but how do we know that volume should be used as one of the axis if they did not specify any of the axis tho

Well you're told that the volume needs to be at least 900cm³.

And you were already asked to explain that the box's volume is x(20 - 2x)(30 - 2x)

So, how do you get the value of x you need,to satisfy the condition for the volume and for the mass of the box to be as small as possible ?(which in turn implies the smallest possible area of the box, since the mass per unit area was already given)

You got to plot volume against x of course!



Inference...

And if you're not given any data points to plot, then use the equations to find some!

A lot of students left this question blank in 2018 just because they were expecting values to be given to them to plot, and got stunned when nothing was given.

Another thing to note :

The thickness of the material is irrelevant in this question. We're only given that the mass per square metre of the box is 210 grams.

No thickness is given and we don't need it to solve the question.

thx :) i understand how to solve the 2 parts now

hi, may i know in what circumstances are we supposed to flip the inequality sign? it's confusing for negative signs as sometimes i see the inequality sign being flipped sometimes its not flipped

for example - 10<= 7-2x<1
-10 < 7-2x
2x < 17 - why is the inequality sign not flipped since the negative sign is being removed here


7-2x < 1
-2x < - 6
x>3
while over here, the inequality sign is being flipped

Your first example :

-10 < 7 - 2x

You do not need flip the negative sign if you do the following :

① adding 10 to both sides.

-10 + 10 < 7 - 2x + 10
0 < 17 - 2x

② adding 2x to both sides.

0 + 2x < 17 - 2x + 2x
2x < 17

x < 17/2
x < 8.5

For this way, we did not actually multiply both sides by a negative or divide both sides by a negative. So there is no need to flip.

However, if you chose this way instead :

① Subtract 7 from both sides,

-10 - 7 < 7 - 2x - 7
-17 < -2x

Here you flip it. (Taking away the negative sign is just a shorcut for multiplying or dividing both sides by -1)

-17(-1) > -2x(-1)
17 > 2x


Example to demonstrate why you have to flip it.

Let's say if we take x = 7
2x = 14

17 > 14, which satisfies the inequality 17 > 2x

But we can't say -17 > -14 , can we?

That's why we flip it.

-17 < -14, which satisfies -17 < -2x

The other one : 7 - 2x < 1

If you do these :

① add 2x to both sides,

7 - 2x + 2x < 1 + 2x
7 < 1 + 2x

② subtract 1 from both sides,

7 - 1 < 1 + 2x - 1
6 < 2x
6/2 < 2x/2
3 < x
Which means x > 3



But if you do this :

① subtract 7 from both sides

7 - 2x - 7 < 1 - 7
-2x < -6

② divide or multiply both sides by -1

-2x/-1 > -6/-1
2x > 6

Then,
x > 6/2
x > 3

OR

-2x(-1) > -6(-1)
2x > 6

Then,
x > 6/2
x > 3


Flipping is needed.

For the first example, I could also do it this way,

-7 < 10 - 2x

Multiply both sides by -1 (or divide both sides by -1. I will show the former only)

-7(-1) > (10 - 2x)(-1)

7 > -10 + 2x
7 > 2x - 10


Then,

7 + 10 > 2x - 10 + 10
17 > 2x
8.5 > x

Rewrite as x < 8.5


Same result.