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SQUARES IN RECTANGLES (30 marks)
You are advised to spend no more than 50 minutes on this part.
This investigation looks at finding the total number of squares inside a rectangle drawn on a grid.
In this investigation:
• the sides of the rectangles are on the grid lines
• the length of a rectangle is never less than its width.
Rectangles of width 1
Total = 1 square Total = 2 squares Total = 3 squares
Complete the statements.
The number of squares in a rectangle of width 1 and length 4 is ..............................................
The number of squares in a rectangle of width 1 and length $L$ is .............................................. [1]
529
569
557
(a) Draw lines on these rectangles and write the number of squares under each one to show there is a total of 11 squares.
Length 4
................... squares of side 1 + ................... squares of side 2 + ................... square of side 2
Total = 11 squares [3]
(b) (i) Complete the table. You may use the grid below the table to help you.
[Table_1]
Rectangles of width 2
| Length of rectangle | Total number of squares |
| --- | --- |
| 2 | 5 |
| 3 | 8 |
| 4 | 11 |
| 5 | |
[Table_1]
[2]
(ii) Find an expression, in terms of $L$, for the total number of squares in a rectangle of width 2 and length $L$.
................................................................. [2]
(iii) Calculate the total number of squares in a rectangle of width 2 and length 170.
................................................................. [2]
522
558
519
(a) Draw lines on these rectangles and write the number of squares under each one to find the total number of squares in a rectangle of width 3 and length 4. You may not need to use all the rectangles.
(b) (i) Complete the table. You may use the grid to help you.
[Table_1]
| Rectangles of width 3 | |
|-------------------------|--------------------------|
| Length of rectangle | Total number of squares |
| 3 | 14 |
| 4 | |
| 5 | 26 |
| 6 | 32 |
| 7 | |
(ii) Find an expression, in terms of $L$, for the number of squares in a rectangle of width 3 and length $L$.
Total marks: (a) 4 + (b)(i) 2 + (b)(ii) 2 = 8
542
588
590
(a) Complete the table.
Use your answers to Question 1, Question 2(b)(ii) and Question 3(b)(ii) to help you.
[Table_1]
Width of rectangle, $w$ | Expression for total number
| of squares in terms of $L$
1 |
2 |
3 |
4 | $10L - 10$
5 |
6 | $21L - 35$
[3]
(b) The expressions in the table have two terms.
(i) The first term is $kL$, where $k$ is an integer.
Find an expression for $k$ in terms of $w$.
........................................................... [3]
(ii) The second term in the expression is a constant.
The constant is in the form $aw^3 + bw$, where $a$ and $b$ are both fractions.
Find the value of $a$ and the value of $b$.
Write down the expression for the constant in terms of $w$.
$a =$ ........................................
$b =$ ........................................
........................................................... [4]
(iii) Use your expressions in part (i) and part (ii) to find the total number of squares in a rectangle of width 10 and length 11.
........................................................... [2]
538
594
556
This task looks at the dimensions of storage boxes.
A company stores metal bolts in closed boxes which are square-based cuboids.
The boxes are made of 5 mm thick cardboard.
The external side of the base is $L$ cm and the external height is $H$ cm.
$$ \text{The volume of a box is all the space that the box takes up.} $$
$$ \text{The capacity of a box is all the space inside it.} $$
(a) Show that the internal dimensions of the box are $(L - 1)$ cm and $(H - 1)$ cm. [2]
(b) Write down a formula for $C$, the capacity of the box, in terms of $L$ and $H$. [2]
(c) The bolts are so heavy that an extra piece of cardboard of thickness 5 mm is placed in the bottom of the box to increase its strength.
Change your model for $C$ in part (b) to include this extra piece of cardboard. [1]
(d) The external area of the base of the box is $900 \text{ cm}^2$.
The height of the box is 5 cm longer than the length of the base.
(i) Use your model in part (c) to calculate the capacity of the box.
Write down all the figures on your calculator. [4]
(ii) Calculate the difference between the capacity and the volume of the box. [3]
580
566
564
The net of a 3D shape is what it looks like when it is opened out flat. This is a net for a box with the extra piece of cardboard.
(a) Show that the formula for $A$, the area of cardboard including the extra piece, is $$A = 3L^2 + 4LH - 2L + 1.$$ You may use the diagram to help you.
(b) The box with external dimensions $L\text{ cm}, L\text{ cm and } H\text{ cm}$ has a volume of $12\ 500\text{ cm}^3$.
(i) Write an expression for $H$ in terms of $L$.
............................................................ [1]
(ii) Show that the model for $A$ in part (a) in terms of $L$ is $$A = 3L^2 + \frac{50000}{L} - 2L + 1.$$ [1]
(iii) Sketch the model in part (ii) on the axes, for $0 < L \leq 60$.
[2]
(iv) Box A has the minimum area of cardboard.
Write down the length of the base of the box and the area of the cardboard.
Length = ..........................................................
Area = ............................................................. [2]
(v) Box B is a cube with volume $12\ 500\text{ cm}^3$.
Find the area of the cardboard when the box is a cube. [3]
(vi) Find which box, A or B, has the greater height. Calculate the difference in height. [3]
(vii) Find which box, A or B, has the greater capacity. Calculate the difference in capacity. [3]
587
502
516
