+118609 
+118609 Hi Rauhan
9. Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns:
Row 1
Row 2
Row 3
She notices that 4 sticks are required to make the single square in the first row,
7 sticks to make 2 squares in the second row
and in the third row she needs 10 sticks to make 3 squares.
(a) Find an expression, in terms of n, for the number of sticks required to make a similar arrangement of n squares in the nth row.
| row (r) | 1 | 2 | 3 | ... | n |
| match sticks (m) | 4 | 7 | 10 | 3n+1 |
Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.
(b) Find the total number of sticks Ann uses in making these 10 rows.
m(10)=3*10+1=31 match sticks
Ann started with 1750 sticks.
Given that Ann continues the pattern to complete k rows but does not have sufficient sticks to complete the (k + 1)th row,
(c) show that k satisfies (3k – 100)(k + 35) < 0.
The total number of sticks in k rows is
\(4+7+10+.......(3k+1) \qquad \text{Sum of an AP}\\ =\frac{n}{2}(a+L)\\ =\frac{k}{2}(4+3k+1)\\ =\frac{k}{2}(3k+5)\\ now\\ \frac{k}{2}(3k+5)<1750\\ k(3k+5)<3500\\ 3k^2+5k<3500\\ 3k^2+5k-3500<0\\ \qquad 3*-3500=-10500\\ \qquad \text{I need 2 numbers that multiply to -10500 and add to +5}\\ \qquad \text{Those numbers are 105 and -100}\\ 3k^2+105k-100k-3500<0\\ 3k(k+35)-100(k+35)<0\\ (3k-100)(k+35)<0 \)
(d) Find the value of k.
if you graph y=(3k-100)(k+35)
it will be a concave up parabola and y will be less then 0 between the two zeros.
3k-100=0
3k=100
k=33 and a 1/3
and
k=35=0
k=-35
k cant be negative so Ann will have enough sticks for 1 to 33 rows.
She will not have enough sticks for 34 rows.
So
k=33
Any questons, just ask :)
