Math Is Fun Forum

Problem # n+4
(1) How many diagonals would a polygon of 'n' sides contain?
(2) What would be the sum of the internal angles of a polygon of 'n' sides?

Well done, tt! You got it right!

Here are the squares and square roots of all squares up to 31.
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100
11² = 121
12² = 144
13² = 169
14² = 256
15² = 225
16² = 256
17² = 289
18² = 324
19² = 361
20² = 400
21² = 441
22² = 484
23² = 529
24² = 576
25² = 625
26² = 676
27² = 729
28² = 784
29² = 841
30² = 900
31² = 961

Remember,
(1) The last digit of a perfect square is always 0, 1, 5, or 6.
(2) Finding square roots is easier by prime factorisation.
eg. 784 = 2 x 2 x 2 x 7 x 7 x 7
Therefore, the square root of 784 is 2 x 2 x 7, i.e. 28

Problem #n+3
A rectangular metal sheet has dimensions 30m, 14m.
Four equal squares are cut off from the four corners of the rectangle.
Then, the sheet is folded to form a cuboidal box with top open.
What is the maximum volume of the cuboid you can get?

You are right, Mathsy, but the number of bags required is 10!
They would contain $1, $2, $4, $8, $16, $32, $64, $128, $256 and $489. For any amount less than or equal to $511, the first 9 bags can be used. For amount greater than $511, one bag of $489 and remaining from one or more of the first nine bags can be used.

Good post, and Welcome, Robin!
Subsequent differences, yes, sometimes they do help in find the next number in sequences.
But, what you have pointed out is, I feel, not acceptable.
You have found the difference, and then added that to the previous term.
In mathematical terms, you have found the difference between 2 terms, a and b, then added the difference to a, which would have to be b!

Problem #n+2

Divide $1,000 (in whole $ increments) into a number of bags so that I can ask for any amount between $1 and $1,000 and you can give me the proper amount by giving me a certain number of these bags without opening them. What is the minimum number of bags you will require?

This is an Arithmetic Progression (AP).
An Arithmetic Progression is something like
1,2,3,4,5,6,7,8 or
2,4,6,8,10,12, or
9,12,15,18,21,24 or
16, 13, 10, 7...........
where the difference between two successive terms is the same.
This is called 'd' or common difference.
If 'a' is the first term of an AP and 'd' is the common difference,
the nth term would be a+(n-1)d
In your case, d is 3 minutes. n is 23, and a is also 3.
Therefore, on the 23rd day, you have trusted me for
3 + (23-1)3 = 3 + (22)3 = 3 + 66 = 69 minutes.

If you require the sum of all the terms, the formula is
n/2 [2a+(n-1)d]
In the example given by you, it would be
23/2 [6 + 22(3)] = 23/2[6+66] = 23/2[72]=23*36 = 828 minutes
Is that clear?

Superlative work, Mathsy
How did you do it?

Mathsy is right This is the complete solution.
Assuming my normal speed is 's' and the normal time taken is 't',
Distance/Time = Speed (or) Distance/Speed = Time
We get 840/s = t ---------------------(1)
The second equation would be 840/(s+5) = t-3
                                             840 = (t-3)(s+5)
                                             840 = st+5t-3s-15 -------------(2)
Subtracting (1) from (2)         840 = st     we get
                                                 0 = 5t-3s-15
Putting t = 840/s (from (1),      0 = 5(840/s) -3s - 15
                                                0 = 4200/s - 3s - 15
                                                0 = 4200 - 3s² -15s
                        3s² +15s - 4200 = 0
Dividing by 3        s² + 5s - 1400 = 0
                                                 s = [-5 ± √(25+5600)]/2
                                                 s = [-5 ± √(5625)]/2
                                                 s = (-5 ± 75 )/2
Taking the positive value,           s = 35
Therefore, my normal speed is 35 kilometres/hour.

Problem #n
The distance between two cities is 840 kilometres.
If I start driving from one city to the other at a speed
5 kilometres/hour more than my normal speed,
I save 3 hours. What is my normal speed?

You can expand the terms as Mathsisfun said.
Or, you may remember the following formulae:-

1. (a+b)² = a² + 2ab + b²
2. (a-b)² = a² - 2ab + b²
3. (a+b)(a-b) = a² - b²
4. (a+b)³ = a³ + 3a²b+3ab²+ b³
5. (a-b)³ = a³ -3a²b +3ab² - b³
6. a³ + b³ = (a+b)(a² -ab +b²)
7. a³ - b³ = (a-b)(a² +ab + b²)