Euler found four whole numbers such that the sum of any two of the
numbers is a perfect square. Three of the numbers that he found are
a = 18530, b=65570, c=45986. Find the fourth number, x. You could
do this by trial and error, and a spreadsheet would be a good tool
for such work. Write down a+x = P^2, b+x = Q^2, c+x = R^2, and then
focus on Q^2-R^2=b-c which is known. Moreover you know that Q >
sqrtb and R > sqrtc . Use this to show that Q-R is less than or
equal to 41 . Use a spreadsheet to calculate values of Q+R , Q and
x for values of Q-R from 1 to 41 , and hence to find the value of x
for which a+x is a perfect square.
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
Find the frequency distribution for ordinary English, and use it to help you crack the code.
But differentiation is not really Stage 4 mathematics, though
very close, so we don't use it in solutions because that would be
unfair to students who haven't seen the idea before, it would be
like suddenly changing into a language you haven't had a chance to
the height, which will be determined by our choice of radius, so
that we get the chosen target volume
the slant length, which we'll find using r and h and
and the surface area, for which there's a great little
Incidentally, if you don't know where that surface area formula
comes from it may be good to take a moment to look at that. Flatten
the curved surface out to get a sector (how do you know it's a
sector ?). The radius will be the cone's slant length, so you can
calculate the area of the whole circle. To know the proportion that
the sector is of that circle compare the sector arc, which is the
cone's base circumference, with the circumference of this new
circle, radius s.
Can you see what each column does ? Click on a cell and check