(1) The radius of sphere A is twice that of sphere B. How many times larger is t

Question

(1) The radius of sphere A is twice that of sphere B. How many times larger is the volume of A compared

to that of B? Note, the volume of a sphere of radius r is: V =4/3?r^3

.

(2) When a pile of pennies that are worth less than $1 are arranged in groups of 4, there is one left over.

Similarly, when those same pennies are arranged in groups of 5 or 6, there is one left over. How

many pennies are there?

(3) There are 2 diagonals in a square. There are 5 diagonals in a pentagon. How many diagonals are

there in a 30-sided polygon?

Explanation / Answer

1) Radius of a sphere is given by 4/3*pi*r^3
Let radius of sphere B be r
then radius of sphere A will be 2r

Volume of sphere A = 4/3*pi*(2r)^3 = 8*4/3*pi*r^3
Volume of sphere B = 4/3*pi*r^3

Volume A / Volume B = 8 (Answer)

therefore sphere A is 8 times larger than sphere A

2) Now when number of pennies is divided by 4,5 and 6 each time 1 penny remain therefore number of pennies will be a multiple of (LCM of (4,5 and 6) + 1 )
(LCM of (4,5 and 6) = 60
therefore number of pennies will be a multiple of 61 Since its value is less than 1$ therefore
Number of pennies will be 61 (Answer)

3) There are 2 diagonals in a square. There are 5 diagonals in a pentagon
According to combinations of choosing 2 points for drawing a diagonal from n points an n sided polynomial will be C(n 2)
From this n sides have to be subtracted as they cannot be the diagonal

therefore number of diagonals = C(n 2) - n = (n)(n-1)/3

Now if n = 30 then
number of diagoals = 30*27/3 = 270 diagonals (Answer)

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