Math in Daily Life: Canned Goods

Geometry Level 2

You're ordering canned foods to send to charity. You buy:

100 cans of one-inch radius and one-inch height,
81 cans of two-inch radius and two-inch height,
64 cans of three-inch radius and three-inch height,
49 cans of four-inch radius and four-inch height,
36 cans of five-inch radius and five-inch height,
25 cans of six-inch radius and six-inch height,
16 cans of seven-inch radius and seven-inch height,
9 cans of eight-inch radius and eight-inch height,
4 cans of nine-inch radius and nine-inch height, and finally
1 can of ten-inch radius and ten-inch height.

But, I also need to keep some of the cans for myself. I can't just give all my food to charity! I need to keep 1748π for myself.

If the total volume of all cans added together equals xπ, what is the value of x-1748?

The answer is 27776.

1 solution

William Pan
Dec 24, 2015

Calculate the volume of all cans:

$100({ 1 }^{ 2 }\times 1)\\ 81({ 2 }^{ 2 }\times 2)\\ 64({ 3 }^{ 2 }\times 3)\\ 49({ 4 }^{ 2 }\times 4)\\ 36({ 5 }^{ 2 }\times 5)\\ 25({ 6 }^{ 2 }\times 6)\\ 16({ 7 }^{ 2 }\times 7)\\ 9({ 8 }^{ 2 }\times 8)\\ 4({ 9 }^{ 2 }\times 9)\\ 1({ 10 }^{ 2 }\times 10)$

Which is basically:

$100\quad \quad \Rightarrow \quad 100\\ 81({ 2 }^{ 3 })\quad \Rightarrow \quad 648\\ 64({ 3 }^{ 3 })\quad \Rightarrow \quad 1728\\ 49({ 4 }^{ 3 })\quad \Rightarrow \quad 3136\\ 36({ 5 }^{ 3 })\quad \Rightarrow \quad 4500\\ 25({ 6 }^{ 3 })\quad \Rightarrow \quad 5400\\ 16({ 7 }^{ 3 })\quad \Rightarrow \quad 5488\\ 9({ 8 }^{ 3 })\quad \Rightarrow \quad 4608\\ 4({ 9 }^{ 3 })\quad \Rightarrow \quad 2916\\ { 10 }^{ 3 }\quad \quad \Rightarrow \quad 1000$

So,

$100+648+1728+3136+4500+5400+5488+4608+2916+1000\quad =\quad 29524\quad =\quad x$

Don't forget that we also need to subtract 1748 for ourselves:

$29524\quad -\quad 1748\quad =\quad 27776$

$\therefore \quad x\quad -\quad 1748\quad = \quad 27776$

Math in Daily Life: Canned Goods

The answer is 27776.

1 solution

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