Question

A shop owner has 500 cans of tomatoes that are near their expiration date.She decides to display them in a triangular pyramid and lower the price to try and sell them quickly. How many cans will be on the bottom layer, and will there be any cans left over?

Answer 1

Answer:

How many cans will be on the bottom layer, and will there be any cans left over?

On the bottom layer will be 31 cans  and there will be 4 cans left over

Step-by-step explanation:

To resolve this problem we need to know Gauss sum of natural numbers, from k = 1 to n:

∑$k=\frac{n*(n+1)}{2}$ (1)

We want to know the value of n when the sum is equal or close to 500, so we replace 500 in the equation (1) to find n:

$500=\frac{n*(n+1)}{2}$

$500*2=n*(n+1)$

$1000=n^2+n$

$n^2+n-1000=0$ (2)

We need to use the quadratic formula to resolve the equation (2)

$n=\frac{-b+/-\sqrt{b^{2} -4ac} }{2a}$ (3)

The coefficients according to equation (2) are:

a=1

b=1

c= 1000

Now, using the quadratic formula we can find n :

Positive sign before the square root,

$n_1=\frac{-1+\sqrt{1^{2} -4(1)(-1000)} }{2(1)}$

$n_1=\frac{-1+\sqrt{1 + 4000} }{2}$

$n_1=\frac{-1+63.25}{2}$

$n_1=\frac{62.25}{2}$

$n_1=31.13$

Negative sign before the square root,

$n_2=\frac{-1-\sqrt{1^{2} -4(1)(-1000)} }{2(1)}$

$n_2=\frac{-1-\sqrt{1 + 4000} }{2}$

$n_2=\frac{-1-63.25}{2}$

$n_2=\frac{-64.25}{2}$

$n_2=-32.13$

The negative term doesn't have sense in this case, We will use the positive term, the whole number closest to the answer we got when clearing n is 31, we will replace n=31 in the Gauss sum of natural numbers equation. (1)

∑$=\frac{n*(n+1)}{2}$

∑$=\frac{31*(31+1)}{2}$

∑$=\frac{31*(32)}{2}$

∑$=\frac{992}{2}$

∑$=496$

Now, we know the triangular pyramid will be 31 cans on the bottom layer and there will be 4 cans left over (500-496=4)