Question

a grain of wheat weighs approximately 0.000008 kg. find the total weight of the wheat requested by sissa ben dahir. (round your answer to the nearest whole number. one trillion equals 1,000,000,000,000.)
resources
read it

Explanation:

Step1: Recall the story of Sissa Ben Dahir

In the story, the number of wheat grains is \(2^{64}-1\) (since he asked for 1 grain on the first square, 2 on the second, 4 on the third, ..., \(2^{63}\) on the 64th square, so total is \( \sum_{i = 0}^{63}2^{i}=\frac{2^{64}-1}{2 - 1}=2^{64}-1\)).

Step2: Calculate the number of grains

First, calculate \(2^{64}-1\). We know that \(2^{10}=1024\approx10^{3}\), so \(2^{64}=2^{60}\times2^{4}=(2^{10})^{6}\times16\approx(10^{3})^{6}\times16 = 10^{18}\times16 = 1.6\times10^{19}\) (more accurately, \(2^{64}=18446744073709551616\), so \(2^{64}-1 = 18446744073709551615\)).

Step3: Calculate total weight

Each grain weighs \(0.000008\space kg=8\times10^{-6}\space kg\). So total weight \(W=(2^{64}-1)\times8\times10^{-6}\space kg\).

Substitute \(2^{64}-1\approx1.8446744073709551615\times10^{19}\) (exact value is \(18446744073709551615\)):

\(W = 18446744073709551615\times8\times10^{-6}\)

First, multiply \(18446744073709551615\times8 = 147573952589676412920\)

Then divide by \(10^{6}\) (since \(10^{-6}=\frac{1}{10^{6}}\)): \(147573952589676412920\div10^{6}=147573952589676.41292\space kg\)

Now, convert to trillion kg. One trillion is \(10^{12}\), so divide by \(10^{12}\):

\(\frac{147573952589676.41292}{10^{12}}=147.57395258967641292\)

Round to the nearest whole number: \(148\)

Answer:

\(148\)