Answer:
(a) 512 dimes
(b)[tex]D(n) = 2^{n-1}[/tex]
(c) 9.22337 x 10¹⁸ dimes
(d) 9.22337 x 10¹² km
(e) roughly 61,489.13 times longer than the distance from the earth to the sun.
Step-by-step explanation:
b. Starting with 1 dime in the first square, if the number of dimes per each square is the double of the previous square, the general equation for the number of dimes in each square 'n' can be found by:
[tex]D(1) = 1\\D(2) = 2^1\\D(3) = 2^2\\D(4) = 2^3\\D(n) = 2^{n-1}[/tex]
a. On the 10th square:
[tex]D(10) = 2^{10-1}\\D(10) = 512\ dimes[/tex]
c. On the 64th square:
[tex]D(64) = 2^{64-1}\\D(10) = 9.22337 *10^{18}\ dimes[/tex]
d. If a dime is 1 mm thick, the 64th pile will be:
[tex]h= 9.22337*10^{18}*0.0001\ m\\h= 9.22337*10^{15}\ m\\h= 9.22337*10^{12}\ km[/tex]
e. The ratio between the height of the last pile (h) to the distance from the earth to the sun is:
[tex]R=\frac{9.22337*10^{12}\ km}{150,000,000\ km}\\ R=61,489.13[/tex]
The height of the last pile is roughly 61,489.13 times longer than the distance from the earth to the sun.
