Given that the initial grant in 2012 is $4000 and it increases by 5% each year, we can model this as a geometric sequence where the first term \(a = 4000\) and the common ratio \(r = 1.05\).

(i) To find the value of the grant in 2022, which is the 11th term of the sequence, we use the formula for the nth term of a geometric sequence:

\(a_n = ar^{n-1}\)

Substituting the values, we get:

\(a_{11} = 4000 \times 1.05^{10}\)

\(a_{11} = 4000 \times 1.62889 \approx 6516\)

Rounding to the nearest dollar, the value is $6520.

(ii) To find the total amount received from 2012 to 2022, we use the formula for the sum of the first n terms of a geometric sequence:

\(S_n = \frac{a(r^n - 1)}{r - 1}\)

Substituting the values, we get:

\(S_{11} = \frac{4000(1.05^{11} - 1)}{0.05}\)

\(S_{11} = \frac{4000(1.71034 - 1)}{0.05}\)

\(S_{11} = \frac{4000 \times 0.71034}{0.05}\)

\(S_{11} = 56827.2\)

Rounding to the nearest dollar, the total amount is $56800.