= Solution
Let $X$ denote the monetary reward. Its <expected value> and <expected utility> are different quantities:
$$
\mathbb E X=\frac{89}{100}+\frac{10}{10}=\boxed{1.89},\qquad
\mathbb E[u(X)]=\frac{89}{100}+\frac{100}{10}=\frac{1089}{100}=10.89.
$$
Thus indifference to a sure nonnegative reward requires $k^2=10.89$, giving \b[$k=3.30$ dollars]. This is the <certainty equivalent>, hence the <lottery>'s monetary value in the question's comparison with a sure reward. The <utility function> $u(x)=x^2$ is increasing and <strictly convex> on nonnegative monetary amounts. Its <certainty equivalent> exceeds its <expected value>, $3.30>1.89$, demonstrating \b[<risk seeking>]; the same conclusion follows from the <Jensen inequality>.
The wording about paying an entry fee needs a qualification. A <certainty equivalent> compares a gross <lottery> with a sure gross reward; a literal <utility indifference price> also accounts for the fee's effect on wealth. If $u$ instead describes final wealth, with initial wealth $w$ and fee $c$, the comparison is $\mathbb E[(w-c+X)^2]=w^2$, on a domain where the <utility function> is increasing. Here $\operatorname{Var}(X)=7.3179$, so the nonnegative-wealth branch gives $c=w+1.89-\sqrt{w^2-7.3179}$ whenever that branch is admissible. An initial wealth and an admissible payment domain are not specified. Therefore \b[$3.30$ is the determined <certainty equivalent>], while a literal wealth-dependent maximum fee is not uniquely determined. This is the distinction in <certainty equivalent and willingness to pay>. If negative rewards were allowed without requiring an increasing <utility function>, the equation for $k$ also has the algebraic root $-3.30$; the usual nonnegative-reward interpretation selects the positive root.
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