So much that you’d waste bandwidth and memory only to do this with jQuery rather than plain JS? ;-) Anyway, here’s a vanilla JS solution:
var answer = 'sample text';
var button = document.getElementById('done');
var textarea = document.getElementById('right-answer');
button.addEventListener('click', function() {
textarea.value = answer;
});
If you’re a bit familiar with JS and jQuery, rewriting it accordingly should be rather trivial. If not, I’d strongly suggest to learn those JS basics first, and then start exploring where it makes sense to use jQuery instead.