"Are you interested in British history and culture?"
Whether or not 感兴趣 is accepted depends on how you structure the sentence. Think of 感兴趣 as "interested" and 兴趣 as "interest"; to use 感兴趣 we would have to remove 有 and structure the sentence as follows 你对英国的历史和文化感兴趣吗.
Another way to think of it is that "你对英国的历史和文化有兴趣吗" could be translated as "Do you have interest in British history and culture?" and "你对英国的历史和文化感兴趣吗" could be translated as "Are you interested in British history and culture?. The meaning is the same so both should be accepted.
It is the choice of words but both make up the sentences that literally have the same meaning. Both solutions should be accepted.
I thought so too! Reported.
But when is it 感兴趣 (gǎnxìngqù) and when just 兴趣 (xìngqù) ?