Kinematics - Archer shooting arrow at apple

[Parallel]

Hello all.
First I want to say that I'm really new to physics,started like a week ago.
and that this is NOT homework(i'm studying by myself)

An archer shoots an arrow with a velocity of 45.0 m/s at an
angle of 50.0° with the horizontal. An assistant standing on
the level ground 150 m downrange from the launch point
throws an apple straight up with the minimum initial
speed necessary to meet the path of the arrow. (a) What is
the initial speed of the apple? (b) At what time after the
arrow launch should the apple be thrown so that the arrow
hits the apple?

I didn't get the same answers as the book.
here's what I've done,I think it's correct.

I'll just explain what I did: I computed the time it takes for the arrow to get to the target(takes 5.2 sec),computed the altitude at this time(47m). computed the time it takes for an apple to get to this altitude..
IF you throw it at the same time the archer shoots the arrow(so actually the apple will be on it's way down when the arrow hits it).so the speed is 34.47 m/s

so I just wanted to know if my reasoning and solution are correct,even though you could've thrown the apple at a later time(which yields the same answer as the book)

help would be appreciated
thanks

 

[berkeman]

So do you get the book's answer if you throw the apple so it just makes it up to the 47m altitude? If so, then you've gotten multiple correct answers. It looks like the book didn't specify that it wanted the shortest possible time.