can I lose 10 pounds by June 10th

Turn 1600 as your minimum cause under 1500 is considered starving/starvation for a teen and I'd not want your post to be locked/warned by a mod. But anyways back to the original question:

Not healthily, I too am 5'8"  but I am 27 and at 145lbs I am trying to lose 10lbs by July 1st and I think I'll be lucky with 5-7lbs. It is the whole last 10lbs deal, when you are healthy (especially mid-ish) BMI they last 10lbs are really hard to lose compared to the first bit of weight.

Even aiming for 5lbs a month, I got 3-4lbs for the last 2 months (and now I am plateauing *blah!*). Aiming for any more than 5lbs a month is too aggressive during a healthy BMI range. I'd say 5lbs lost would be max and if your lucky and you may do it by June 10th, so aim for a pound a week but you may have a better chance at losing 5lbs by next month than me :)

10 lbs in one month at your age and current weight is too much. Try for a max of 1lbs a week.