SOLUTION: Jim is an avid fisherman. He varies the depth at which he fishes according to the following: D(t)=-t^2+10t where t is measured in hours. Estimate the time when he fishes at the gre

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: Jim is an avid fisherman. He varies the depth at which he fishes according to the following: D(t)=-t^2+10t where t is measured in hours. Estimate the time when he fishes at the gre      Log On


   

Question 202718: Jim is an avid fisherman. He varies the depth at which he fishes according to the following: D(t)=-t^2+10t where t is measured in hours. Estimate the time when he fishes at the greatest depth and tell me that depth.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
To start with, it helps if you can recognize the equation of D(t) as being the equation of a parabola, because of the t^2 term, which opens downward, because of the negative coefficient in front of the t^2 term. A graph of this equation is provided below.

graph%28200%2C+200%2C+-5%2C+25%2C+-5%2C+25%2C+-x%5E2+%2B+10x%29
From looking at this we can tell that the maximum depth (the highest value of D(t)) will be the vertex of the parabola. For parabolas in general the x-coordinate of the vertex can be found at %28-b%29%2F%282a%29 where "a" and "b" are taken from the standard form for a parabola: at%5E2+%2B+bt+%2Bc.

In your equation the "a" is -1 and the "b" is 10. So the x-coordinate of the vertex is %28-%2810%29%29%2F%282%2A%28-1%29%29+=+%28-10%29%2F%28-2%29+=+5

So the maximum depth will be when the hour is 5 and the maximum depth will be D(5):
D%285%29+=+-%285%29%5E2+%2B+10%285%29+=+-%2825%29+%2B+10%285%29+=+-25+%2B+50+=+25