Question

A pomegranate is thrown from a ground level straight up into the air at time t = 0 with velocity 67 feet per second. Its height at time t seconds is f(t)=-16^2+67t.

(a) Find the time it hits the ground and (b) the time it reaches its highest point. (c) What is the maximum height?
Please round to two decimal places.

Answer 1

a) when it hits the ground, f(t) = 0
0 = -16t^2 + 67t
16t^2 - 67t = 0
t(16t - 67) = 0
t = 0 or t = 67/16 = appr 4.19 seconds
t=0 would be the start of the toss

so the answer you want is t = 4.19 sec

b) since you labeled it "pre-calculus" I assume you cannot at this point take the derivative, too bad

so we have to complete the square

f(t) = -16t^2 + 67t
= -16(t^2 - (67/16)t <b>+ 4489/1024 - 4489/1024</b>)
= -16( (t - 67/32)^2 - 4489/1024)
= -16(t - 67/32)^2 + 4489/64

it will reach the maximum height at 67/32 or appr 2.09 sec, and that max height is 4489/64 or 70.14 ft

Notice we could have used the properties of the parabola to shorten this up a bit
since the x-intercepts are 0 and 4.19, the vertex must lie midway between them
This would be 2.09 , the answer I got when completing the square. Subbing 2.09 in for f(2.09) gives us
-16(2.09)^2 + 67(2.09) = 70.14