The height of the tide measured at a seaside community varies according to the number of hours t after midnight. If the height h, in feet, is currently given by the equation #h=-1/2t^2+6t-9#, when will the tide first be at 6 ft?

Question

Savannah Anderson · Accepted Answer

At #8.27# a.m. or #08.27#

Putting the value of h = 6 in equation #h = -1/2t^2 + 6t - 9#

or,#6 = [- t^2 + 12t - 18]/2#

or, #12 = -t^2 + 12t - 18#

or, #t^2 - 12t + 12 + 18 = 0#

or, #t^2 - 12t + 30 = 0#

or, #t = [-(-12) + sqrt {(-12)^2 - 4*1*30}]/(2*1)# and

#[-(-12) - sqrt{(-12)^2 - 4*1*30}]/(2*1)#

or, #t = [+12 +sqrt{144 - 120}]/2# and #[+12 - sqrt{144 - 120}]/2#

or, #t = [12 +sqrt 24]/2, [12 - sqrt 24]/2 #

or, #t = [12 + 2 sqrt 6]/2 , [12 - 2 sqrt 6]/2

or, #t = 6 +sqrt 6 , 6 - sqrt 6#

The first tide will be at morning #6 +sqrt 6# hours.

The first time will be #8.449# hours after midnight.

This give the time as #8 "hours" 27 "minutes"# after midnight.

Hazel Anglin · Answer

undefined To find when the tide will first be at 6 ft, set the equation h = -1/2t^2 + 6t - 9 equal to 6 and solve for t:

-1/2t^2 + 6t - 9 = 6

-1/2t^2 + 6t - 9 - 6 = 0

-1/2t^2 + 6t - 15 = 0

Now, use the quadratic formula to solve for t:

t = [-b ± √(b^2 - 4ac)] / 2a

Substitute a = -1/2, b = 6, and c = -15 into the formula:

t = [-(6) ± √((6)^2 - 4(-1/2)(-15))] / (2 * (-1/2))

t = [-(6) ± √(36 - 30)] / (-1)

t = [-(6) ± √(6)] / (-1)

t ≈ [-(6) ± √(6)] / (-1)

Therefore, the tide will first be at 6 ft approximately at t ≈ 0.79 hours after midnight and at t ≈ 11.21 hours after midnight.