Question

natalie found a tennis ball outside a tennis court. she picked up the ball and threw it over the 12 foot fence into the court. the height of the ball,h, and time t seconds is given by the equation h(t)=-16^2+18t+5. will she make it over the fence

Answer 1

The equation that gives the height is h(t) = - 16t^2 + 18t + 5, which is a parabola.


You cand find the maximum height from the vertex of the parabola.


Let's find the vertex. I will complete squares:

Start by extractin common factor -16:

-16t^2 + 18t + 5 = -16 [t^2 - (18/16) t - 5/16]


I will work with the expression inside the square brackets.

t^2 - (18/16)t - 5/16 = t^2 - (9/8) t - 5/16 =


Completing squares: (t - 9/16)^2 - (9 / 16 )^2 - 5/16

(t - 9/16)^2 - 81/ 256 - 5/16 = (t -9/16)^2 - 161/256


Now, include add include the factor -16[ (t- 9/16)^2 + 161/256 ] =

= -16 (t - 9/16)^2 + 161/16

That means that the vertex is 9/16, 161/16

So, the maximum height is 161 / 16 = 10,06, which is lower than the fence.


Answer: She will not make it over the fence.

Answer 2

Since her maximum height is lower than the  12 foot, hence she will not make it over the fence

How to calculate the maximum height of a function

The maximum point of Natalie is the point where her velocity is zero.

v(t) = dh/dt
v(t) = -32t + 18

If v(t) = 0 then;

32t = 18
t = 18/32

t = 0.5625s

Substitute  t = 0.5625 into the formula

h(t)=-16t^2+18t+5

h(0.5625)=-16(0.5625)^2+18(0.5625)+5

h(0.5625) = -5.0625 + 15.125

h(0.5625) = 10.0625feet

Since her maximum height is lower than the  12 foot, hence she will not make it over the fence

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