Answer:
1a) 230 feet
1b) 3.75s
1c) t=7.54s
2) t=2s
3) t=2.32s
Step-by-step explanation:
The equation that models Brett's last home run is [tex]h(t)=-16t^2+120t+5[/tex]
We need to complete the square to obtain the function in the vertex form:
[tex]h(t)=-16(t^2-7.5t)+5[/tex]
[tex]h(t)=-16(t^2-7.5t+3.75^2)+5+-16(-3.75)^2[/tex]
[tex]h(t)=-16(t-3.75)^2+230[/tex]
1a) The vertex is (3.75,230).The maximum value is the y-coordinate of the vertex, which is 230 feet.
1b) The time it takes to reach the maximum value is the x-coordinate of the vertex. It reach the maximum heigth after t=3.75 seconds
1c) To find the time the ball hit the ground, we equate the h(t)=0 and solve for t.
[tex]-16(t-3.75)^2+230=0[/tex]
[tex]-16(t-3.75)^2=-230[/tex]
[tex](t-3.75)^2=14.375[/tex]
[tex]t-3.75=\pm \sqrt{14.375}[/tex]
[tex]t=3.75\pm \sqrt{14.375}[/tex]
[tex]t=3.75\pm 3.79[/tex]
[tex]t=3.75-3.79\:\:or\:t=3.75+3.79[/tex]
[tex]t=3.75-3.79\:\:or\:t=3.75+3.79\\t=-0.04\:\:or\:t=7.54[/tex]
The time is positive, so the ball hit the ground after 7.54 seconds.
Question 2)
The function that models the amusement ride is [tex]h(t)=-16t^2+64t+60[/tex]
We want to find the time it takes for the riders to reach the maximum height.
This time is given by: [tex]t=-\frac{b}{2a}[/tex]
Comparing [tex]h(t)=-16t^2+64t+60[/tex] to [tex]h(t)=at^2+bt+c[/tex] we have a=-16, b=64, c=60.
We substitute to obtain:
[tex]t=-\frac{64}{2*-16} \\t=-\frac{64}{-32} \\t=2[/tex]
Hence it took 2 seconds to rech the maximum height.
Question 3)
The equation that models the height f the catridge is [tex]h(t)=-16t^2+35t+5[/tex]
To find the time that the catridge will land, we equate the function to zero and solve for t.
[tex]-16t^2+35t+5=0[/tex]
This is a quadratic equation with =-16, b=35, an c=5
The solution is given by:
[tex]t=\frac{-b\pm \sqrt{b^2-4ac} }{2a}[/tex]
We substitute the values to get:
[tex]t=\frac{-35\pm \sqrt{35^2-4*-16*5} }{2*-16}[/tex]
This gives:
[tex]t=\frac{35-\sqrt{1545} }{32} \:or\: t=\frac{35+\sqrt{1545} }{32}[/tex]
This simplifies to:
[tex]t=-0.13\:or\: t=2.32[/tex]
The time it takes to land must be positive thefore t=2.32 seconds
