A ball is projected upward from the top of a 90-foot building at a velocity of 64 feet per second. The ball's height above the ground below the building is described by the function h(t)=-16t^2+64t+90, with t being the time in seconds after the ball is projected upward.
a) Determine the amount of the vertical intercept, and interpret what this means in the context of the problem(in terms of seconds and feet above the ground).

b)Determine the numerical value of all horizontal intercepts (if any), and interpret what they mean in the context of the problem (in terms of seconds and feet above the ground).

c)Write the coordinates of the vertex. Interpret what these numbers mean in the context of the problem.

d)If nothing stops the ball before, then how much time elapses until the ball hits the ground below the building?

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yungsherman yungsherman · Answer 1 · 2019-08-02T08:59:29+02:00

Answer:

a) At t=0, the height is 90. This means that initial height of the ball is 90.

The ball is 90 feet high at t=0

b) [tex]t = \frac{8+\sqrt{154} }{4}[/tex]

or 5.10

This value (5.10) represents the time at which the ball hits the ground

c) the vertex is (2,154)

This information tells us that the ball reaches the highest height of 154 ft after 2 seconds

d) the ball will hit the ground after 5.10 seconds.

Step-by-step explanation:

a) the vertical intercept is the value when t=0 (when the curve hits the y-axis)

h(t)=-16t^2+64t+90

h(0)=-16(0)^2+64(0)+90

h(0)=90

At t=0, the height is 90. This means that initial height of the ball is 90.

The ball is 90 feet high at t=0

b) The horizontal asymptotes are the values that make h(t)=0 (When the curve hits the x-axis)

solve the equation for zero

0 =-16t^2+64t+90

since we cant factor, we need to use the quadratic equation.

After applying the quadratic equation the answer should be:

[tex]t = \frac{8+\sqrt{154} }{4}[/tex]

or 5.10

This value (5.10) represents the time at which the ball hits the ground

C) to find the vertex we need to convert to vertex form

The vertex form is:

-16(t - 2)^2+154

From this we can tell that the vertex is (2,154)

This information tells us that the ball reaches the highest height of 154 ft after 2 seconds

d) the ball will hit the ground after 5.10 seconds. We found this out once we fold for h(t)=0 in problem b.