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Hi! Please help... a woman drops a front door key to her husband from their apartment window several stories above the ground. the function h=-16t^2 +64 gives the height h of the key in feet, t seconds after she releases it.

A/ how long does it take the key to reach the ground?

B/ what are the reasonable domain and range for the function h? ...?

Verified Answer :

AWhen the key reaches the ground, h = 0.

   0 = -16t^2 + 64
   16t^2 = 64
   t^2 = 4
   t = 2 seconds.

B. The reasonable domain of the function is [0, 2], since the function does not hold when the key reaches the ground. The range of the function is [0, 64] which corresponds to the domain.

Answer:

It would take 2 seconds to the key to reach the ground.

The domain the function is: 0≤t≤2 or in interval notation [0,2].

The range of the function is: 0≤h≤64 or in interval notation [0,64].

Step-by-step explanation:

The provided function is: \(h=-16t^2 +64\)

Where h represents the height and t represents the time.

Part(A) how long does it take the key to reach the ground?

When key will be at ground, the height will be 0.

Substitute the value of h=0 in above equation and solve for t.

\(-16t^2 +64=0\)

\(-16(t^2 -4)=0\)

\((t^2 -2^2)=0\)

\((t+2)(t-2)=0\)

\(t+2=0\ or\ t-2=0\)

\(t=-2\ or\ t=2\)

Ignore the negative value of t.

Hence, it would take 2 seconds to the key to reach the ground.

Part (B) what are the reasonable domain and range for the function h?

Time and Distance should be a positive number.

Thus, the value of t and h should be greater or equal to 0.

From the part (a) we know that the after 2 second the key will hit the ground so the value of t can be greater or equal to 0 but should be less or equal to 2.

Thus, the domain the function is: 0≤t≤2 or in interval notation [0,2].

The maximum height of the key will be at t=0.

Thus the maximum value of h can be 64 and minimum value can be 0.

The range of the function is: 0≤h≤64 or in interval notation [0,64].

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