Question

A bird drops a stick to the ground from a height of 85 ft. The function h=-16t^2+85 gives the sticks approximate height h above the ground, in feet, after t seconds. At about what time does the stick hit the ground?

Answer 1

The stick will hit the ground approximately 2.3 seconds after being dropped, as determined by setting the height function h=-16t^2+85 equal to 0 and solving for t using the quadratic formula.

Step-by-step explanation:

The student asks when a stick dropped from a height of 85 ft will hit the ground, based on the function h=-16t2+85. To find when the stick hits the ground, we set h equal to 0 and solve for t:

0 = -16t2 + 85

This is a quadratic equation, which we can solve by using the quadratic formula or by factoring if possible. Applying the quadratic formula:

t = (-b ± √(b2 - 4ac)) / (2a)

Where a = -16, b = 0, and c = 85.

New equation:

t = √(85/16)

t ≈ 2.3 seconds (only the positive root is considered as time cannot be negative)

The stick will hit the ground approximately 2.3 seconds after it is dropped.

Answer 2

To find the time at which the stick hits the ground, we need to set the height function h(t) equal to zero and solve for t:

`\[ h(t) = -16t^2 + 85 \]`

Set h(t) to zero:

`\[ -16t^2 + 85 = 0 \]`

Now, solve for t:

`\[ -16t^2 = -85 \]`

Divide both sides by -16:

`\[ t^2 = (85)/(16) \]`

Take the square root of both sides:

`\[ t = \sqrt{(85)/(16)} \]`

Now, calculate the numerical value for t:

`\[ t \approx √(5.3125) \]`

`\[ t \approx 2.307 \]`

So, at approximately
`\( t \approx 2.307 \)` seconds, the stick hits the ground.