Question

A person is standing on a second-floor balcony drops keys to a friend standing below the balcony. The keys are dropped from a height of 10 feet. The height in feet of the keys as they fall is given by the function h(t) = 16t² + 10, where t is the time in seconds since the keys were dropped. The friend catches the keys at a height of 4 feet. Find the elapsed time before the keys are caught.

a) 0.5 seconds
b) 1 second
c) 1.5 seconds
d) 2 seconds

Answer 1

After correcting the equation to 4 = -16t² + 10 and solving for t, we find that the keys take approximately 0.61 seconds to be caught at a height of 4 feet.

Step-by-step explanation:

The height in feet of the keys as they fall is given by the function h(t) = 16t² + 10. To find the elapsed time before the keys are caught at a height of 4 feet, we need to solve for t when h(t) = 4.

To do this, set the equation to 4 and solve for t:

4 = 16t² + 10

Subtract 10 from both sides to get:

-6 = 16t²

Divide by 16 to isolate t²:

t² = -6 / 16

t² = -3 / 8

However, we cannot have a negative time, and it seems there is a mistake. The correct initial setup should be:

4 = 16t² + 10

Subtract 10 from both sides:

-6 = 16t²

Divide both sides by 16:

t² = -3/8

This is not possible because time cannot be negative or imaginary. The correct equation should be:

4 = -16t² + 10

Solving for t gives us:

16t² = 10 - 4

16t² = 6

t² = 6/16

t² = 0.375

t = √0.375

t ≈ 0.612

Thus, the keys take approximately 0.61 seconds to reach a height of 4 feet, which means the closest answer would be b) 1 second, if we round to the nearest whole number.