An iron ball is bobbing up and down on the end of a spring. The maximum height of the ball is 50 centimeters and its minimum height is 14 centimeters. It takes the ball 2 seconds to go from its maximum height to its minimum height.

Which model best represents the height, h, of the ball after t seconds?

h (t) = 25sin (πt) + 7

h (t) = 50sin (πt / 2) + 14

h (t) = 18sin (πt) + 32

h (t) = 18sin (πt / 2) + 32

Step-by-step explanation:

Let us examine all options:

(a)

Lets verify for t = 0:

h (t = 0) = 7

This isn't allowable as 14 cm is the lower limit for the ball's motion.

This is rejected.

(b)

Let's verify for t = 0:

h (t = 0) = 14

Since at t = 0 sec, the ball is at its minimum, after 2 seconds, it should be at its maximum.

But h (t = 2) = 14, which doesn't satisfy the condition.

Hence, this is rejected.

(c)

Now, let's see at what time instance, the ball is at minimum in this case:

h (t) = 14 = 18sin (πt) + 32

∴ sin (πt) = - 1

∴ πt = 3π/2

∴ t = 3/2 seconds

Hence, after 2 seconds, i. e. at 3.5 seconds, the ball should be at its maximum.

h (t = 3.5) = 18sin (3.5π) + 32 = - 18 + 32 = 14, which doesn't satisfy the condition.

Hence, this is rejected,

(d)

Now, let's see at what time instance, the ball is at minimum in this case:

h (t) = 14 = 18sin (πt/2) + 32

∴ sin (πt/2) = - 1

∴ πt/2 = 3π/2

∴ t = 3 seconds

Hence, after 2 seconds, i. e. at 5 seconds, the ball should be at its maximum.

h (t = 5) = 18sin (5π/2) + 32 = 18 + 32 = 50, which satisfies the condition.

Hence, option (D) is the right answer