Question

The height h(t) (in feet) of the seat of a child’s swing above ground level is given by the equation

below where t is the time in seconds after the swing is set in motion.
ℎ() = −1.1 cos ((2/3) ) + 3.1
a. Find the maximum and minimum height of the swing.
b. When is the first time after t = 0 that the swing is at a height of 3 feet?
c. When is the second time after t = 0 that the swing is at a height of 3 ft?

Answer 1

Answer: See below

Explanation:

The function of the seat’s height from the ground level is given as,

`h(t)=-1.1 \cos \left((2 \pi)/(3) t\right)+3.1`

Here, t denotes the time.

(a) The height will be maximum or minimum when the derivative of the function of height is equal to zero.

`\begin{aligned}h^(\prime)(t) &=0 \\(d)/(d t)\left(-1.1 \cos \left((2 \pi)/(3) t\right)+3.1\right) &=0 \\-1.1 * (2 \pi)/(3)\left(-\sin \left((2 \pi)/(3) t\right)\right) &=0 \\t &=0,1.5\end{aligned}`

The height of the seat at time t = 0 s can be determined as,

`\begin{aligned}h(0) &=-1.1 \cos \left((2 \pi)/(3)(0)\right)+3.1 \\&=2 \mathrm{ft}\end{aligned}`

Therefore, the maximum height of the swing is 4.2 ft and the minimum height of the swing is 2 ft.

(b) The height of the swing is given as,

`\begin{aligned}h &=3 \mathrm{ft} \\-1.1 \cos \left((2 \pi)/(3) t\right)+3.1 &=3 \\t &=0.7 \mathrm{~s}\end{aligned}`

Therefore, the first time after t = 0 s that the swing’s height of 3 ft is 0.7 s.

(c) The height of the swing is given as,

`\begin{aligned}h &=3 \mathrm{ft} \\-1.1 \cos \left((2 \pi)/(3) t\right)+3.1 &=3 \\(2 \pi)/(3) t &=1.47976+2 \pi \\t &=3.7 \mathrm{~s}\end{aligned}`

Therefore, the second time after t = 0 s that the swing’s height of 3 ft is 3.7 s.