#### To determine

**To find:** The rate at which the height of the rider is rising when his seat is 16 m above the ground level.

#### Answer

The rate at which the height of the rider rises is dhdt=8π m/min.

#### Explanation

**Given:**

The Ferris wheel of radius 10 m is rotating 1 revolution in every 2 minutes.

**Formula used:**

(1) Chain rule: dydx=dydu⋅dudx

(2) Pythagorean Theorem

**Calculation:**

Let us assume that O be the centre of the Ferris wheel and B be the lowest point on the circumference of the Ferris wheel and A be the position of the rider seat which is h m from the centre of the wheel and θ be the angle to the rider seat from the horizontal as shown in the Figure 1 given below.

Since the angle θ and the height h increase with the time t, the angle θ and height h are the function of the time t.

dθdt=2π rad2 minutes=π rad/min

From the above triangle,

sinθ=h10h=10sinθ

When the height of the rider 16 m from the ground that is h=6 m.

x=102−62 [QBy Pyhtagorean Theorem]=100−36=64=8 m

Differentiate h with respect to the time t.

ddt(h)=ddt(10sinθ)dhdt=10cosθdθdt [Qdydx=dydu⋅dudx]

Substitutes cosθ=810 and dθdt=π in dhdt .

dhdt=10(810)(π)=8π rad/min

Therefore, the rate at which the height of the rider rises is dhdt=8π m/min.