91Ó°ÊÓ

A spherical balloon is inflated so that its volume is increasing at the rate of \(3 \mathrm{ft}^{3} / \mathrm{min} .\) How fast is the diameter of the balloon increasing when the radius is \(1 \mathrm{ft} ?\)

Grain pouring from a chute at the rate of \(8 \mathrm{ft}^{3} / \mathrm{min}\) forms a conical pile whose altitude is always twice its radius. How fast is the altitude of the pile increasing at the instant when the pile is \(6 \mathrm{ft}\) high?

Sand pouring from a chute forms a conical pile whose height is always equal to the diameter. If the height increases at a constant rate of \(5 \mathrm{ft} / \mathrm{min}\), at what rate is sand pouring from the chute when the pile is 10 ft high?

The perceived loudness \(\beta\) of a sound in decibels (dB) is related to its intensity \(l\) in watts/square meter \(\left(\mathrm{W} / \mathrm{m}^{2}\right)\) by the equation $$\beta=10 \log \left(1 / I_{0}\right)$$ where \(I_{0}=10^{-12} \mathrm{W} / \mathrm{m}^{2} .\) Damage to the average ear occurs at 90 dB or greater. Find the decibel level of each of the following sounds and state whether it will cause ear damage. $$\begin{array}{lclcl} & \text { SOUND } & I \\ \text { (a) } & \text { Jet aircraft (from 500 ft) } & 1.0 \times 10^{2} \mathrm{W} / \mathrm{m}^{2} \\\ \text { (b) } & \text { Amplified rock music } & 1.0 \mathrm{W} / \mathrm{m}^{2} \\ \text { (c) } & \text { Garbage disposal } & 1.0 \times 10^{-4} \mathrm{W} / \mathrm{m}^{2} \\ \text { (d) } & \text { TV (mid volume from 10 } \mathrm{ft} \text { ) } & 3.2 \times 10^{-5} \mathrm{W} / \mathrm{m}^{2}\end{array}$$