Question

In a certain town, the amount of sulfur oxide in the air, S, in tons, is related to the town’s population, P, in people. The relationship is S2 = 1 10 P2 + 20P + 600 The town’s population is growing at a rate, with respect to time, of 10 people/month. At what rate, with respect to time, is the amount of sulfur oxide in the air changing, when the population is 100?

Answer 1

Change in sulfur oxide in the air =
`\frac{\textup{110010}}{\textup{S}}`

Explanation:

Data provided in the question:

Relation between the amount of sulfur oxide in the air and the population as:

S² = 110P² + 20P + 600

Population growth rate,
`\frac{\textup{dP}}{\textup{dt}}` = 10 people per month

Now,

change in sulfur oxide with time i.e
`\frac{\textup{dS}}{\textup{dt}}`

differentiating the given relation with respect to time 't'

we have

`2S\frac{\textup{dS}}{\textup{dt}}` =
`2*110P\frac{\textup{dP}}{\textup{dt}}` + 20

at P = 100 and
`\frac{\textup{dP}}{\textup{dt}}` = 10 people per month

we have

`2S\frac{\textup{dS}}{\textup{dt}}` = 2 × 110 × 100 × 10 + 20

or

`2S\frac{\textup{dS}}{\textup{dt}}` = 220020

or

`\frac{\textup{dS}}{\textup{dt}}` =
`\frac{\textup{220020}}{\textup{2S}}`

or

Change in sulfur oxide in the air =
`\frac{\textup{110010}}{\textup{S}}`