Chapter 12.5, Problem 46E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Sales and pricing Suppose that a chain of auto service stations, Quick-Oil, Inc., has found that the relationship between its price p for an oil change and its monthly sales volume y in thousands of dollars is d y d p = − 1 2 ( y p + 4 ) Solve this differential equation if y = 18 when p   = $60 . To determine To calculate: The simplified form of differential equation dydp=12(yp+4) if y=18, when p=$60.

Explanation

Given information:

The relationship between its price p for an oil change and its monthly sales volume y in thousands of dollars is dydp=12(yp+4). If y=18 when p=$60. Formula used: Suppose that a chain of auto service stations, quick oil, Inc., has found that the relationship between its price p for an oil change and its monthly sales volume y in thousands of dollars is: dydp=12(yp+4) When a differential equation can be equivalently expressed in the form: g(y)dy=f(x)dx This is known as variable separable. Calculation: Consider the provided information is: If y=18 when p=$60. Suppose that a chain of auto service stations, quick oil, Inc., has found that the relationship between its price p for an oil change and its monthly sales volume y in thousands of dollars is dydp=12(yp+4).

When a differential equation can be equivalently expressed in the form:

g(y)dy=f(x)dx

Suppose that a chain of auto service stations, quick oil, Inc., has found that the relationship between its price p for an oil change and its monthly sales volume y in thousands of dollars is:

dydp=12(yp+4)

Now rearrange the terms containing y on left side and p on right side as:

dyy=12dpp+4

Integrate both side of the equation as:

dyy</

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