Questions: Effect of Price on Demand for Tires Suppose the quantity demanded weekly of the Super Titan radial tires is related to its unit price by the equation [ p+x^2=289 ] where p is measured in dollars and x is measured in units of a thousand. How fast is the quantity demanded weekly changing when x=10, p=189, and the price per tire is increasing at the rate of 2 / week nearest tire.)

Solution Steps

To solve this problem, we need to find the rate at which the quantity demanded weekly is changing. This involves differentiating the given equation with respect to time and then substituting the given values to find the desired rate.

1. Start with the given equation \( p + x^2 = 289 \).
2. Differentiate both sides of the equation with respect to time \( t \) to find the relationship between the rates of change of \( p \) and \( x \).
3. Substitute the given values \( x = 10 \), \( p = 189 \), and \( \frac{dp}{dt} = 2 \) into the differentiated equation to solve for \( \frac{dx}{dt} \).

The given equation is: \[ p + x^2 = 289 \]

Differentiate both sides of the equation with respect to time \( t \): \[ \frac{d}{dt}(p + x^2) = \frac{d}{dt}(289) \] \[ \frac{dp}{dt} + 2x \frac{dx}{dt} = 0 \]

Substitute the given values \( x = 10 \), \( p = 189 \), and \( \frac{dp}{dt} = 2 \) into the differentiated equation: \[ 2 + 2(10) \frac{dx}{dt} = 0 \] \[ 2 + 20 \frac{dx}{dt} = 0 \]

Solve the equation for \( \frac{dx}{dt} \): \[ 20 \frac{dx}{dt} = -2 \] \[ \frac{dx}{dt} = -\frac{2}{20} \] \[ \frac{dx}{dt} = -\frac{1}{10} \]

Final Answer