The rate of change of the unit price $p$ (in dollars) of Apex women's boots is given by $$ p^{\prime}(x)=\frac{-250 x}{\left(16+x^{2}\right)^{3 / 2}} $$ where $x$ is the quantity demanded daily in units of a hundred. Find the demand function for these boots if the quantity demanded daily is 300 pairs $(x=3)$ when the unit price is $\$ 50 /$ pair.

          The rate of change of the unit price $p$ (in dollars) of Apex women's boots is given by
$$
p^{\prime}(x)=\frac{-250 x}{\left(16+x^{2}\right)^{3 / 2}}
$$
where $x$ is the quantity demanded daily in units of a hundred. Find the demand function for these boots if the quantity demanded daily is 300 pairs $(x=3)$ when the unit price is $\$ 50 /$ pair.

Added by Janet P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Eleni Katirtzoglou

04/16/2022

Step 1

$$ p(x) = \int p'(x) dx = \int \frac{-250 x}{\left(16+x^{2}\right)^{3 / 2}} dx $$ This is a standard form of integral that can be solved by substitution. Let $u = 16 + x^2$, then $du = 2x dx$. The integral becomes: $$ p(x) = -125 \int \frac{du}{u^{3/2}} = -125 Show more…

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The rate of change of the unit price $p$ (in dollars) of Apex women's boots is given by $$ p^{\prime}(x)=\frac{-250 x}{\left(16+x^{2}\right)^{3 / 2}} $$ where $x$ is the quantity demanded daily in units of a hundred. Find the demand function for these boots if the quantity demanded daily is 300 pairs $(x=3)$ when the unit price is $\$ 50 /$ pair.