Respuesta :
Answer:
The marginal revenue is
[tex]\dfrac{dR}{dx}=\dfrac{55ln(x)-55}{ln(x)^2}[/tex]
For x=20, the marginal revenue is R'=12.
Step-by-step explanation:
The marginal revenue is defined as the additional revenue that yields an additional unit sale of the good.
Mathematically, is equivalent to the first derivative of the revenue function. In this case, we are not considering the additional cost, so we use the demand function.
The revenue then becomes
[tex]R=p\cdot q=\dfrac{55x}{ln(x)}[/tex]
We can calculate this first derivative as:
[tex]\dfrac{dR}{dx}=\dfrac{d}{dx}[55x/ln(x)] \\\\\\ u=55x\\\\v=ln(x)\\\\ (u/v)'=(u'v-uv')/v^2=(55ln(x)-55x/x)/ln(x)^2 \\\\(u/v)'=(55ln(x)-55)/ln(x)^2 \\\\\\\dfrac{dR}{dx}=\dfrac{55ln(x)-55}{ln(x)^2}[/tex]
The marginal revenue for x=20 is:
[tex]\dfrac{dR}{dx}(20)=\dfrac{55ln(20)-55}{ln(20)^2}\\\\\\ \dfrac{dR}{dx}(20)=\dfrac{55(3)-55}{3^2}=\dfrac{110}{9}= 12.22\approx 12[/tex]
