Question

question suppose a small company sells speakers. the company determines that the supply function for speakers is given by $s(x)=100x - 50$, and the demand function for speakers is given by $d(x)=500 - 20x$, where $x$ is the speaker price. how many items will be produced at the equilibrium price? give your answer to the nearest integer. provide your answer below:

Explanation:

Step1: Set supply equal to demand

$S(x)=D(x)$
$100x - 50=500 - 20x$

Step2: Solve for x (equilibrium price)

Add $20x$ to both sides: $100x+20x - 50=500 - 20x+20x$
$120x - 50=500$
Add 50 to both sides: $120x - 50 + 50=500+50$
$120x=550$
$x=\frac{550}{120}=\frac{55}{12}\approx4.58$

Step3: Substitute x into supply or demand function

We use the supply function $S(x)=100x - 50$. Substitute $x = \frac{55}{12}$
$S(\frac{55}{12})=100\times\frac{55}{12}-50=\frac{5500}{12}-50=\frac{5500 - 600}{12}=\frac{4900}{12}\approx408.33$
Or use the demand function $D(x)=500 - 20x$. Substitute $x=\frac{55}{12}$
$D(\frac{55}{12})=500-20\times\frac{55}{12}=500-\frac{1100}{12}=\frac{6000 - 1100}{12}=\frac{4900}{12}\approx408.33$
Rounding to the nearest integer gives 4.

Answer:

4