Chapter 3.1, Problem 52PPSE

Solution

a.

To identify the dependent and independent variables.

It is found that $p$ is the independent variable and $n$ is the dependent variable.

Given:

Given that the variable $p$ represents the price and that the variable $n$ represents the number of widgets.

Calculation:

Observe that the number of widgets sold increases as the price of the widget decrease.

Thus, the number of widgets sold $n$ is dependent on its price $p$ .

That is, $p$ is the independent variable and $n$ is the dependent variable.

Conclusion:

It is found that $p$ is the independent variable and $n$ is the dependent variable.

b.

To write a linear equation that relates price and quantity demanded.

The linear relation connecting the demand $n$ and price $p$ :

  $n=-1600p+14800$.

Given:

Given that $2000$ widgets sold when its price is $8$ dollars.

That is, $n=2000$ when $p=8$ .

Again, it is given that $10,000$ widgets sold when its price is $3$ dollars.

That is, $n=10,000$ when $p=3$ .

Concept Used:

Equation of a line passing through two points:

The equation of the line passing through the points $\left(p_1,n_1\right)$ and $\left(p_2,n_2\right)$ :

  $n-n_1=\frac{n-n_1}{p_2-p_1}\cdot \left(p-p_1\right)$ .

Calculation:

Observe that $n=2000$ when $p=8$ and $n=10,000$ when $p=3$ .

That is, the graph of the linear equation representing the relation would pass through the points $\left(p_1,n_1\right)=\left(8,2000\right)\text{and }\left(p_2,n_2\right)=\left(3,10000\right)$ .

Now, the equation of the line:

  $\begin{array}{c}n-n_1=\frac{n-n_1}{p_2-p_1}\cdot \left(p-p_1\right)\\ n-2000=\frac{10000-2000}{3-8}\left(p-8\right)\\ n-2000=\frac{8000}{-5}\left(p-8\right)\\ n-2000=-1600\left(p-8\right)\\ n-2000=-1600\left(p-8\right)\\ n-2000=-1600p+12800\\ n=-1600p+12800+2000\\ n=-1600p+14800\end{array}$ .

Thus, the linear relation existing between the demand $n$ and price $p$ :

  $n=-1600p+14800$ .

Conclusion:

The linear relation connecting the demand $n$ and price $p$ :

  $n=-1600p+14800$.

c.

To write a linear relation connecting the price and quantity supplied.

Given:

Given that the shop can make $2000$ widgets for $\$5$ each and $20000$ widgets for $\$2$ each.

Concept Used:

Equation of a line passing through two points:

The equation of the line passing through the points $\left(p_1,s_1\right)$ and $\left(p_2,s_2\right)$ :

  $s-s_1=\frac{s-s_1}{p_2-p_1}\cdot \left(p-p_1\right)$ .

Calculation:

Suppose that $s$ represents the supply and $p$ represents the price.

Now, since the shop can make $2000$ widgets for $\$5$ each and $20000$ widgets for $\$2$ each, the graph of the equation of the line representing the relation passes through the points $\left(p_1,s_1\right)=\left(5,2000\right)$ and $\left(p_2,s_2\right)=\left(2,20000\right)$ .

Now, the equation of the line:

  $\begin{array}{c}s-s_1=\frac{s-s_1}{p_2-p_1}\cdot \left(p-p_1\right)\\ s-2000=\frac{20000-2000}{2-5}\cdot \left(p-5\right)\\ s-2000=\frac{18000}{-3}\left(p-5\right)\\ s-2000=-6000\left(p-5\right)\\ s-2000=-6000p+30000\\ s=-6000p+30000+2000\\ s=-6000p+32000\end{array}$ .

Thus, the relationship existing between supply $s$ and price $p$ :

  $s=-6000p+32000$ .

Conclusion:

The relationship existing between supply $s$ and price $p$ :

  $s=-6000p+32000$ .

d.

To find the equilibrium point and to explain its meaning.

The equilibrium price of the widget is about $3.91$ dollars.

The equilibrium quantity of the widgets is about $8544$ .

It has the implication that, when the price of the widget is $3.91$ dollars the demand equals supply and the equilibrium quantity will be $8544$ .

Calculation:

The demand function is found in part b:

  $n=-1600p+14800$ .

The supply function is found in part c:

  $s=-6000p+32000$ .

Equilibrium occurs when the demand of the product equals its supply.

That is, the equilibrium occurs when $n=s$ .

Solve the equation $n=s$ for the equilibrium price $p$ :

  $\begin{array}{c}n=s\\ -1600p+14800=-6000p+32000\\ -1600p+6000p=32000-14800\\ 4400p=17200\\ p=\frac{17200}{4400}\\ p\approx 3.91\end{array}$ .

Thus, the equilibrium price of the widget is about $3.91$ dollars.

Substitute $p=3.91$ in any of the equations, say $n=-1600p+14800$ , to find the equilibrium quantity:

  $\begin{array}{l}n=-1600\cdot 3.91+14800\\ n=8544\end{array}$ .

Thus, the equilibrium quantity of widgets is $8544$ .

It has the implication that, when the price of the widget is $3.91$ dollars the demand equals supply and the equilibrium quantity will be $8544$ .

Conclusion:

The equilibrium price of the widget is about $3.91$ dollars.

The equilibrium quantity of the widgets is about $8544$ .

It has the implication that, when the price of the widget is $3.91$ dollars the demand equals supply and the equilibrium quantity will be $8544$ .