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The demand for a certain product is given by pplus3qequals355​, and the supply for this product is given by pminus8qequalsnegative 30​, where p is the price and q is the number of products. Complete parts​ (a) and​ (b) below.

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Part 1
a. Find the price at which the quantity demanded equals the quantity supplied.
​$
  
250
Part 2
b. Find the equilibrium quantity.

1 answer

To find the price at which the quantity demanded equals the quantity supplied, let's solve the equations given:

  1. Demand equation: \[ p + 3q = 355 \] (Rearranging gives: \( p = 355 - 3q \))

  2. Supply equation: \[ p - 8q = -30 \] (Rearranging gives: \( p = 8q - 30 \))

Now, we set the two equations for \( p \) equal to each other to find the equilibrium quantity \( q \):

\[ 355 - 3q = 8q - 30 \]

Now, combine like terms:

\[ 355 + 30 = 8q + 3q \] \[ 385 = 11q \]

Now, solving for \( q \):

\[ q = \frac{385}{11} = 35 \]

Now that we have the equilibrium quantity, we can substitute this value of \( q \) back into one of the equations to find the equilibrium price \( p \). Using the demand equation:

\[ p = 355 - 3(35) \] \[ p = 355 - 105 \] \[ p = 250 \]

Thus:

  • The price at which quantity demanded equals quantity supplied is \( $250 \).
  • The equilibrium quantity is \( 35 \) products.

Final results: a. Price: \( $250 \)
b. Equilibrium quantity: \( q = 35 \)