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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.

1 answer

To find the price at which the quantity demanded equals the quantity supplied, we need to solve the two equations you've provided for \( p \) and \( q \):

  1. Demand: \[ p + 3q = 355 \]

  2. Supply: \[ p - 8q = -30 \]

We can rearrange and solve these equations simultaneously. Let's express \( p \) in terms of \( q \) from both equations.

From the demand equation: \[ p = 355 - 3q \]

From the supply equation: \[ p = 8q - 30 \]

Now, we can set these two expressions for \( p \) equal to each other: \[ 355 - 3q = 8q - 30 \]

Next, we solve for \( q \):

  1. Combine like terms: \[ 355 + 30 = 8q + 3q \] \[ 385 = 11q \]

  2. Divide both sides by 11: \[ q = \frac{385}{11} = 35 \]

Now that we have \( q \), we can find \( p \) using either equation. Let's use the demand equation: \[ p = 355 - 3(35) \] \[ p = 355 - 105 \] \[ p = 250 \]

Thus, the price at which the quantity demanded equals the quantity supplied is: \[ \boxed{250} \]