Question

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.

Answer 1

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:
\[
\boxed{250}
\]