A supplier of portable hair dryers will make x hundred units of hair dryers available in the market when the unit price is p = V16 + 4.2x dollars. Determine the producers' surplus if the market price is set at $10/unit. (Round your answer to two decimal places.) $ 50.24
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
![### Producer's Surplus Calculation for Portable Hair Dryers
**Problem Statement:**
A supplier of portable hair dryers will make \( x \) hundred units of hair dryers available in the market when the unit price is
\[ p = \sqrt{16 + 4.2x} \]
dollars. Determine the producer's surplus if the market price is set at \( \$10 \) per unit. Round your answer to two decimal places.
**Solution:**
1. **Find the quantity of hair dryers, \( x \), when the market price is $10/unit:**
Given the price function \( p = \sqrt{16 + 4.2x} \) and \( p = 10 \):
\[
10 = \sqrt{16 + 4.2x}
\]
Squaring both sides:
\[
100 = 16 + 4.2x
\]
\[
100 - 16 = 4.2x
\]
\[
84 = 4.2x
\]
\[
x = \frac{84}{4.2} \approx 20 \text{ hundred units} \text{ (or } 2,000 \text{ units)}
\]
2. **Determine the supply function:**
Rearrange the price function to solve for \( x \):
\[
p = \sqrt{16 + 4.2x}
\]
\[
p^2 = 16 + 4.2x
\]
\[
x = \frac{p^2 - 16}{4.2}
\]
3. **Calculate the total revenue (area of the rectangle formed by the market price and quantity supplied):**
\[
\text{Total Revenue} = 10 \times 2000 = 20000 \text{ dollars}
\]
4. **Calculate the producer's surplus (shaded area under the supply curve, above the price line):**
\[
\text{Producer's Surplus} = \text{Total Revenue - Cost}
\]
The cost is the area under the supply curve from \( x = 0 \) to \( x = 20 \):
\[
\text{Total Cost} = \int_0^{2000](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F456d1733-597b-47a7-bff4-881d2af765b2%2Fb91e444e-cdba-4af5-8978-35554da22a72%2Fz0clx6a.png&w=3840&q=75)
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