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.
Please answer both questions and be detailed, thank you in advance!
![**Problem**: Sketch the region and determine the area of the region in the first quadrant bounded by \( f(x) = \frac{6 - x}{x} \), the x-axis, and \( x = 2 \).
### Explanation:
To solve this problem, we will follow these steps:
1. **Sketch the Graph**:
- Plot the function \( f(x) = \frac{6 - x}{x} \) in the first quadrant. The function can be simplified to \( f(x) = \frac{6}{x} - 1 \).
- Identify the asymptotes and intercepts.
- The vertical asymptote is at \( x = 0 \).
- The horizontal asymptote is at \( y = -1 \).
- Plot the x-intercept by setting \( f(x) = 0 \). Solving \( \frac{6 - x}{x} = 0 \) gives \( x = 6 \).
2. **Determine the Bounded Region**:
- The region is bounded by \( f(x) \), the x-axis (y = 0), and the vertical line \( x = 2 \).
3. **Set Up the Integral**:
- The area \( A \) of the region can be found using the definite integral:
\[
A = \int_{a}^{b} f(x) \, dx
\]
where \( a = 0 \) and \( b = 2 \).
4. **Calculate the Area**:
- Integrate the function from \( x = 0 \) to \( x = 2 \):
\[
A = \int_{0}^{2} \left( \frac{6 - x}{x} \right) \, dx
\]
- Split the integral:
\[
A = \int_{0}^{2} \left( \frac{6}{x} - 1 \right) \, dx = \int_{0}^{2} \frac{6}{x} \, dx - \int_{0}^{2} 1 \, dx
\]
- Compute the integrals separately:
\[
\int_{0}^{2} \frac{6}{x} \, dx - \int_{0}^{2} 1 \, dx =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a908816-dcaf-4aa4-8dd6-7dd2f0aea84f%2F19e5df83-87eb-42c9-b563-8ae8b9d3ebb9%2F64u5vy8.png&w=3840&q=75)
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