Sketch the region enclosed by the curves given below.
y=7cosx
y=7-(14x/pi)

To sketch the region enclosed by the curves \( y = 7\cos(x) \) and \( y = 7 - \frac{14x}{\pi} \), we can follow these steps:

### 1. **Identify the curves:**
   - \( y = 7\cos(x) \) is a cosine wave with an amplitude of 7, oscillating between \( y = -7 \) and \( y = 7 \).
   - \( y = 7 - \frac{14x}{\pi} \) is a linear equation with a y-intercept of 7 and a slope of \( -\frac{14}{\pi} \).

### 2. **Determine the points of intersection:**
   - To find the points where the curves intersect, set the equations equal to each other:
     \[
     7\cos(x) = 7 - \frac{14x}{\pi}
     \]
     Simplifying:
     \[
     \cos(x) = 1 - \frac{2x}{\pi}
     \]
   - This equation needs to be solved numerically or graphically to find the points of intersection.

### 3. **Sketch the curves:**
   - **\( y = 7\cos(x) \)**:
     - This curve oscillates with a period of \( 2\pi \), reaching maximum at \( x = 0 \), \( y = 7 \) and minimum at \( x = \pi \), \( y = -7 \).
   - **\( y = 7 - \frac{14x}{\pi} \)**:
     - This line starts at \( (0, 7) \) and decreases linearly with a negative slope.

### 4. **Plot the intersection points and enclose the region:**
   - These points will tell us where the region of interest lies between the two curves.

Here is a sketch of the enclosed region:

1. **Draw the cosine wave** \( y = 7\cos(x) \).
2. **Plot the straight line** \( y = 7 - \frac{14x}{\pi} \).
3. **Mark the intersection points** and shade the region enclosed between the two curves.

Let me know if you want to see a graphical sketch, or need further explanation. 

Would you like the intersection points calculated or have any other questions?

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**Relative Questions:**
1. How do you calculate the area of the region enclosed between the two curves?
2. Can you solve the equation \( \cos(x) = 1 - \frac{2x}{\pi} \) numerically?
3. How does the period of the cosine function affect the intersection points?
4. What is the slope of the line \( y = 7 - \frac{14x}{\pi} \) and how does it relate to the cosine function?
5. How would the sketch change if the amplitude of the cosine function was different?

**Tip:** When sketching regions between curves, it's helpful to first determine where the curves intersect, as this tells you the limits of integration or the bounds of the enclosed area.

Sketch the region enclosed by the curves y=7cos(x) and y=7-(14x/pi).

Learn how to sketch the region enclosed by the trigonometric curve y = 7cos(x) and the linear equation y = 7 - (14x/pi). This problem involves understanding the intersection of functions, graphical analysis, and calculus concepts. Suitable for students in Grades 11-12.

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