Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem:**
The velocity (in feet/second) of a projectile \( t \) seconds after it is launched from a height of 10 feet is given by \( v(t) = -15.6t + 147 \). Approximate its height after 3 seconds using 6 rectangles.
**Solution:**
First, we need to use the given velocity function to approximate the height of the projectile after 3 seconds. The velocity function is \( v(t) = -15.6t + 147 \).
To approximate the height after 3 seconds using 6 rectangles, we need to use the concept of Riemann sums where the number of rectangles (n) is 6.
Below are the steps to proceed:
1. **Divide the interval \([0, 3]\) into 6 subintervals:**
- The width of each subinterval, \(\Delta t\) = \(\frac{3 - 0}{6} = 0.5\) seconds.
2. **Calculate the midpoints of each subinterval:**
- For each subinterval, the midpoint, \( t_i \), can be calculated as follows:
- \( t_1 = \frac{0 + 0.5}{2} = 0.25 \)
- \( t_2 = \frac{0.5 + 1.0}{2} = 0.75 \)
- \( t_3 = \frac{1.0 + 1.5}{2} = 1.25 \)
- \( t_4 = \frac{1.5 + 2.0}{2} = 1.75 \)
- \( t_5 = \frac{2.0 + 2.5}{2} = 2.25 \)
- \( t_6 = \frac{2.5 + 3.0}{2} = 2.75 \)
3. **Evaluate the velocity function at each midpoint:**
- \( v(t_1) = v(0.25) = -15.6(0.25) + 147 \)
- \( v(t_2) = v(0.75) = -15.6(0.75) + 147 \)
- \( v(t_3) = v(1.25) = -15.6](https://content.bartleby.com/qna-images/question/8f26cfff-a5a0-4f1a-b4dd-ba875949e336/560fa4a9-a94b-4e9c-8030-6009a67f262d/dznk86a_processed.jpeg)
Transcribed Image Text:**Problem:**
The velocity (in feet/second) of a projectile \( t \) seconds after it is launched from a height of 10 feet is given by \( v(t) = -15.6t + 147 \). Approximate its height after 3 seconds using 6 rectangles.
**Solution:**
First, we need to use the given velocity function to approximate the height of the projectile after 3 seconds. The velocity function is \( v(t) = -15.6t + 147 \).
To approximate the height after 3 seconds using 6 rectangles, we need to use the concept of Riemann sums where the number of rectangles (n) is 6.
Below are the steps to proceed:
1. **Divide the interval \([0, 3]\) into 6 subintervals:**
- The width of each subinterval, \(\Delta t\) = \(\frac{3 - 0}{6} = 0.5\) seconds.
2. **Calculate the midpoints of each subinterval:**
- For each subinterval, the midpoint, \( t_i \), can be calculated as follows:
- \( t_1 = \frac{0 + 0.5}{2} = 0.25 \)
- \( t_2 = \frac{0.5 + 1.0}{2} = 0.75 \)
- \( t_3 = \frac{1.0 + 1.5}{2} = 1.25 \)
- \( t_4 = \frac{1.5 + 2.0}{2} = 1.75 \)
- \( t_5 = \frac{2.0 + 2.5}{2} = 2.25 \)
- \( t_6 = \frac{2.5 + 3.0}{2} = 2.75 \)
3. **Evaluate the velocity function at each midpoint:**
- \( v(t_1) = v(0.25) = -15.6(0.25) + 147 \)
- \( v(t_2) = v(0.75) = -15.6(0.75) + 147 \)
- \( v(t_3) = v(1.25) = -15.6
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