Give step-by-step solution with explanation and final answer:Problem 16. The force on a sailboat mast can be represented by the following function:
" 3
F=f 00 (3) ea:

where = = the elevation abov

Question

Give step-by-step solution with explanation and final answer:Problem 16. The force on a sailboat mast can be represented by the following function:
" 3
F=f 00 (3) ea:

where = = the elevation above the deck and H = the height of the mast. For this problem
compute the force Fon the sail for a mask height H = 30.
(a) Using the Romberg integration algorithm to a tolerance of ¢, = 0.5%. For compu-
tational convenience, the first column of the trapezoidal approximations are included to
a sufficient level of accuracy in the table below. That is, you will not need to compute
beyond the level of accuracy of Ry. Take = Ry; — Rig1l/ Ry
(b) Use MATLAB's built-in integral function with a long format.
(c) Take the exact (true) value to be the value given by MATLAB's built-in integral
function in part (b). Compute the absolute true relative error |e for Ri

[7 [ ou [3] [ow [ Of) ]

[1] 31800501104 X xX X

[2 [100173121965 | Ris X 3

[3 [132058477875 | Raa Ria x |

[4 1435.00704356 | Rua Ris Ria

VIPSolutions · Accepted Answer

Let's break down the problem step by step.

### Given:
The force $ F $ is given by:
$$
F = \int_{0}^{H} 200 \left( \frac{z}{5+z} \right) e^{-2z/H} dz
$$
where $ H = 30 $.

---

## (a) **Romberg Integration Algorithm**

**Step 1: Define the function**

Let:
$$
f(z) = 200 \frac{z}{5+z} e^{-2z/30}
$$

**Step 2: Romberg Table**

Given values in the first column (trapezoidal approximations):

| i | $ O(h_i^2) $        | $ O(h_i^4) $ | $ O(h_i^6) $ | $ O(h_i^8) $ |
|---|-----------------------|----------------|----------------|----------------|
| 1 | 348.0501404           | X              | X              | X              |
| 2 | 1001.7312965          | $ R_{2,2} $  | X              | X              |
| 3 | 1320.58478775         | $ R_{3,2} $  | $ R_{3,3} $  | X              |
| 4 | 1435.0704356          | $ R_{4,2} $  | $ R_{4,3} $  | $ R_{4,4} $  |

**Romberg Integration Formula:**
$$
R_{k,j} = R_{k,j-1} + \frac{R_{k,j-1} - R_{k-1,j-1}}{4^{j-1} - 1}
$$

---

### **Let's compute the required entries:**

#### **First, copy the given values:**

- $ R_{1,1} = 348.0501404 $
- $ R_{2,1} = 1001.7312965 $
- $ R_{3,1} = 1320.58478775 $
- $ R_{4,1} = 1435.0704356 $

---

#### **Compute $ R_{2,2} $:**

$$
R_{2,2} = R_{2,1} + \frac{R_{2,1} - R_{1,1}}{4-1}
= 1001.7312965 + \frac{1001.7312965 - 348.0501404}{3}
$$

$$
= 1001.7312965 + \frac{653.6811561}{3}
= 1001.7312965 + 217.8937187
= \boxed{1219.6250152}
$$

---

#### **Compute $ R_{3,2} $:**

$$
R_{3,2} = R_{3,1} + \frac{R_{3,1} - R_{2,1}}{4-1}
= 1320.58478775 + \frac{1320.58478775 - 1001.7312965}{3}
$$

$$
= 1320.58478775 + \frac{318.85349125}{3}
= 1320.58478775 + 106.28449708
= \boxed{1426.86928483}
$$

---

#### **Compute $ R_{3,3} $:**

$$
R_{3,3} = R_{3,2} + \frac{R_{3,2} - R_{2,2}}{16-1}
= 1426.86928483 + \frac{1426.86928483 - 1219.6250152}{15}
$$

$$
= 1426.86928483 + \frac{207.24426963}{15}
= 1426.86928483 + 13.81628464
= \boxed{1440.68556947}
$$

---

#### **Compute $ R_{4,2} $:**

$$
R_{4,2} = R_{4,1} + \frac{R_{4,1} - R_{3,1}}{4-1}
= 1435.0704356 + \frac{1435.0704356 - 1320.58478775}{3}
$$

$$
= 1435.0704356 + \frac{114.48564785}{3}
= 1435.0704356 + 38.16188262
= \boxed{1473.23231822}
$$

---

#### **Compute $ R_{4,3} $:**

$$
R_{4,3} = R_{4,2} + \frac{R_{4,2} - R_{3,2}}{16-1}
= 1473.23231822 + \frac{1473.23231822 - 1426.86928483}{15}
$$

$$
= 1473.23231822 + \frac{46.36303339}{15}
= 1473.23231822 + 3.09086889
= \boxed{1476.32318711}
$$

---

#### **Compute $ R_{4,4} $:**

$$
R_{4,4} = R_{4,3} + \frac{R_{4,3} - R_{3,3}}{64-1}
= 1476.32318711 + \frac{1476.32318711 - 1440.68556947}{63}
$$

$$
= 1476.32318711 + \frac{35.63761764}{63}
= 1476.32318711 + 0.5656771
= \boxed{1476.88886421}
$$

---

### **Table with all values:**

| i | $ O(h_i^2) $        | $ O(h_i^4) $     | $ O(h_i^6) $     | $ O(h_i^8) $     |
|---|-----------------------|--------------------|--------------------|--------------------|
| 1 | 348.0501404           | X                  | X                  | X                  |
| 2 | 1001.7312965          | 1219.6250152       | X                  | X                  |
| 3 | 1320.58478775         | 1426.86928483      | 1440.68556947      | X                  |
| 4 | 1435.0704356          | 1473.23231822      | 1476.32318711      | 1476.88886421      |

---

#### **Check if the tolerance is met for $ R_{4,4} $:**

$$
\epsilon_a = \left| \frac{R_{4,4} - R_{3,3}}{R_{4,4}} \right| \times 100\%
$$

$$
= \left| \frac{1476.88886421 - 1440.68556947}{1476.88886421} \right| \times 100\%
= \left| \frac{36.20329474}{1476.88886421} \right| \times 100\%
= 2.45\%
$$

This is slightly above the 0.5% tolerance, so you would continue, but for the scope of the table and the problem, these are the values you fill in.

---

## (b) **MATLAB's `integral` function**

You would use:
```matlab
f = @(z) 200.*(z./(5+z)).*exp(-2*z/30);
F_exact = integral(f, 0, 30, 'AbsTol', 1e-12, 'RelTol', 1e-12);
```
Suppose this gives: $ F_{\text{exact}} = 1477.0 $ (for illustration; actual value may differ slightly)

---

## (c) **Absolute true relative error for $ R_{4,4} $**

$$
\epsilon_t = \left| \frac{F_{\text{exact}} - R_{4,4}}{F_{\text{exact}}} \right|
$$

Assume $ F_{\text{exact}} = 1477.0 $:

$$
\epsilon_t = \left| \frac{1477.0 - 1476.88886421}{1477.0} \right|
= \left| \frac{0.11113579}{1477.0} \right|
= 7.53 \times 10^{-5}
$$

---

## **Final Answers (Summary)**

### (a) **Romberg Table:**
| i | $ O(h_i^2) $        | $ O(h_i^4) $     | $ O(h_i^6) $     | $ O(h_i^8) $     |
|---|-----------------------|--------------------|--------------------|--------------------|
| 1 | 348.0501404           | X                  | X                  | X                  |
| 2 | 1001.7312965          | 1219.6250152       | X                  | X                  |
| 3 | 1320.58478775         | 1426.86928483      | 1440.68556947      | X                  |
| 4 | 1435.0704356          | 1473.23231822      | 1476.32318711      | 1476.88

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