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The image you uploaded contains three parts of a mathematics problem. Let's go through each one step by step.

### Part (a)
The problem asks to evaluate the summation:

\[
\sum_{x=1}^{6} x^2 \ln(1+x)^{\frac{1}{2}}
\]

You are required to use a non-programmable calculator and round the answer to **three significant figures**. Let's break it down:

1. **For each \( x \) from 1 to 6**, calculate \( x^2 \) and \( \ln(1 + x) \), and then take the square root of \( \ln(1 + x) \).
2. Multiply \( x^2 \) by \( \sqrt{\ln(1 + x)} \) for each value of \( x \).
3. Sum up the results.

Let's compute it manually step by step.

### Part (b)
The population of Mtwara follows the model:

\[
p(t) = p_0 e^{\lambda t}
\]

where \( \lambda = 0.034657 \). You are asked to find the time \( t \) when the population is **three times** the initial population \( p_0 \). To find \( t \):

1. Set up the equation:
   \[
   3p_0 = p_0 e^{\lambda t}
   \]

2. Simplify:
   \[
   3 = e^{\lambda t}
   \]

3. Take the natural logarithm of both sides:
   \[
   \ln(3) = \lambda t
   \]

4. Solve for \( t \):
   \[
   t = \frac{\ln(3)}{\lambda}
   \]

Use \( \lambda = 0.034657 \) to find \( t \).

### Part (c)
You are given a set of dataThe image you uploaded contains three parts of a mathematics problem. Let's go through each one step by step.

### Part (a)
The problem asks to evaluate the summation:

\[
\sum_{x=1}^{6} x^2 \ln(1+x)^{\frac{1}{2}}
\]

You are required to use a non-programmable calculator and round the answer to **three significant figures**. Let's break it down:

1. **For each \( x \) from 1 to 6**, calculate \( x^2 \) and \( \ln(1 + x) \), and then take the square root of \( \ln(1 + x) \).
2. Multiply \( x^2 \) by \( \sqrt{\ln(1 + x)} \) for each value of \( x \).
3. Sum up the results.

Let's compute it manually step by step.

### Part (b)
The population of Mtwara follows the model:

\[
p(t) = p_0 e^{\lambda t}
\]

where \( \lambda = 0.034657 \). You are asked to find the time \( t \) when the population is **three times** the initial population \( p_0 \). To find \( t \):

1. Set up the equation:
   \[
   3p_0 = p_0 e^{\lambda t}
   \]

2. Simplify:
   \[
   3 = e^{\lambda t}
   \]

3. Take the natural logarithm of both sides:
   \[
   \ln(3) = \lambda t
   \]

4. Solve for \( t \):
   \[
   t = \frac{\ln(3)}{\lambda}
   \]

Use \( \lambda = 0.034657 \) to find \( t \).

### Part (c)
You are given a set of 

\[
6, 6, 10, 10, 8, 8, 14, 11, 11, 14, 8, 10, 8, 14, 14, 14, 11, 11, 14, 11
\]

You are asked to find:
1. **Mean**: Sum the numbers and divide by the total count.
2. **Variance**: Use the formula for variance (either sample or population variance, depending on the context).
3. **Sample standard deviation**: Take the square root of the sample variance.

---

Let me know if you'd like me to solve a specific part, or provide any further clarifications!

Using a non-programmable calculator, evaluate the summation, find population growth, and compute mean, variance, and sample standard deviation for given data.

This problem involves evaluating a summation, modeling population growth with exponential functions, and calculating the mean, variance, and sample standard deviation of given data. Suitable for advanced high school or college-level math students.

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