https://file.aiquickdraw.com/mathbot/file/172736621927286h1rf3e.jpg

Let's analyze and solve the problem step by step based on the image.

### Given:
The equation for the average price \( P \) of a second-hand car in relation to its age \( t \) is given as:
\[
P = \frac{50}{a + b e^{t/4}}
\]
where \( a \) and \( b \) are constants. The data provided is:

\[
\begin{array}{|c|c|}
\hline
t \ (\text{yrs}) & P \ (\text{lac}) \\
\hline
2 & 20.50 \\
4 & 17.25 \\
6 & 14.50 \\
8 & 11.75 \\
\hline
\end{array}
\]

### Part i) Estimate the values of \( a \) and \( b \) (rounded to 3 significant figures).

To find \( a \) and \( b \), we can use the non-linear least squares fitting method since this is a non-linear equation. The MATLAB function `lsqcurvefit` can be used to solve this, but we can manually solve this using optimization or numerical methods in a Python environment as well.

### Part ii) Estimate the value of a car of this type that is 10 years old and estimate the original price.

Once we find \( a \) and \( b \), we'll substitute \( t = 10 \) into the equation to get the price after 10 years. The original price (new car price) corresponds to \( t = 0 \).

### Part iii) Using MATLAB function `lsqcurvefit`.

In MATLAB, the function `lsqcurvefit` requires the following syntax:
```matlab
a0 = [initial_guess_for_a, initial_guess_for_b];
[x,resnorm] = lsqcurvefit(@(x,t) 50./(x(1) + x(2)*exp(t/4)), a0, tdata, Pdata);
```
Where:
- `a0` is the initial guess for \( a \) and \( b \),
- `tdata` is the vector of ages \([2, 4, 6, 8]\),
- `Pdata` is the corresponding price vector \([20.50, 17.25, 14.50, 11.75]\).

#### Let's proceed by solving parts i) and ii) numerically first.

I'll estimate the values of \( a \) and \( b \) based on the given data using Python.### Results:

i) The estimated values of the constants are:
- \( a = 1.93 \) (rounded to 3 significant figures),
- \( b = 0.330 \) (rounded to 3 significant figures).

ii) Using the estimated values of \( a \) and \( b \):
- The estimated price for a car that is 10 years old is approximately **8.40 lac**,
- The estimated original price (when \( t = 0 \)) is approximately **22.11 lac**.

### MATLAB code for Part iii:

```matlab
% Define the function
P_model = @(x, t) 50 ./ (x(1) + x(2) * exp(t / 4));

% Given data
t_data = [2, 4, 6, 8];
P_data = [20.50, 17.25, 14.50, 11.75];

% Initial guess for a and b
a0 = [1, 1];

% Curve fitting using lsqcurvefit
[a_fit, resnorm] = lsqcurvefit(P_model, a0, t_data, P_data);

% Display the results
disp('Estimated values of a and b:');
disp(a_fit);
```

This MATLAB code will give similar estimates for \( a \) and \( b \).

Would you like further details or help with any specific part? 

Here are 5 related questions:
1. How does non-linear curve fitting work in general?
2. What are other possible models for estimating depreciation?
3. Can this model be applied to other items besides cars?
4. How would you interpret the original price in real-world terms?
5. How does changing \( t \) affect the price estimation?

**Tip:** Always provide good initial guesses for \( a \) and \( b \) when using optimization methods like `lsqcurvefit` for faster convergence.

Average price, P, of a certain type of second-hand car is believed to be related to its age, t years, by an equation of the form P = 50 / (a + b * exp(t/4)), where a and b are constants. The following data for the average price in Taka (lac) is provided for different ages (years): t (yrs) = [2, 4, 6, 8], P (lac) = [20.50, 17.25, 14.50, 11.75]. Estimate the values of a and b, estimate the value of the car at 10 years old, and use MATLAB function lsqcurvefit to fit the general form.

Learn how to estimate the constants a and b in the exponential model for second-hand car prices using MATLAB's lsqcurvefit function. This problem demonstrates key concepts of non-linear curve fitting and exponential decay, with provided data for various car ages (years 2-8) and their corresponding prices (in lac Taka).

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