To solve the exponential equation of the form $y = -98.4 \cdot e^{(0.044x - 5)} + 28.254716$ using the given points $(0, 27.5917)$, $(75.34961, 10)$, $(85.27772, 0)$, and $(57.31236, 20)$, we need to verify if these points satisfy the equation or make adjustments to the equation's parameters to ensure a proper fit.

Step 1: Understand the equation

The general form of the exponential equation is:

$y = a \cdot e^{(bx - c)} + d$

where:

• $a = -98.4$ (this represents the amplitude and is negative, which inverts the curve).
• $b = 0.044$ (this represents the growth/decay rate of the exponential function).
• $c = 5$ (this represents a horizontal shift).
• $d = 28.254716$ (this is a vertical shift, affecting the y-intercept).

Step 2: Substituting Points into the Equation

We'll check whether each of the given points satisfies the equation.

Point 1: $(0, 27.5917)$

Substitute $x = 0$ into the equation:

$y = -98.4 \cdot e^{(0.044 \cdot 0 - 5)} + 28.254716$

Simplifying:

$y = -98.4 \cdot e^{-5} + 28.254716$

Calculate $e^{-5}$:

$e^{-5} \approx 0.006737947$

Substituting:

$y \approx -98.4 \cdot 0.006737947 + 28.254716$

$y \approx -0.663876 + 28.254716 \approx 27.59084$

The calculated $y$ value is close to 27.5917, which means the point $(0, 27.5917)$ approximately satisfies the equation.

Point 2: $(75.34961, 10)$

Substitute $x = 75.34961$:

$y = -98.4 \cdot e^{(0.044 \cdot 75.34961 - 5)} + 28.254716$

Calculate the exponent:

$0.044 \cdot 75.34961 - 5 \approx 3.315785$

Then calculate $e^{3.315785}$:

$e^{3.315785} \approx 27.606$

Substitute:

$y = -98.4 \cdot 27.606 + 28.254716$

$y \approx -2717.5424 + 28.254716 \approx -2689.287684$

This $y$ value does not satisfy the equation well. Therefore, the initial equation likely needs adjustments to better fit the given points.

Step 3: Revising the Equation

Given that the equation doesn't fit the points well, we'll adjust the parameters $a$, $b$, $c$, and $d$ to fit the data points better.

We can start by using numerical methods (like regression) to fit the exponential curve to the data points. However, this involves iterative techniques beyond simple algebraic solutions.

Step 4: Use Graphical/Software Tools

Given that manual adjustments might be challenging, graphing software (like Excel, Python, or a graphing calculator) can be used to fine-tune the equation parameters to better fit the points.

Summary

The provided equation is close but does not perfectly match the given points. A better fit can be achieved by slightly adjusting the parameters $a$, $b$, $c$, and $d$ or using regression tools to optimize the fit to these points.

Would you like to explore the detailed steps of numerical regression, or do you have any questions about the process so far?

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Related Questions:

1. How can you find the best-fit exponential curve for a set of data points using regression?
2. What are the key differences between algebraic and graphical methods for solving equations?
3. How does changing the parameter $b$ in the equation affect the shape of the curve?
4. Why is it important to verify the fit of an equation against multiple points?
5. Can exponential equations be solved exactly for all points, or is approximation sometimes necessary?

Tip:

When working with exponential equations, it's often useful to graph the function and data points to visually assess the fit and guide adjustments to the equation's parameters.