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Chapter 3: Problem 27

Find the exponential model that fits the points shown in the graph or table. $$\begin{array}{|c|c|c|} \hline x & 0 & 4 \\ \hline y & 5 & 1 \\ \hline \end{array}$$

Short Answer

Expert verified
The exponential model that fits the given points is \(y = 5 * (\sqrt[4]{1/5})^{x}\).

Step by step solution

01

Formulate the Exponential Equation for Both Points

We can write the exponential model as \(y = ab^{x}\). From the given table, we have two points, (0,5) and (4,1). We substitute these coordinates into our exponential model to set up two equations. From (0,5), the equation becomes \(5 = ab^{0}\), and from (4,1), the equation becomes \(1 = ab^{4}\).
02

Solve the First Equation

Right away, from \(5 = ab^{0}\) we can solve for a because any number (except zero) to the power of zero is 1. Therefore, this gives us \(a = 5\).
03

Substitute the Value of a into the Second Equation

Using the value of a from Step 2, we can solve the second equation \(1 = ab^{4}\) which simplifies to \(1 = 5b^{4}\). By re-arranging this equation, we can solve for b. Divide both sides by 5: \(b^{4} = 1/5\). Then take the fourth root of both sides to get \(b = \sqrt[4]{1/5}\).
04

Verify the Solution

To verify that a = 5 and b = \(\sqrt[4]{1/5}\) satisfy the exponential model, you substitute the values of a and b back into the model formula \(y = 5 * (\sqrt[4]{1/5})^{x}\) and check if it gives the y-values according to the given x-values in the table.

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