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The exponential decay graph shows the expected depreciation for a new boat, selling for $3500, over 10 years.
A coordinate graph is shown. The horizontal axis extends from 0 to 12 years. The vertical axis extends from 0 to 9500 with an axis label of 'Value' in dollars. A curve is graphed which begins at 0 comma 3500, then decreases passing through approximately 1 comma 2700
Write an exponential function for the graph. Use the function to find the value of the boat after 9.5 years.
To find the exponential function for the graph, we need to determine the equation in the form y = ab^x, where y is the value of the boat, x is the number of years, a is the initial value of the boat, and b is the decay factor.
From the given information, we know that the boat starts at a value of $3500 when x = 0. Therefore, a = 3500.
Also, we know that the boat passes through the point (1, 2700). Plugging these values into the exponential function equation, we get:
2700 = 3500 * b^1
Simplifying, we divide both sides by 3500:
b = 2700/3500
b ≈ 0.7714
So, the exponential function for the graph is:
y = 3500 * 0.7714^x
Now, we can find the value of the boat after 9.5 years by plugging x = 9.5 into the equation:
y = 3500 * 0.7714^9.5
y ≈ 2079.79
Therefore, the value of the boat after 9.5 years is approximately $2079.79.