Answer :
The formula for f(x) an exponential function is f(x) =1760(2.71828)^(0.01732x).
Let f(x) be an exponential function such that f(-1) = 220 and f(40) = 704.
The exponential function is of the form
f(x) = abx,
where a is the initial value, b is the base, and x is the exponent. We can use these values to solve for a and b.
Let's first solve for b. We have:
f(40) = ab
40 = 704
Dividing both sides by 220, we get:
f(40) / f(-1) = (ab40) / (ab-1)
= 704 / 220
Simplifying the left side, we get:
b41 = 704 / 220
= 8
Dividing both sides by b, we get:
b40 = 8 / b
Now, we can substitute this into our equation for f(40):
f(40) = ab
40 = a(8 / b)
= 8a / b
Next, we can solve for a using the value of f(-1):
f(-1) = ab-1 = 220
Substituting in b40 = 8 / b, we get:
ab-1 = 220
a(8 / b)-1 = 220
a(b / 8) = 220
Multiplying both sides by 8/b, we get:
ab = 220(8/b)
= 1760/b
Now, we can substitute both of these values into our equation for f(x):
f(x) = ab
x = (1760/b)
(bx) = 1760(x / 40)
So the formula for f(x) is
f(x) = 1760(2.71828)^(0.01732x).
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