Answer :
The value of f(8.5) is 172.32.
Finding f(8.5) for the given exponential function:
We know that f(x) is an exponential function, but its exact form is unknown. However, we have two data points to work with:
f(1) = 8
f(5.5) = 69
The general form of an exponential function is:
f(x) = a * b^x
where:
a is the initial value (y-intercept)
b is the growth factor (base raised to the power of x)
1. Find the initial value (a)
Using the point f(1) = 8, we can plug in x = 1 and y = 8 into the general equation:
8 = a * b^1
8 = a * b
Since b is unknown, we cannot directly solve for a. However, we can keep this equation for later.
2. Find the growth factor (b)
Using the point f(5.5) = 69, we can plug in x = 5.5 and y = 69 into the general equation:
69 = a * b^5.5
Divide both sides by a (which we don't know yet, but it's not zero since f(1) is not zero):
69/a = b^5.5
Now, we can use the equation we obtained in step 1:
8 = a * b
Substitute b from this equation into the equation above:
69/(8) = (a * b)^5.5
Simplify:
69/8 = (8 * b)^5.5
8.625 = 32 * b^5.5
Take the fifth root of both sides:
(8.625)^(1/5.5) = (32 * b^5.5)^(1/5.5)
b ≈ 1.2247
3. Find f(8.5)
Now that we know b ≈ 1.2247, we can plug it back into the general equation along with the initial value from step 1 (a = 8):
f(8.5) = 8 * (1.2247)^(8.5)
≈ 172.32 (rounded to two decimal places)
Therefore, f(8.5) ≈ 172.32.
The answer is f(8.5) ≈ 172.32.
Finding f(8.5) for the given exponential function:
We know that f(x) is an exponential function, but its exact form is unknown. However, we have two data points to work with:
f(1) = 8
f(5.5) = 69
The general form of an exponential function is:
f(x) = a * b^x
where:
a is the initial value (y-intercept)
b is the growth factor (base raised to the power of x)
1. Find the initial value (a)
Using the point f(1) = 8, we can plug in x = 1 and y = 8 into the general equation:
8 = a * b^1
8 = a * b
Since b is unknown, we cannot directly solve for a. However, we can keep this equation for later.
2. Find the growth factor (b)
Using the point f(5.5) = 69, we can plug in x = 5.5 and y = 69 into the general equation:
69 = a * b^5.5
Divide both sides by a (which we don't know yet, but it's not zero since f(1) is not zero):
69/a = b^5.5
Now, we can use the equation we obtained in step 1:
8 = a * b
Substitute b from this equation into the equation above:
69/(8) = (a * b)^5.5
Simplify:
69/8 = (8 * b)^5.5
8.625 = 32 * b^5.5
Take the fifth root of both sides:
(8.625)^(1/5.5) = (32 * b^5.5)^(1/5.5)
b ≈ 1.2247
3. Find f(8.5)
Now that we know b ≈ 1.2247, we can plug it back into the general equation along with the initial value from step 1 (a = 8):
f(8.5) = 8 * (1.2247)^(8.5)
≈ 172.32 (rounded to two decimal places)
Therefore, f(8.5) ≈ 172.32.
