Find a formula for the exponential function that satisfies h(2)=22 and h(4)=14.08
if h(x) = ae^bx,
just plug in your values:
22 = ae^2b
14.08 = ae^4b
divide one by the other, and the a's cancel
14.08/22 = .64 = e^2b
e^b = .8
22 = a*.64
a = 34.375
h(x) = 34.375 e^bx = 34.375 * 0.8^x
Well, I don't want to ruffle any mathematician's feathers, but it seems like we have a bit of an exponential identity crisis here. You see, exponential functions usually have this amazing ability to grow and expand, like a jack-in-the-box on steroids. However, the data you provided seems to be going in the opposite direction. It looks like we have a case of exponential shrinking here!
So, let's put on our imaginary shrinking capes and solve this mystery. We know that the general form of an exponential function is h(x) = a * b^x. Now we just need to determine the values for a and b based on our data.
We can start by plugging in our first data point, h(2) = 22. This gives us the equation 22 = a * b^2. Similarly, plugging in the second data point, h(4) = 14.08, gives us 14.08 = a * b^4.
Now it's time to put on our detective hats and solve this case. We could use advanced mathematical techniques, but why not make it a bit more entertaining? Let's divide the second equation by the first equation to create a new equation: (14.08/22) = (a * b^4) / (a * b^2).
Clearing out the common factors, we get 0.64 = b^2. Taking the square root of both sides, we find b = 0.8 (approximately).
Now that we've found b, we can substitute it back into one of the original equations (let's use the first one) to find a. Plugging in 0.8 for b and solving for a, we get 22 = a * (0.8)^2, which gives us a = 41.25 (approximately).
So, after some clown-like detective work, our formula for the exponential function that satisfies the given conditions is h(x) = 41.25 * (0.8)^x. Now you have an equation that captures both the shrinking and the humor of this situation. Happy math-ing!
To find a formula for the exponential function that satisfies the given conditions, let's assume the general form of an exponential function is h(x) = a * b^x, where a and b are constants to be determined.
We are given two conditions:
1. h(2) = 22
2. h(4) = 14.08
Using condition 1, substitute x = 2 and h(x) = 22 into the general form of the exponential function:
22 = a * b^2 ----(1)
Similarly, using condition 2, substitute x = 4 and h(x) = 14.08 into the general form:
14.08 = a * b^4 ----(2)
Now, we can solve these two equations simultaneously to find the values of a and b.
Divide equation (2) by equation (1):
(14.08 / 22) = (a * b^4) / (a * b^2)
Simplify:
0.64 = b^2
Take the square root of both sides:
√(0.64) = √(b^2)
0.8 = b
Now substitute the value of b into equation (1):
22 = a * (0.8)^2
22 = a * 0.64
Divide both sides by 0.64:
a = 22 / 0.64
a ≈ 34.375
Therefore, the formula for the exponential function that satisfies h(2) = 22 and h(4) = 14.08 is:
h(x) = 34.375 * (0.8)^x
To find a formula for the exponential function that satisfies h(2)=22 and h(4)=14.08, we can use the general form of an exponential function:
h(x) = a * b^x
where 'a' is the initial value or the value of the function when x = 0, and 'b' is the base of the exponential function.
Using the given information, we have two equations:
h(2) = a * b^2 = 22 ---(1)
h(4) = a * b^4 = 14.08 ---(2)
We can solve this system of equations to find the values of 'a' and 'b'.
Dividing equation (2) by equation (1), we get:
(h(4) / h(2)) = (a * b^4) / (a * b^2)
14.08 / 22 = b^4 / b^2
0.64 = b^2
Taking the square root of both sides, we get:
b = ±√0.64
Since we are dealing with exponential growth, we take the positive square root:
b = √0.64 = 0.8
Now, substitute the value of 'b' into either equation (1) or (2) to find 'a'. Let's use equation (1):
a * (0.8)^2 = 22
0.64a = 22
a ≈ 34.375
Therefore, the formula for the exponential function that satisfies h(2)=22 and h(4)=14.08 is:
h(x) = 34.375 * 0.8^x