find 2 exponential functions with asymptote of y=80 , with points (0,176) (1,169) (3,156) (5,146) (20,101) (30,90) (60,81).one should have the base of "e"
I cannot find a function of the form
80 + a*e^(-kx)
which fits more than two of the points, since when x=0, you have
80+a=176, so a=96
and when x=1 you have
80+96e^-k = 169
e^-k = 89/96
k = ln(96/89) = 0.07571
80+96e^(-0.0757x) does not fit any of the other points.
To find exponential functions with an asymptote of y = 80 and given points, we can use the general form of an exponential function:
y = A * (base)^(kx) + C
Where:
- A represents the initial value or amplitude of the function.
- The base represents the base of the exponential function. In one of our cases, it will be the number e.
- k represents the growth or decay rate of the function.
- x represents the independent variable (in this case, time).
- C represents any vertical shift of the function.
We'll use the given points to form a system of equations that we can solve to find the specific values of A, k, and C for each exponential function.
For the first exponential function with the base e, we can write the equation using the point (0,176):
176 = A * e^(0 * k) + C
Since anything raised to the power of 0 equals 1, this simplifies to:
176 = A + C
Using the same logic, we can write two more equations by substituting the remaining points:
169 = A * e^(1 * k) + C
156 = A * e^(3 * k) + C
Now we have a system of three equations with three unknowns (A, k, and C). Solving this system will give us the values we need for the first exponential function with the base of e.
For the second exponential function, we can use the same process, but this time, the base will be different. However, we only need to find the appropriate values for A, k, and C. The base will remain e.
Let's solve this system of equations step-by-step:
1. To make the calculations easier, let's simplify the equations by substituting A + C from the first equation:
169 = A * e^k + (176 - A)
156 = A * e^(3k) + (176 - A)
2. Subtract the second equation from the first equation:
169 - 156 = A * e^k - A * e^(3k) + A - (176 - A)
13 = A * (e^k - e^(3k)) + A - 176
3. Combine like terms:
13 = A * (e^k - e^(3k)) - 176
4. Rearrange the equation:
A * (e^k - e^(3k)) = 13 + 176
A * (e^k - e^(3k)) = 189
5. Divide both sides by (e^k - e^(3k)):
A = 189 / (e^k - e^(3k))
At this point, we have obtained an expression for A in terms of k. By substituting this value back into any of the initial equations, we can find the specific values for A, k, and C.
Please note that solving the above system of equations may involve numerical methods like using software or calculators with equation-solving capabilities.