Find a formula for the exponential function of the form
A=f(t)=IB^t passing through the points (2.1, 21) and (4.9, 11)
A=f(t)
To find the specific formula for the exponential function A=f(t)=IB^t passing through the given points, we can use the two points to create a system of equations.
Let's start with the first point (2.1, 21):
f(2.1) = IB^(2.1) = 21
Now, let's consider the second point (4.9, 11):
f(4.9) = IB^(4.9) = 11
We can divide the second equation by the first equation to eliminate the constant I:
IB^(4.9)/IB^(2.1) = 11/21
Simplifying this expression, we get:
B^(4.9-2.1) = 11/21
B^2.8 = 11/21
Now, we can take the logarithm (base B) of both sides to solve for B:
log(B^2.8) = log(11/21)
2.8 log(B) = log(11/21)
log(B) = log(11/21) / 2.8
Finally, we can solve for B using the property of logarithms:
B = 10^(log(11/21) / 2.8)
Now that we have the value of B, we can substitute it back into one of our original equations (let's use the first point):
21 = I * B^(2.1)
Substituting the value of B we found:
21 = I * (10^(log(11/21) / 2.8))^(2.1)
Simplifying further:
21 = I * (10^(2.1 * (log(11/21) / 2.8)))
Now, solving for I:
I = 21 / (10^(2.1 * (log(11/21) / 2.8)))
Putting it all together, the formula for the exponential function of the form A=f(t)=IB^t passing through the points (2.1, 21) and (4.9, 11) is:
A = (21 / (10^(2.1 * (log(11/21) / 2.8)))) * (10^(log(11/21) / 2.8))^t
To find the formula for the exponential function of the form A = f(t) = IB^t passing through the points (2.1, 21) and (4.9, 11), we can use the property that two points uniquely determine a line (or curve).
Step 1: Substitute the coordinates of the first point (2.1, 21) into the equation A = IB^t:
21 = IB^(2.1)
Step 2: Substitute the coordinates of the second point (4.9, 11) into the equation A = IB^t:
11 = IB^(4.9)
Now we have a system of two equations with two unknowns (I and B). We can solve this system of equations simultaneously to find the values of I and B.
Step 3: Divide the two equations to eliminate the unknown I:
(21 / 11) = (IB^(2.1)) / (IB^(4.9))
Step 4: Simplify the equation:
(21 / 11) = B^(2.1 - 4.9)
Step 5: Simplify the exponent:
(21 / 11) = B^(-2.8)
Step 6: Take the logarithm (base B) on both sides of the equation to solve for the exponent:
log(B) [(21 / 11)] = log(B) [B^(-2.8)]
Step 7: Simplify the right side of the equation using the logarithmic property:
log(B) [(21 / 11)] = -2.8
Step 8: Solve for log(B) by isolating it:
log(B) = -2.8 / log(B) [(21 / 11)]
Step 9: Use the logarithmic property to convert log(B) into B:
B = 10^[-2.8 / log(B) [(21 / 11)]]
Now we have the value of B. To find the value of I, we can substitute B into one of the initial equations. Let's use the first equation (21 = IB^(2.1)).
Step 10: Substitute B into the equation:
21 = I (10^[-2.8 / log(B) [(21 / 11)]])^(2.1)
Step 11: Solve for I:
I = 21 / [10^[-2.8 / log(B) [(21 / 11)]])^(2.1)]
Now we have the values of I and B, which can be used to form the exponential function A = f(t) = IB^t:
A = (21 / [10^[-2.8 / log(B) [(21 / 11)]])^(2.1)]) * B^t
f(t) = a*b^t
a*b^2.1 = 21
a*b^4.9 = 11
divide to get
b^2.8 = 11/21
now finish it off