The equation of the line of best fit is lnM = -0.12t + 4.67. Given that M = ab^t, find the value of b.
I got as far as -0.12t + 4.67 = lna + tlnb but I don't know where to go from there.
of with -0.12t + 4.67 = lna + tlnb
-0.12t + 4.67 - lna = tlnb
ln b = (-0.12t + 4.67 - lna)/t
b = e^ ( (-0.12t + 4.67 - lna)/t )
ln M = - 0.12 t + 4.67
M = e ^ ( - 0.12 t + 4.67 )
M = e ^ ( - 0.12 t ) * e ^ 4.67
M = [ e ^ ( - 0.12 ) ] ^ t * e ^ 4.67
M = 0.88692 ^ t * 106.69774
M = 106.69774 * 0.88692 ^ t
So:
M = a * b ^ t
a = 106.69774 , b = 0.88692
Excellent! I like that.
To find the value of b, we need to rearrange the equation lnM = -0.12t + 4.67 in terms of M = ab^t.
Starting from your equation: -0.12t + 4.67 = ln(a) + tln(b), let's isolate ln(b) by moving the other terms to the other side of the equation:
ln(b) = -0.12t + 4.67 - ln(a).
Now, to eliminate the natural logarithm, we can exponentiate both sides of the equation using the property e^(ln(x)) = x. In this case, since we have ln(b) on the left side, we will use base e:
e^(ln(b)) = e^(-0.12t + 4.67 - ln(a)).
Simplifying further, we get:
b = e^(-0.12t + 4.67 - ln(a)).
So the value of b is given by this equation, where t and a are the given variables in the equation M = ab^t.
Note: Remember to substitute the appropriate values for t and a when finding the numerical value of b.