Linearize the following equations into the form y = mx + c: R = R0 exp(-λt), θ = A(t − B), P = P0 (1 - kh)ⁿ.

Question

ScanSolve AI Tutor · Accepted Answer

here are the linearized forms of the equations from the paper:

1. For the equation $R = R_0 \exp(-\lambda t)$ ()

Take the natural logarithm ( $\ln$ ) of both sides of the equation.
$\ln(R) = \ln(R_0 \exp(-\lambda t))$
Use logarithm properties $\ln(AB) = \ln(A) + \ln(B)$ and $\ln(e^x) = x$ .
$\ln(R) = \ln(R_0) + \ln(\exp(-\lambda t))$ $\ln(R) = \ln(R_0) - \lambda t$
Rearrange into the linear form $y = mx + c$ .
Let $y = \ln(R)$ and $x = t$ .
The linearized equation is:
$\boxed{\ln(R) = (-\lambda)t + \ln(R_0)}$
Here, the slope $m_{slope} = -\lambda$ and the y-intercept $c_{intercept} = \ln(R_0)$ .

2. For the equation $\theta = A(t - B)$ ()
Expand the right side of the equation.
$\theta = At - AB$
Rearrange into the linear form $y = mx + c$ .
Let $y = \theta$ and $x = t$ .
The linearized equation is:
$\boxed{\theta = At - AB}$
Here, the slope $m_{slope} = A$ and the y-intercept $c_{intercept} = -AB$ .

3. For the equation $P = P_0 (1 - kh)^n$ ()
Take the natural logarithm ( $\ln$ ) of both sides of the equation.
$\ln(P) = \ln(P_0 (1 - kh)^n)$
Use logarithm properties $\ln(AB) = \ln(A) + \ln(B)$ and $\ln(X^n) = n \ln(X)$ .
$\ln(P) = \ln(P_0) + \ln((1 - kh)^n)$ $\ln(P) = \ln(P_0) + n \ln(1 - kh)$
Rearrange into the linear form $y = mx + c$ .
Let $y = \ln(P)$ and $x = \ln(1 - kh)$ .
The linearized equation is:
$\boxed{\ln(P) = n \ln(1 - kh) + \ln(P_0)}$
Here, the slope $m_{slope} = n$ and the y-intercept $c_{intercept} = \ln(P_0)$ .

Linearize the following equations into the form y = mx + c: R = R0 exp(-λt), θ = A(t - B), P = P0 (1 - kh)ⁿ.