This equation takes the shape of a straight line in x-y coordinates, y = b + xm. The slope m equals (elt-1) and the intercept equals (D/S) 0. This is known as the "starting ratio." The value for elt is obtained by dividing the length of the chronology by its number of segments.

The calculation of the starting point starts with the determination of the divergence time. For this purpose, an outgroup is needed that is sufficiently close to both species being analyzed. The closer it is related to both species, the more reliable the result will be. In our example, we choose **the bird genus** Alticola as the outgroup. The divergence time can then be calculated using the formula t = c(el-1)/(xm - el+1), where el is the number of events in the lower part of the chronology and xm is the number of synonymous substitutions per site between **the two genes** used for the calculation. In our case, there are five events in the lower part of the chronology and the average distance between two consecutive points is about 15 kilobases (kb). This results in a divergence time of about 75 million years ago (Mya).

Isoclines are lines in the (x, y) plane generated by making f (x, y) equal to a constant in an equation of the form y' = f (x, y). This results in a sequence of lines (for various constants) along which the solution curves have the same gradient. Isoclines can be used to understand how changes in one variable affect other variables.

When calculating isoclines, it's useful to know that the x- and y-coordinates of the points where the line meets the surface f (x, y)=c are given by solving the equation f (x, y)=c for x or y. If necessary, solve **this equation** using the fact that f (x, y) is linear in both x and y; for example, if f is linear in y, then the line must pass through the point (x, y), where x solves f (x, y)=c, because f(x, y)=f for all x or y. Similarly, if f is linear in x, the line must pass through the point (x, y), where y solves f (x, y)=c.

In general, isoclines can be calculated by finding the values x and y such that f (x, y)=k where k is a constant. The line passing through these points has the same gradient as the curve defined by f (x, y).

The isocost line represents a firm's budget limitation while purchasing production inputs. To compute **the isocost line** for a company, first solve **the total cost equation**, TC = (W x L) + (r x K), and then solve for **K. W = wages**, L = labor, r = rent (what you pay for capital usage), and K = capital. Total capital must be less than or equal to the isocost line.

The isocost line also represents the maximum amount of debt that a company can incur before entering into financial difficulty. If a company's net income falls short of its isocost goal, it will need to cut back on **its manufacturing operations** or close down completely.

When a company files for bankruptcy, the court will examine its overall financial situation to determine if there is any hope of it being restored to health. If the company has exceeded its isocost line, it means that it was already in poor financial condition even before going bankrupt. No one will want to invest in such a company because they cannot be trusted not to exceed their budget limitations.

Isocost calculations are useful for managers to identify problems with production processes before they cause financial harm. For example, if a company's isocost line is reached before its profit line, this would indicate that management needs to find ways to reduce costs without reducing output too much. Financial trouble may still lie ahead but at least managers know what problems need to be addressed first.

D = D0 + D* as a consequence, D = D0 + N (e l t-1) alternatively, for small l t, D = D0 + N l t This is the fundamental radioactive decay equation that is used to calculate the ages of rocks, minerals, and isotopes. D and N can be measured, and l has been found experimentally for **almost all known unstable nuclides**. Thus, radioactive dating is one of **the most reliable methods** in geology for determining the age of objects with respect to time.

Radioactive decay is the loss of energy from an atom or molecule after its nucleus has undergone a nuclear transition. Atoms decay via emission of radiation including alpha particles, beta electrons, gamma rays, and neutrons. The rate of this decay is called the half-life. The average lifetime of a nucleus is about 10 billion years, but some isotopes are stable and will not decay anymore even if they are very heavy. Some isotopes may decay by particle emissions instead of photons; these include **electron-capture decays** and neutron-deficient isotopes produced in stars.

Radioactive dating relies on the fact that certain elements are naturally occurring in well-defined ratios. For example, oxygen exists in earth's crust in nearly constant ratio to silicon, aluminum, iron, magnesium, calcium, etc. Because these other elements do not change their ratio to oxygen, it can be used as a clock to date ancient rock formations. Other elements that remain constant over time are uranium, thorium, and potassium.