Question

Lincoln accepted a new job at a company with a contract guaranteeing annual raises. Let S

represent Lincoln's salary after working for n years at the company. A graph of S is shown
below. Write an equation for S then state the slope of the graph and determine its interpretation
in the context of the problem.

Answer 1

The slope of the graph represents the initial rate of change in salary adjusted by the effect of the annual raise rate.

Let's assume that Lincoln's initial salary (at n = 0 years) is represented by

S_0 , and the annual raise is a constant rate represented by r.

The equation for Lincoln's salary (S) in terms of the number of years worked (n) can be expressed as:

S(n)=S_0 ⋅(1+r)^n

Here,

S(n) is Lincoln's salary after working for n years.

S_0 is the initial salary.

r is the annual raise rate.

The slope of the graph is given by the derivative of S(n) with respect to n. Taking the derivative, we get:

dS/ dn =S_0 ⋅ln(1+r)⋅(1+r)^n

The slope of the graph is S_0 ⋅ln(1+r). The interpretation of the slope in the context of the problem is as follows:

S_0 is the initial salary, so the product S_0 ⋅ln(1+r) represents the initial rate of change in Lincoln's salary.

The term

ln(1+r) is the natural logarithm of the annual raise rate plus 1. It influences the rate at which Lincoln's salary grows each year.