A certain mountain has an elevation of 19,545 feet.

Question

In 1918, the glacier on this peak covered 9 acres. By 2002 this glacier had melted to only 1 acre.

(a) Assume that this glacier melted at a constant rate each year find this yearly rate? (Type a integer or a decimal and round to the nearest thousandth

(b) Use the answer to part (a) to write a linear equation that gives the acreage A of this glacier t years past 1918.

Answer 1

(a) The average rate of change is defined as the change in the value of a quantity (which, for a given function, would be the y-values) divided by the elapsed time (usually the x-values, or t-values if you are using the variable t for time). For the problem in question, the rate at which the glacier is melting would be given as follows:

     rate of change = change in acres/change in time (in years)

                              = (1 acre - 9 acres)/(2002 - 1918)

                              = (-8 acres)/(84 years) = (-8/84) acres/year

                             = -0.095 acres/year

(b) Since we are looking to find the linear equation that represents this function and we calculated the rate of change, which is also known to be the slope of a line, then we can use the slope-intercept form of a linear equation as plug in our known values. The slope-intercept form of a linear equation is as follows:

   y = mx + b ,

   where m is the slope of the line and b is the y-intercept which is the value of y when x=0

Since we are using the variable t for our x-values, the formula is:   y = mt + b

Since we are looking at the time in years from 1918 to 2002, which is 84 years, then we can assign 1918 as year 0 (t = 0) and 2002 as year 84 (t = 84). With this, we know the y-intercept (b) for this linear equation is the y-value for year 0 (t = 0) which is 9 acres (i.e., the y-intercept is b = 9).  We also found that the slope (m), which is the rate of change, is -0.095. So.....

.....   y = mt + b   ,     m = -0.095     and     b = 9

     y = -0.095t + 9 

We can check the answer by plugging in   t=84   for the year 2002 to make sure we approximately 1 acre (or, y=1):

     y = -0.095t + 9   ,     t = 84

     y = -0.095(84) + 9

     y = -7.98 + 9 

     y = 1.02 ≈ 1