(a) To sketch a graph of D, we can plot points for various values of x and connect them with a smooth curve. Since the function D(x) is linear, the graph will be a straight line.
We know that the intercept of D(x) is 20, which means that when x = 0 (i.e., in the year 1980), the depth of the sediment layer was 20 cm. We can plot this point on the graph as (0, 20).
To find another point on the graph, we can plug in a value of x and calculate the corresponding value of D(x). For example, if we let x = 10 (i.e., 10 years after 1980), then:
D(10) = 20 + 0.24(10) = 22.4
So we can plot the point (10, 22.4) on the graph.
We can repeat this process for other values of x to get more points on the graph. Once we have enough points, we can connect them with a straight line to get the graph of D(x).
(b) The slope of the graph is equal to the coefficient of x in the equation of the line. In this case, the equation of the line is:
D(x) = 20 + 0.24x
So the slope is 0.24.
(c) The sediment layer is increasing at a rate of 0.24 cm per year. This is equal to the slope of the graph, which tells us the rate at which the depth of the sediment layer is increasing for each additional year since 1980.