A student stuffs envelopes for extra income during her spare time. Her initial cost to obtain the necessary information for the job was ​$120. Each envelope costs ​$0.06 and she gets paid ​$0.07 per envelope stuffed. Let x represent the number of envelopes stuffed.

Question

​(a) Express the cost C as a function of x.
​(b) Express the revenue R as a function of x.
​(c) Determine the value of x for which revenue equals cost.
​(d) Graph the equations y=​C(x) and y=​R(x) on the same​ axes, and interpret the graph.

Answer 1

(a) The cost, C, can be expressed as the sum of the initial cost to obtain the necessary information, $120, and the cost per envelope stuffed, which is $0.06 multiplied by the number of envelopes stuffed, x. Therefore, the cost C as a function of x is:

C(x) = $120 + ($0.06 * x)

(b) The revenue, R, can be expressed as the amount earned per envelope stuffed, which is $0.07 multiplied by the number of envelopes stuffed, x. Therefore, the revenue R as a function of x is:

R(x) = $0.07 * x

(c) To determine the value of x for which revenue equals cost, we need to set the two functions equal to each other and solve for x:

$120 + ($0.06 * x) = $0.07 * x

Simplifying the equation:

$120 = $0.01 * x
x = $120 / $0.01
x = 12,000

Therefore, the value of x for which revenue equals cost is 12,000 envelopes stuffed.

(d) To graph the equations y = C(x) and y = R(x), we can plot the cost and revenue values for different values of x on the same axes. The x-axis represents the number of envelopes stuffed, and the y-axis represents the cost and revenue.

The graph will show that initially, as the number of envelopes stuffed increases, the cost increases linearly while the revenue also increases linearly. However, eventually, the revenue will exceed the cost and continue to increase at a faster rate than the cost.

The point of intersection on the graph represents the number of envelopes stuffed, 12,000, at which the revenue equals the cost. This point represents the break-even point, where the student will start making a profit from stuffing envelopes.

Note: The graph would display a positive slope for both the cost and revenue lines, as the number of envelopes stuffed increases.