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.
(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.
(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.
(a) The cost C as a function of x can be calculated by multiplying the cost per envelope with the total number of envelopes, which is given by:
C(x) = $0.06 * x
(b) The revenue R as a function of x can be calculated by multiplying the revenue per envelope with the total number of envelopes, which is given by:
R(x) = $0.07 * x
(c) To determine the value of x for which revenue equals cost, we need to set the cost C(x) equal to the revenue R(x) and solve for x:
0.06x = 0.07x
0.01x = 0
x = 0
So, when the revenue equals the cost, the value of x is 0.
(d) Graphing the equations y = C(x) and y = R(x) on the same axes:
The graph will be a straight line passing through the origin (0,0), with a positive slope for both equations. The line for C(x) will have a steeper slope (0.06) than the line for R(x) (0.07).
Interpretation of the graph:
The graph shows the relationship between the number of envelopes stuffed (x) and the cost (y) and revenue (y) associated with it. As the number of envelopes stuffed increases, both the cost and revenue increase. The revenue line is slightly steeper than the cost line, indicating that the revenue increases at a slightly higher rate than the cost. The point where the two lines intersect (x = 0) represents the break-even point, where the revenue equals the cost.
(a) The cost C for the student to stuff x number of envelopes consists of two parts: the initial cost to obtain necessary information and the cost per envelope stuffed. The initial cost is $120, and the cost per envelope is $0.06.
Therefore, the expression for the cost C as a function of x is:
C(x) = $120 + ($0.06 * x)
(b) The revenue R for the student, which represents the total amount she gets paid for stuffing x number of envelopes, is calculated by multiplying the payment per envelope by the number of envelopes stuffed. The payment per envelope is $0.07.
Therefore, the expression for the revenue R as a function of x is:
R(x) = $0.07 * x
(c) To determine the value of x for which the revenue equals the cost, we need to set the equations for cost and revenue equal to each other and solve for x.
$120 + ($0.06 * x) = $0.07 * x
Simplifying the equation, we get:
$120 = $0.01 * x
Dividing both sides of the equation by $0.01, we find:
x = $120 / $0.01
x = 12,000
Therefore, the value of x for which revenue equals cost is 12,000.
(d) To graph the equations y = C(x) and y = R(x) on the same axes, we plot the points for various values of x. Let's consider a range of x from 0 to 15,000, as x = 12,000 is the value for which revenue equals cost.
For the cost equation C(x):
- At x = 0, C(0) = $120, which is the initial cost.
- At x = 12,000, C(12,000) = $120 + ($0.06 * 12,000) = $840, which is the final cost.
For the revenue equation R(x):
- At x = 0, R(0) = 0, as no envelopes are stuffed and no revenue is earned.
- At x = 12,000, R(12,000) = $0.07 * 12,000 = $840, which is the revenue obtained.
Plotting these points on the same graph, with x on the horizontal axis and y on the vertical axis, we can observe the relationship between cost and revenue as the number of envelopes stuffed increases. The graph shows that initially, the revenue is lower than the cost due to the initial investment, but as more envelopes are stuffed, the revenue surpasses the cost.
Interpreting the graph, we can see that the break-even point, where revenue equals cost, occurs at x = 12,000. Beyond this point, the revenue continues to increase, resulting in a profit once the initial investment is recovered.