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.
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Part 1
(a) C(x)=enter your response here (C in dollars)
(Use integers or decimals for any numbers in the expression.)
Part 2
(b) R(x)=enter your response here (R in dollars)
(Use integers or decimals for any numbers in the expression.)
Part 3
(c) Revenue equals cost when x=enter your response here.
(Type an integer or a decimal.)
Part 4
(d) Choose the correct graph below that represents y=C(x) and y=R(x).
A.
0
20,000
0
1500
x
y
A coordinate system has a horizontal x-axis labeled from 0 to 20000 in increments of 2000 and a vertical y-axis labeled from 0 to 1500 in increments of 150. A line that rises from left to right passes through the points (0, 120) and (10000, 720). A line that rises from left to right passes through the points (0, 0) and (10000, 1050). The lines intersect at approximately (2667, 280).
B.
0
20,000
0
1500
x
y
A coordinate system has a horizontal x-axis labeled from 0 to 20000 in increments of 2000 and a vertical y-axis labeled from 0 to 1500 in increments of 150. A line that falls from left to right passes through the points (0, 120) and (2000, 0). A line that rises from left to right passes through the points (0, 0) and (10000, 1050). The lines intersect at approximately (727, 76).
C.
0
20,000
0
1500
x
y
A coordinate system has a horizontal x-axis labeled from 0 to 20000 in increments of 2000 and a vertical y-axis labeled from 0 to 1500 in increments of 150. A line that falls from left to right passes through the points (0, 120) and (2000, 0). A line that rises from left to right passes through the points (0, 0) and (10000, 700). The lines intersect at approximately (923, 65).
D.
0
20,000
0
1500
x
y
The correct choice is D. A coordinate system has a horizontal x-axis labeled from 0 to 20000 in increments of 2000 and a vertical y-axis labeled from 0 to 1500 in increments of 150. A line that falls from left to right passes through the points (0, 120) and (2000, 0). A line that rises from left to right passes through the points (0, 0) and (10000, 700). The lines intersect at approximately (923, 65).
This graph represents the cost function, C(x), and revenue function, R(x). The intersection point of the two lines represents the value of x for which revenue equals cost. The graph shows that initially, as the number of envelopes stuffed increases, the cost increases linearly, while the revenue increases at a slower rate. However, at a certain point, the revenue surpasses the cost, and the graph begins to show a positive slope. This indicates that after a certain number of envelopes stuffed, the student will start making a profit.
(a) The cost C as a function of x can be calculated by adding the initial cost to the cost of individual envelopes:
C(x) = 120 + 0.06x
(b) The revenue R as a function of x can be calculated by multiplying the number of envelopes stuffed by the payment per envelope:
R(x) = 0.07x
(c) To find the value of x for which revenue equals cost, we can set the cost function equal to the revenue function:
C(x) = R(x)
120 + 0.06x = 0.07x
Simplifying the equation:
0.07x - 0.06x = 120
0.01x = 120
x = 120 / 0.01
x = 12000
Therefore, revenue equals cost when x = 12000.
(d) The correct graph representation is the one that shows the line rising from left to right and passing through the points (0, 120) and (10000, 720) for C(x) and the line rising from left to right and passing through the points (0, 0) and (10000, 1050) for R(x). The lines intersect at approximately (2667, 280).
The correct graph is graph A.
(a) To express the cost C as a function of x, we need to consider the initial cost to obtain the necessary information, which is $120, and the cost per envelope, which is $0.06.
The cost per envelope is constant, so we can multiply it by the number of envelopes stuffed (x) to get the cost function.
C(x) = $0.06 * x + $120
(b) To express the revenue R as a function of x, we need to consider the income for each envelope stuffed, which is $0.07.
The income per envelope is constant, so we can multiply it by the number of envelopes stuffed (x) to get the revenue function.
R(x) = $0.07 * x
(c) Revenue equals cost when the income from stuffing envelopes equals the cost of obtaining the necessary information.
R(x) = C(x)
$0.07 * x = $0.06 * x + $120
Simplifying the equation:
$0.01 * x = $120
Divide both sides of the equation by $0.01:
x = $12,000
Therefore, revenue equals cost when x = $12,000.
(d) The graph represents the cost function (y = C(x)) and the revenue function (y = R(x)) on the same axes.
The x-axis represents the number of envelopes stuffed (x) and the y-axis represents the cost/revenue in dollars (y).
From the given options, the correct graph is option A. It shows a line that rises from left to right passing through the points (0, 120) and (10,000, 720) for the Cost function (y = C(x)), and a line that rises from left to right passing through the points (0, 0) and (10,000, 1050) for the Revenue function (y = R(x)).
The intersection point of the two lines represents the value of x where revenue equals cost, which is approximately (2667, 280). This means that when approximately 2667 envelopes are stuffed, the revenue generated will be equal to the cost incurred.
Interpreting the graph, we can see that initially, the revenue is lower than the cost, but as the number of envelopes stuffed increases, the revenue surpasses the cost, indicating profitability.