A student stufits envelopes for extra income during her spare time. Her Inltial cost to obtain the necessary information for the job was $\mathbf{\$ 1 3 0}$. Each envelope costs 50.03 and she gets paid 50.04 per envelope stufied. 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) $C(x)=\square($ Cin dollars) (Use integers or decimals for any numbers in the expression.) (b) $\mathbf{R}(x)=\square(R \ln$ dollars $)$ (Use integers or decimals for any numbers in the expression.) (c) Revenue equals cost when $x=\square$. (Type an integer or a decimal.) (d) Choose the correct graph below that represents $y=C(x)$ and $y=R(x)$. A. B. c. D.
Solution
Step 1 ：The cost function, \(C(x)\), is the initial cost plus the cost per envelope times the number of envelopes. So, \(C(x) = 130 + 0.03x\).
Step 2 ：The revenue function, \(R(x)\), is the amount she gets paid per envelope times the number of envelopes. So, \(R(x) = 0.04x\).
Step 3 ：To find the value of \(x\) for which revenue equals cost, we set \(C(x) = R(x)\) and solve for \(x\).
Step 4 ：\(130 + 0.03x = 0.04x\)
Step 5 ：\(130 = 0.01x\)
Step 6 ：\(\boxed{x = 13000}\)
Step 7 ：So, revenue equals cost when she stuffs 13000 envelopes.
Step 8 ：The graph of \(y = C(x)\) is a straight line with a slope of 0.03 and y-intercept of 130. The graph of \(y = R(x)\) is a straight line with a slope of 0.04 and no y-intercept. The two lines intersect at the point (13000, 520), which represents the number of envelopes she needs to stuff for her revenue to equal her cost.
