Answer:
a) $825
b) $205.10
c) $47.5
d) R(51) = 872.5, R(52) = 920, R(53) = 967.5
Step-by-step explanation:
We are given the following information in the question:
![R(x) = 0.006x^3 + 0.02x^2 + 0.5x](https://tex.z-dn.net/?f=R%28x%29%20%3D%200.006x%5E3%20%2B%200.02x%5E2%20%2B%200.5x)
where x is the number of lawn chairs.
a) Current daily revenue
![R(50) = 0.006(50)^3 + 0.02(50)^2 + 0.5(50) = 825](https://tex.z-dn.net/?f=R%2850%29%20%3D%200.006%2850%29%5E3%20%2B%200.02%2850%29%5E2%20%2B%200.5%2850%29%20%3D%20825)
Thus, $825 is the current daily revenue when Pierce sells 50 lawn chairs daily.
b) Revenue increase if 54 lawn chairs were sold each day
![R(54)-R(50)\\= 0.006(54)^3 + 0.02(54)^2 + 0.5(54) - 825\\= 205.10](https://tex.z-dn.net/?f=R%2854%29-R%2850%29%5C%5C%3D%200.006%2854%29%5E3%20%2B%200.02%2854%29%5E2%20%2B%200.5%2854%29%20-%20825%5C%5C%3D%20205.10)
Thus, revenue increase $205.10 if 54 lawn chairs were sold each day.
c) Marginal revenue
![R'(x) = \displaystyle\frac{dR}{dx} = \frac{d(0.006x^3 + 0.02x^2 + 0.5x)}{dx}\\\\R'(x) = 0.018x^2 + 0.04x+0.5\\R'(50) = 0.018(50)^2 + 0.04(50)+0.5 = 47.5](https://tex.z-dn.net/?f=R%27%28x%29%20%3D%20%5Cdisplaystyle%5Cfrac%7BdR%7D%7Bdx%7D%20%3D%20%5Cfrac%7Bd%280.006x%5E3%20%2B%200.02x%5E2%20%2B%200.5x%29%7D%7Bdx%7D%5C%5C%5C%5CR%27%28x%29%20%3D%200.018x%5E2%20%2B%200.04x%2B0.5%5C%5CR%27%2850%29%20%3D%200.018%2850%29%5E2%20%2B%200.04%2850%29%2B0.5%20%3D%2047.5)
Thus, $47.5 is the marginal revenue when 50 lawn chairs are sold daily.
d) R(51), R(52), and R(53)
![R(51) = R(50) + 1R'(50) = 825 + 1(47.5) = 872.5\\R(52) = R(50) + 2R'(50) = 825 + 2(47.5) = 920\\R(53) = R(50) + 3R'(50) = 825 + 3(47.5) = 967.5](https://tex.z-dn.net/?f=R%2851%29%20%3D%20R%2850%29%20%2B%201R%27%2850%29%20%3D%20825%20%2B%201%2847.5%29%20%3D%20872.5%5C%5CR%2852%29%20%3D%20R%2850%29%20%2B%202R%27%2850%29%20%3D%20825%20%2B%202%2847.5%29%20%3D%20920%5C%5CR%2853%29%20%3D%20R%2850%29%20%2B%203R%27%2850%29%20%3D%20825%20%2B%203%2847.5%29%20%3D%20967.5)