Sure! Let's solve this step-by-step.
### Given:
Acme's monthly revenue function from selling [tex]\( x \)[/tex] carry-on suitcases is:
[tex]\[ R(x) = 0.004x^3 - 0.7x^2 + 120x \][/tex]
Acme is currently selling 28 suitcases monthly.
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### Part a) Current Monthly Revenue
To find the current monthly revenue when 28 suitcases are sold, we substitute [tex]\( x = 28 \)[/tex] into the revenue function [tex]\( R(x) \)[/tex].
[tex]\[ R(28) = 0.004(28)^3 - 0.7(28)^2 + 120(28) \][/tex]
Using the given numbers:
[tex]\[ R(28) = 2899.008 \][/tex]
So, the current monthly revenue is:
[tex]\[ \boxed{2899.008} \][/tex]
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### Part b) Revenue Increase if Sales Increase from 28 to 30 Suitcases
First, we find the revenue at 30 suitcases by substituting [tex]\( x = 30 \)[/tex] into the revenue function.
[tex]\[ R(30) = 0.004(30)^3 - 0.7(30)^2 + 120(30) \][/tex]
Then, we subtract the revenue at 28 suitcases from the revenue at 30 suitcases.
[tex]\[ R(30) - R(28) \][/tex]
Using the given numbers:
[tex]\[ R(30) = 3078.0 \][/tex]
Revenue increase:
[tex]\[ 3078.0 - 2899.008 = 178.992 \][/tex]
So, the revenue would increase by:
[tex]\[ \boxed{178.992} \][/tex]
---
### Part c) Marginal Revenue when 28 Suitcases are Sold
Marginal revenue is the derivative of the revenue function with respect to [tex]\( x \)[/tex], evaluated at [tex]\( x = 28 \)[/tex].
First, find the derivative of the revenue function, [tex]\( R'(x) \)[/tex].
[tex]\[ R'(x) = \frac{d}{dx}(0.004x^3 - 0.7x^2 + 120x) \][/tex]
After calculating the derivative, evaluate it at [tex]\( x = 28 \)[/tex]:
Using the given numbers:
[tex]\[ R'(28) = 90.208 \][/tex]
So, the marginal revenue when 28 suitcases are sold is:
[tex]\[ \boxed{90.208} \][/tex]
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### Part d) Estimate Revenue for Selling 29 Suitcases
To find the estimated revenue when 29 suitcases are sold, we substitute [tex]\( x = 29 \)[/tex] into the revenue function [tex]\( R(x) \)[/tex].
[tex]\[ R(29) = 0.004(29)^3 - 0.7(29)^2 + 120(29) \][/tex]
Using the given numbers:
[tex]\[ R(29) = 2988.856 \][/tex]
So, the estimated revenue for selling 29 suitcases is:
[tex]\[ \boxed{2988.856} \][/tex]
---
These are the detailed solutions for each part of the problem.
