vThe profit for a product is increasing at a rate of $5600 per week. The demand and cost functions for the product are given by p = 6000 − 25x and C = 2400x + 5200, where x is the number of units produced per week. Find the rate of change of the sales with respect to Larson, Ron. Algebra and Trigonometry (p. 158). Cengage Learning. Kindle Edition.

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abiolataiwo2015 abiolataiwo2015 · Answer 1 · 2021-06-25T10:13:57+02:00

Answer:

4 units per week

Explanation:

Calculation to Find the rate of change of sales

First step

dP/dt=5600

Second step

Since the revenue is the product of demand and sales

Hence,

R(x)=px

=(6000-25x)x

=6000x-25x²

Third step is to determine the profit which is the difference of revenue and cost.

Hence,

P(x)=R(x)-C(x)

=6000x-25x²-(2400x+ 5200)

=6000x- 25x² -2400x-5200

=3600x-25x²-5200

Fourth step is to Differentiate the profit with respect to time

dP/dt=3600 dx/dt- 50 dx/dt-0

=50(3600/50-x) dx/dt

=50(72-x) dx/dt

Now let Find the rate of change of sales when dP/dt=5600 and x =44

5600=50(72-44) dx/dt

5600=50(28) dx/dt

5600=1400 dx/dt

dx/dt=5600/1400

dx/dt= 4 units per week

Therefore the rate of change of sales is 4 units per week