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10. When a wholesale produce market has x crates of lettuce available on a given...
Dec 29, 2023
10. When a wholesale produce market has x crates of lettuce available on a given day, it charges p dollars per crate as determined by the supply equation px - 20p- 6x + 40 = 0. If the daily supply is decreasing at the rate of 8 crates per day, at what rate is the price changing when the supply is 100 crates?
(A) not changing
(B) increasing at 0.10perday(C)decreasingat0.10 per day
(D) increasing at 1.00perday(E)decreasingat1.00 per day
C
Key Concept
Implicit Differentiation and Related Rates
Explanation
To find the rate at which the price is changing, we use implicit differentiation with respect to time on the supply equation px−20p−6x+40=0. Differentiating both sides with respect to time t, we get p′x+px′−20p′−6x′=0. Given that x′=−8 crates/day (since the supply is decreasing) and x=100 crates, we can solve for p′, the rate at which the price is changing. Substituting the known values and solving for p′ will give us the rate of change of the price when the supply is 100 crates.
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