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Vesna [10]
3 years ago
10

If the price is increasing at a rate of 2 dollars per month when the price is 10 dollars, find the rate of change of the demand.

Mathematics
1 answer:
densk [106]3 years ago
8 0

Answer:

The demand reduces by $7.12 per month

<em></em>

Step-by-step explanation:

Given

p\to price

x \to demand

2x^2+5xp+50p^2=24800.

p =10; \frac{dp}{dt} = 2

Required

Determine the rate of change of demand

We have:

2x^2+5xp+50p^2=24800.

Differentiate with respect to time

4x\frac{dx}{dt} + 5x\frac{dp}{dt} + 5p\frac{dx}{dt} + 100p\frac{dp}{dt} = 0

Collect like terms

4x\frac{dx}{dt} + 5p\frac{dx}{dt} = -5x\frac{dp}{dt}  - 100p\frac{dp}{dt}

Factorize

\frac{dx}{dt}(4x + 5p) = -5(x  + 20p)\frac{dp}{dt}

Solve for dx/dt

\frac{dx}{dt} = -\frac{5(x  + 20p)}{4x + 5p}\cdot \frac{dp}{dt}

Given that: 2x^2+5xp+50p^2=24800. and p = 10

Solve for x

2x^2 + 5x * 10 + 50 * 10^2 = 24800

2x^2 + 50x + 5000 = 24800

Equate to 0

2x^2 + 50x + 5000 - 24800 =0

2x^2 + 50x -19800 =0

Using a quadratic calculator, we have:

x \approx -113\ and\ x\approx88

Demand must be greater than 0;

So: x=88

So, we have: x=88; p =10; \frac{dp}{dt} = 2

The rate of change of demand is:

\frac{dx}{dt} = -\frac{5(88  + 20*10)}{4*88 + 5*10} * 2

\frac{dx}{dt} = -\frac{5(288)}{402} * 2

\frac{dx}{dt} = -\frac{2880}{402}

\frac{dx}{dt} \approx -7.16

<em>This implies that the demand reduces by $7.12 per month</em>

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