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The price (in dollars) p and the quantity demanded q are related by the equation: p^2+2q^2=1100.If R is revenue, dR/dt can be expressed by the following equation: dR/dt=Adp/dt,

where A is a function of just q.


A = ?


Find dR/dt when q=15 and dp/dt=5.

dR/dt = The price (in dollars) p and the quantity demanded q are related by the equation: p^2+2q^2=1100.If R is revenue, dR/dt can be expressed by the following equation: dR/dt=Adp/dt,

where A is a function of just q.


A = ?


Find dR/dt when q=15 and dp/dt=5.

dR/dt =

Verified Answer :

FOR THE FIRST PROBLEM:

Differentiating p^2 + 2q^2 = 1100 with respect to time => 2p(dp/dt) + 4q(dq/dt) = 0 

dp/dt = -2q/p(dq/dt) 

R = pq 

dR/dt = p(dq/dt) + q(dp/dt) = -p²/2q(dp/dt) + q(dp/dt) = (2q² - 1100)/2q * (dp/dt) + q(dp/dt) 

A = (2q - 550/q)

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The equation of A which is a function of just q is;

A = [(4q² - 1100)/2q]

The value of dR/dt when q = 15 and dp/dt = 5 is;

dR/dt = -33.33

We are given the relationship between price(p) and quantity(q) as;

p² + 2q² = 1100

Let's differentiate with respect to t to get;

2p(dp/dt) + 4q(dq/dt) = 0

We are told if revenue is R, then dR/dt can be expressed by the equation: dR/dt = Adp/dt,

Let's make dp/dt the subject in 2p(dp/dt) + 4q(dq/dt) = 0 to get;

2p(dp/dt) = -4q(dq/dt)

dp/dt = -2(q/p)(dq/dt)

Now, let's make p the subject from p² + 2q² = 1100.

p² = 1100 - 2q²

p = √(1100 - 2q²)

Thus;

dp/dt = -2[q/(√(1100 - 2q²))](dq/dt)

Now, we know that formula for revenue is;

R = pq

Differentiating both sides with respect to t gives;

dR/dt = p(dq/dt) + q(dp/dt)

From dp/dt = -2(q/p)(dq/dt), we see that;

dq/dt = -(p/2q)(dp/dt)

Thus;

dR/dt = -(p²/2q)(dp/dt) + q(dp/dt)

p = √(1100 - 2q²)

Thus;

dR/dt = [(2q² - 1100)/2q + q](dp/dt)

Thus simplifies to get;

dR/dt = [(4q² - 1100)/2q](dp/dt)

We are told that;

dR/dt = Adp/dt

Thus;

A = [(4q² - 1100)/2q]

We want to find dR/dt when q = 15 and dp/dt = 5.

Thus;

dR/dt = [(4(15)² - 1100)/(2 × 15)] × 5

dR/dt = -(200/30) × 5

dR/dt = -33.33

Read more at; https://brainly.com/question/13185614

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