The annual demand q for bottles of wine from a vineyard when the bottles are priced at p dollars each satisfies the equation
qe0.04p = 6000.
The price is currently $12 per bottle. Find the rate at which demand changes (with respect to time) if the price increases at a rate of $1.20 per year. (Round your answer to the nearest whole number.)
q e^(.04 p) = 6000
so
q = 6000 e^(-.04 p)
dq/dt = 6000 (-.04)e^(-.04 p) dp/dt
p = 12 and dp/dt = 1.2
dp/dt = -240 [e^(-0.48)] (1.2)
= -240 * .619 * 1.2 = - 178
agree
-178 bottles per year
Well, let's see. We need to find the rate at which demand changes with respect to time when the price increases at a rate of $1.20 per year.
First, let's differentiate the equation qe^(0.04p) = 6000 with respect to time.
d(qe^(0.04p))/dt = d(6000)/dt
Now, let's substitute in the given values. The price is currently $12 per bottle, so p = 12. The rate at which the price changes is $1.20 per year, so dp/dt = 1.20.
d(qe^(0.04(12)))/dt = 0
Simplifying, we have:
d(qe^0.48)/dt = 0
Since any number raised to the power of 0 is equal to 1, we have:
d(qe)/dt = 0
Now, let's solve for dq/dt, the rate at which demand changes:
dq/dt = -q * de/dt
Since we know that dq/dt = 0, we can conclude that the rate at which demand changes is 0 bottles per year.
So, the answer is 0 bottles per year. This means that even if the price increases at a rate of $1.20 per year, the demand for bottles of wine from the vineyard does not change. Looks like people really love their wine!
To find the rate at which demand changes with respect to time, we need to differentiate the equation with respect to time.
Given: qe^(0.04p) = 6000
We are given that p = $12 and it is increasing at a rate of $1.20 per year. Let's differentiate both sides with respect to time (t):
Differentiating both sides with respect to t:
d/dt [qe^(0.04p)] = d/dt [6000]
Using the chain rule on the left side:
(dq/dt)e^(0.04p) + qe^(0.04p) * d(0.04p)/dt = 0
Since p is increasing at a constant rate of $1.20 per year, d(0.04p)/dt = 0.04 * 1.20 = 0.048.
Substituting the given values into the equation:
(dq/dt)e^(0.04 * 12) + qe^(0.04 * 12) * 0.048 = 0
Simplifying further:
(dq/dt)e^(0.48) + qe^(0.48) * 0.048 = 0
Now, we can calculate the value of (dq/dt) by rearranging the equation:
(dq/dt) = - qe^(0.48) * 0.048 / e^(0.48)
Simplifying:
(dq/dt) = - q * 0.048
Now, substitute the value of q from the given equation, qe^(0.04p) = 6000, when p = $12:
q * e^(0.04 * 12) = 6000
qe^(0.48) = 6000
q = 6000 / e^(0.48)
Now, substitute the value of q back into the equation:
(dq/dt) = - (6000 / e^(0.48)) * 0.048
Calculating the value:
(dq/dt) ≈ -112.02
Therefore, the rate at which demand changes (with respect to time) is approximately 112 bottles per year.
To find the rate at which demand changes with respect to time, we need to differentiate the demand equation with respect to time.
Given: qe^0.04p = 6000
Differentiating both sides of the equation with respect to time, we get:
(dq/dt)e^0.04p + 0.04q(dq/dt)e^0.04p = 0
We are given that the price increases at a rate of $1.20 per year, which means the rate of change of price with respect to time (dp/dt) is 1.20.
Given: dp/dt = 1.20
We also know that the initial price is $12 per bottle, so p = 12.
Now we can substitute these values into the differentiated equation:
(dq/dt)e^0.04(12) + 0.04q(dq/dt)e^0.04(12) = 0
Simplifying the equation:
(dq/dt)e^0.48 + 0.04qe^0.48(dq/dt) = 0
Factoring out (dq/dt):
(dq/dt)(e^0.48 + 0.04qe^0.48) = 0
To find the rate at which demand changes (dq/dt), we need to solve this equation for dq/dt. Since the equation equals zero, we know that either (dq/dt) or (e^0.48 + 0.04qe^0.48) equals zero.
However, we are looking for the rate of change in demand, so we can rule out dq/dt = 0. Therefore, we must have e^0.48 + 0.04qe^0.48 = 0.
Solving this equation for q, we get:
e^0.48 + 0.04qe^0.48 = 0
0.04qe^0.48 = -e^0.48
q = -e^0.48 / (0.04e^0.48)
q = -1/0.04
q = -25
Now we have the value of q, which represents the current demand. Since demand cannot be negative, we consider the absolute value of q, so the current demand is 25 bottles.
Therefore, the rate at which demand changes (with respect to time) is 25 bottles per year.