A car dealer determines that if gasoline-electric hybrid automobiles are sold for x dollars apiece and the price of gasoline is y cents per gallon, then approximately H hybrid cars will be sold each year, where

H(x,y)=6000−13x^(1/2)+2(0.1y+20)^(3/2).
She estimates that t years from now, the hybrid cars will be selling for 40000+300t dollars apiece and that gasoline will cost 300+20t^(1/2) cents per gallon. At what rate will the annual demand for hybrid cars be changing with respect to time 2 years from now?

I have no clue how to even begin this problem.

dH/dt = ∂H/∂x dx/dt + ∂H/∂y dy/dt

∂H/∂x = -13/(2√x)
∂H/∂y = 0.3√(0.1y+20)

dx/dt = 300
dy/dt = 10/√y

So, plug and chug, with t=2

DEMAND FOR HYBRID CARS A car dealer determines that if gasoline-electric hybrid automobiles are sold for x dollars apiece and the price of gasoline is y cents per gallon, then approximately H hybrid cars will be sold each year, where H(x, y) = 3,500 – 19x/2 + 6(0.1y + 16)³/² She estimates that i years from now, the hybrid cars will be selling for x(1) = 35,050 + 350/ dollars apiece and that gasoline will cost 76. y(1) = 300 + 10(31)/2 cents per gallon. At what rate will the annual demand for hybrid cars be changing with respect to time 3 years from now? Will it be increasing or decreasing?

To find the rate at which the annual demand for hybrid cars will be changing with respect to time 2 years from now, we need to find the derivative of the demand function H with respect to time t.

First, we need to express the demand function H(x,y) in terms of t rather than x and y. We're given that t years from now, hybrid cars will be selling for (40000 + 300t) dollars apiece and the price of gasoline will be (300 + 20t^(1/2)) cents per gallon.

We can substitute these values into the demand function and express it in terms of t:

H(t) = 6000 - 13(40000 + 300t)^(1/2) + 2(0.1(300 + 20t^(1/2)) + 20)^(3/2)

Next, we need to differentiate H(t) with respect to t using the chain rule. The derivative of each term is as follows:

d/dt [6000] = 0

d/dt [13(40000 + 300t)^(1/2)] = 13 * 1/2(40000 + 300t)^(-1/2) * 300

d/dt [2(0.1(300 + 20t^(1/2)) + 20)^(3/2)] = 2 * 3/2(0.1(300 + 20t^(1/2)) + 20)^(1/2) * 0.1 * 1/2(300 + 20t^(1/2))^(-1/2) * 20

Now, substitute the expressions back into the demand function:

H'(t) = 13 * 1/2(40000 + 300t)^(-1/2) * 300 + 2 * 3/2(0.1(300 + 20t^(1/2)) + 20)^(1/2) * 0.1 * 1/2(300 + 20t^(1/2))^(-1/2) * 20

Simplify the expression:

H'(t) = 195/2(40000 + 300t)^(-1/2) + 6(0.1(300 + 20t^(1/2)) + 20)^(1/2) * 0.1(300 + 20t^(1/2))^(-1/2)

Finally, substitute t = 2 into the expression to find the rate at which the demand will be changing 2 years from now:

H'(2) = 195/2(40000 + 300(2))^(-1/2) + 6(0.1(300 + 20(2)^(1/2)) + 20)^(1/2) * 0.1(300 + 20(2)^(1/2))^(-1/2)

Simplify the expression to get your answer.

To find the rate at which the annual demand for hybrid cars will be changing with respect to time 2 years from now, we need to calculate the derivative of the demand function H(x, y) with respect to time (t) and evaluate it at t = 2.

Let's go step by step.

Step 1: Find the derivative of H(x, y) with respect to x.
To do this, treat y as a constant and differentiate with respect to x. The derivative of x^(1/2) with respect to x is (1/2)x^(-1/2). The derivative of a constant times x^(-1/2) is simply the constant times the derivative of x^(-1/2).

So, the derivative of -13x^(1/2) with respect to x is:
d/dx [-13x^(1/2)] = -13 * (1/2)x^(-1/2) = -13/2 * x^(-1/2)

Step 2: Find the derivative of H(x, y) with respect to y.
To do this, treat x as a constant and differentiate with respect to y. The derivative of 0.1y with respect to y is simply 0.1.

So, the derivative of 0.1y with respect to y is:
d/dy [0.1y] = 0.1

Step 3: Find the derivative of H(x, y) with respect to x and y.
To do this, differentiate both terms in the function with respect to y and x respectively.

So, the derivative of 2(0.1y+20)^(3/2) with respect to y is:
d/dy [2(0.1y+20)^(3/2)] = 2 * (3/2) * (0.1y+20)^(3/2 - 1) * 0.1 = 0.3(0.1y+20)^(1/2)

Step 4: Combine the derivatives calculated in steps 1, 2, and 3.
The derivative of H(x, y) with respect to x is -13/2 * x^(-1/2), the derivative with respect to y is 0.1, and the derivative with respect to x and y is 0.3(0.1y+20)^(1/2).

So, the derivative of H(x, y) with respect to x and y is:
dH/dxdy = -13/2 * x^(-1/2) + 0.3(0.1y+20)^(1/2)

Step 5: Substitute the given values for x, y, and t into the derivative obtained in step 4.
We are interested in finding the rate of change of the demand with respect to time when t = 2. Substitute the given values: x = 40000 + 300t and y = 300 + 20t^(1/2).

So, the derivative of H(x, y) with respect to x and y when t = 2 becomes:
dH/dxdy = -13/2 * (40000 + 300t)^(-1/2) + 0.3(0.1(300 + 20t^(1/2)) + 20)^(1/2)

Step 6: Evaluate the derivative at t = 2.
Substitute t = 2 into the expression for the derivative obtained in step 5.

So, the derivative of H(x, y) with respect to x and y at t = 2 becomes:
dH/dxdy = -13/2 * (40000 + 300(2))^(-1/2) + 0.3(0.1(300 + 20(2)^(1/2)) + 20)^(1/2)

Simplify the expression and calculate the value to find the rate at which the annual demand for hybrid cars will be changing with respect to time 2 years from now.