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.
To find the rate at which the annual demand for hybrid cars will be changing with respect to time, we need to differentiate the demand function H(x,y) with respect to time t. Then, we can substitute the values for x and y based on the given information to find the rate of change.
Let's start step by step:
Step 1: Differentiate the demand function H(x, y) with respect to time t.
Since H(x, y) is a function of x and y, we need to use the chain rule when differentiating with respect to t. The chain rule states that if z = f(x, y), where x and y are functions of t, then dz/dt = ∂z/∂x * dx/dt + ∂z/∂y * dy/dt.
In this case, we need to differentiate H(x, y) with respect to t, so we can write it as H(x(t), y(t)). Let's calculate the partial derivatives:
∂H/∂x = -13(1/2)x^(-1/2) = -13/(2√x)
∂H/∂y = 2(3/2)(0.1)(y + 20)^(1/2) = 3(y + 20)^(1/2)
So, dz/dt = ∂H/∂x * dx/dt + ∂H/∂y * dy/dt.
Step 2: Substitute the values for x and y based on the given information.
According to the problem, t years from now, the hybrid cars will be selling for 40000 + 300t dollars, and gasoline will cost 300 + 20t^(1/2) cents per gallon.
We can replace x and y in H(x, y) with the given values:
x = 40000 + 300t
y = 300 + 20t^(1/2)
Step 3: Calculate the rate of change 2 years from now.
To find the rate of change of the demand for hybrid cars with respect to time 2 years from now, substitute t = 2 into dz/dt.
dz/dt = ∂H/∂x * dx/dt + ∂H/∂y * dy/dt
Let's substitute values for x, y, and t:
dz/dt = (∂H/∂x)(dx/dt) + (∂H/∂y)(dy/dt)
= (∂H/∂x)(300) + (∂H/∂y)(10√2)
Now, substitute the calculated values of the partial derivatives:
∂H/∂x = -13/(2√x) = -13/(2√(40000 + 300t))
∂H/∂y = 3(y + 20)^(1/2) = 3(300 + 20t^(1/2) + 20)^(1/2)
So, the final expression for dz/dt is:
dz/dt = (-13/(2√(40000 + 300t)))(300) + (3(300 + 20t^(1/2) + 20)^(1/2))(10√2)
Now, you can simplify the expression and evaluate it at t = 2 to find the rate of change of the annual demand for hybrid cars with respect to time 2 years from now.
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 dH/dt, the derivative of the function H(x,y) with respect to time.
First, let's find the partial derivative of H(x,y) with respect to x:
dH/dx = -13 * (1/2) * x^(-1/2) = -13/2√x
Next, let's find the partial derivative of H(x,y) with respect to y:
dH/dy = 2 * (3/2) * (0.1y + 20)^(1/2) * (0.1) = 3√(0.1y + 20)
Now, let's substitute the values for x and y, given t years from now:
x = 40000 + 300t
y = 300 + 20√t
Substituting these values into the partial derivatives, we get:
dH/dx = -13/2√(40000 + 300t)
dH/dy = 3√(0.1(300 + 20√t) + 20)
Finally, 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 evaluate these derivatives at t = 2:
dH/dx (when t = 2) = -13/2√(40000 + 300(2))
dH/dy (when t = 2) = 3√(0.1(300 + 20√2) + 20)
Simplifying these expressions will give us the specific rate at which the annual demand for hybrid cars will be changing with respect to time 2 years from now.