the demand for a certain type of auto tire is given by x=f(p) 50(1+e^-p/60).
a)if the weekly quantity demanded is 60000 tires find the price?
b)if the current price is $120 per tire and is increasing at a rate of $2 per week, what is the rate of change of the demand per week?
please show few steps to get better idea of how it is calculated.
To find the price when the weekly quantity demanded is 60,000 tires (part a), we can use the demand function x = f(p) = 50(1 + e^(-p/60)). We need to solve this equation for p.
a) If the weekly quantity demanded is 60,000 tires, we have:
60,000 = 50(1 + e^(-p/60))
To solve for p, we'll need to isolate the exponential term. Simplify the equation step by step:
1 + e^(-p/60) = 60,000/50 = 1,200
Now, subtract 1 from both sides of the equation:
e^(-p/60) = 1,200 - 1 = 1,199
Next, take the natural logarithm of both sides:
ln(e^(-p/60)) = ln(1,199)
Simplify the left side of the equation by using the logarithm property: ln(e^x) = x
-p/60 = ln(1,199)
Now, multiply both sides by -60 to isolate p:
p = -60 * ln(1,199)
Using a calculator, find the value of ln(1,199) and multiply it by -60 to get the final result for p.
b) To find the rate of change of the demand per week when the current price is $120 per tire and is increasing at a rate of $2 per week, we need to find dx/dt, the derivative of x with respect to t, while t represents the time in weeks.
Given x = f(p) = 50(1 + e^(-p/60)), we know that p = 120 and dp/dt = 2.
To find dx/dt, we'll use the chain rule of differentiation. Let u = -p/60, so p = -60u, and x = f(u) = 50(1 + e^u).
Now, differentiate x with respect to t:
dx/dt = dx/du * du/dt
To find dx/du, we differentiate x with respect to u:
dx/du = d(50(1 + e^u))/du
= 50 * de^u/du
Since de^u/du = e^u, we have:
dx/du = 50 * e^u
Now, we differentiate u with respect to t:
du/dt = d(-p/60)/dt
= (-1/60) * dp/dt
Substituting the given values, we get:
du/dt = (-1/60) * 2 = -1/30
Finally, we can calculate dx/dt:
dx/dt = dx/du * du/dt
= 50 * e^u * (-1/30)
Now, substitute the value of u by using p = 120:
dx/dt = 50 * e^(-120/60) * (-1/30)
Simplify this expression to find the rate of change of the demand per week.