Sales of televisions grow at a rate proportional to the amount present (t is measured in days)
(a) Set up a differental equation to model this problem
(b) Solve the differential equation if at t=0, there are 20 televiosions sold and after 3 days, 500 televisions are sold.
sales = a(e^(kt))
if t=0, sales = 20
20 = a(e^0_
a = 20
then sales = 20(e^(kt)
when t=3, sales = 500
500 = 20(e^3k))
e^(3k) = 25
3k = ln 25
k = 1.07296
sales = 20(e^(1.07296t)
To set up a differential equation to model this problem, we can let T(t) represent the number of televisions sold at time t.
(a) Let's denote the rate of growth of the television sales as dT/dt (the derivative of T with respect to t). Given that the sales of televisions grow at a rate proportional to the amount present, we can write this as:
dT/dt = k * T,
where k is the constant of proportionality.
(b) To solve the differential equation, we can separate the variables by rearranging the equation:
dT/T = k * dt.
Now, we can integrate both sides of the equation. On the left side, we integrate with respect to T, and on the right side, we integrate with respect to t, yielding:
∫ dT/T = ∫ k * dt.
The integral on the left side gives us ln(T), and the integral on the right side gives us k * t (since the integral of a constant is the constant multiplied by the variable). Therefore, our integrated equation becomes:
ln(T) = k * t + C,
where C is the constant of integration.
Now, we can solve for T by taking the exponential of both sides:
e^(ln(T)) = e^(k * t + C).
By the properties of exponents, e^(ln(T)) simplifies to just T. On the right side, we can apply the rule e^(a + b) = e^a * e^b to rewrite the equation as:
T = e^(k * t) * e^C.
Since e^C is just a constant, we can combine it with k and write it as a new constant, A. Hence, our solution becomes:
T = A * e^(k * t).
Now, we can use the given initial conditions to find the specific values of A and k.
At t = 0, there are 20 televisions sold, so we have:
T(0) = A * e^(k * 0) = A * e^0 = A.
Therefore, A = 20.
After 3 days, 500 televisions are sold, so we have:
T(3) = A * e^(k * 3) = 500.
Substituting A = 20, we get:
20 * e^(k * 3) = 500.
Dividing both sides by 20, we have:
e^(k * 3) = 25.
Taking the natural logarithm of both sides, we get:
k * 3 = ln(25).
Solving for k, we find:
k = ln(25) / 3.
Therefore, the solution to the differential equation is:
T(t) = 20 * e^((ln(25) / 3) * t).