a) At t=0, we can find p(0) by substituting t=0 into the logistic growth model:
p(0) = 0.90 / (1 + 3.5e^(-0.339*0))
= 0.90 / (1 + 3.5e^0)
= 0.90 / (1 + 3.5*1)
= 0.90 / 4.50
= 0.20
So, when Intel's latest coprocessor is first introduced, 20% (or 0.20) of new personal computers sold at Best Buy will have it.
b) To find the maximum proportion of new personal computers sold with Intel's latest coprocessor, we need to find the critical points of the function. We can do this by finding p'(t) and solving for when it equals 0.
p(t) = 0.90 / (1 + 3.5e^(-0.339t))
To find p'(t), we use the quotient rule:
p'(t) = (0.90d/dt(1 + 3.5e^(-0.339t)) - (1 + 3.5e^(-0.339t))d/dt(0.90)) / (1 + 3.5e^(-0.339t))^2
Simplifying, we get:
p'(t) = (0.90 * 3.5 * (-0.339)e^(-0.339t)) / (1 + 3.5e^(-0.339t))^2
Setting p'(t) to 0 and solving for t:
0 = (0.90 * 3.5 * (-0.339)e^(-0.339t)) / (1 + 3.5e^(-0.339t))^2
Since the numerator is never equal to zero, the denominator must be equal to zero:
1 + 3.5e^(-0.339t) = 0
This equation has no real solutions because the exponential term can never be negative. Therefore, there is no maximum proportion of new personal computers sold with Intel's latest coprocessor.
c) To find when 75% of the new personal computers sold at Best Buy will have Intel's latest coprocessor, we need to solve the equation:
0.75 = 0.90 / (1 + 3.5e^(-0.339t))
Multiplying both sides by (1 + 3.5e^(-0.339t)):
0.75(1 + 3.5e^(-0.339t)) = 0.90
0.75 + 2.625e^(-0.339t) = 0.90
2.625e^(-0.339t) = 0.15
Taking the natural log of both sides:
ln(2.625e^(-0.339t)) = ln(0.15)
ln(2.625) - 0.339t = ln(0.15)
-0.339t = ln(0.15) - ln(2.625)
t ≈ (ln(0.15) - ln(2.625)) / -0.339
Using a calculator, we find t ≈ 8.57 months. So, approximately after 8.57 months, 75% of the new personal computers sold at Best Buy will have Intel's latest coprocessor.