The logistic growth model p(t)=0.90/1+3.5e^-0.339t relates the proportion of new personal computers sold @ Best Buy that have Intel's latest coprecessor t months after it has been introduced.
a) what proportion of new personal computers sold @ Best Buy will have Intel's latest coprocessor when it is first introduced (that is, at t=0)
b) determine the max proportion of new personal computers sold @ Best Buy that will have Intel's latest coprocessor
c) when will 75% of the new personal computers sold at Best Buy have Intel's latest copressor.
You have
p(t)=0.90/(1+3.5e-0.339*t)
For a) find p(0).
For b) find p'(t) and use what you've been taught about find max/min values. Set p'(t) to 0 and solve.
For c) solve .75=0.90/(1+3.5e-0.339*t) It looks to be somewhere between 8 and 9 months when I graphed the function.
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.
a) To find the proportion of new personal computers sold at Best Buy that have Intel's latest coprocessor when it is first introduced (t = 0), we need to find p(0) by substituting t = 0 into the equation:
p(0) = 0.90/(1 + 3.5e^(-0.339 * 0))
p(0) = 0.90/(1 + 3.5e^0)
p(0) = 0.90/(1 + 3.5 * 1)
p(0) = 0.90/(1 + 3.5)
p(0) = 0.90/4.5
p(0) = 0.20
Therefore, when the latest coprocessor is first introduced, approximately 20% of the new personal computers sold at Best Buy will have Intel's latest coprocessor.
b) To find the maximum proportion of new personal computers sold at Best Buy that will have Intel's latest coprocessor, we need to find p'(t) and set it equal to zero to find the critical point:
p(t) = 0.90/(1 + 3.5e^(-0.339t))
p'(t) = (-0.90 * 3.5 * -0.339 * e^(-0.339t))/(1 + 3.5e^(-0.339t))^2
p'(t) = (1.20765 * e^(-0.339t))/(1 + 3.5e^(-0.339t))^2
To find the critical point, we set p'(t) equal to zero:
0 = (1.20765 * e^(-0.339t))/(1 + 3.5e^(-0.339t))^2
Since e^(-0.339t) is always positive, we can multiply both sides of the equation by (1 + 3.5e^(-0.339t))^2 to get rid of the denominator:
0 = 1.20765 * e^(-0.339t)
Since e^(-0.339t) is never zero, the only way for the equation to be true is when 1.20765 = 0. Therefore, there are no critical points for this equation.
Since there are no critical points, there is no maximum or minimum for the proportion of new personal computers sold at Best Buy that will have Intel's latest coprocessor. The value of p(t) will just continuously change based on the value of t.
c) To find the time at which 75% of the new personal computers sold at Best Buy will have Intel's latest coprocessor, we set p(t) equal to 0.75 and solve for t:
0.75 = 0.90/(1 + 3.5e^(-0.339t))
To solve this equation, we can first cross-multiply:
0.75(1 + 3.5e^(-0.339t)) = 0.90
Next, distribute the 0.75:
0.75 + 2.625e^(-0.339t) = 0.90
Subtract 0.75 from both sides:
2.625e^(-0.339t) = 0.15
Divide both sides by 2.625:
e^(-0.339t) = 0.057
To solve for t, we can take the natural logarithm of both sides:
ln(e^(-0.339t)) = ln(0.057)
Simplifying:
-0.339t = ln(0.057)
Divide both sides by -0.339:
t = (ln(0.057))/(-0.339)
Using a calculator, we find that:
t ≈ 8.75
Therefore, approximately 8.75 months after the coprocessor is introduced, 75% of the new personal computers sold at Best Buy will have Intel's latest coprocessor.
To solve the given logistic growth model and answer each question:
a) To find the proportion of new personal computers sold at Best Buy that will have Intel's latest coprocessor when it is first introduced (at t=0), you need to find p(0).
Substitute t=0 into the equation p(t):
p(0) = 0.90/(1 + 3.5e^(-0.339*0))
Since anything raised to the power of 0 is 1, the equation simplifies to:
p(0) = 0.90/(1 + 3.5)
p(0) = 0.90/4.5
p(0) ≈ 0.2
Therefore, when the coprocessor is first introduced, approximately 20% of new personal computers sold at Best Buy will have Intel's latest coprocessor.
b) To determine the maximum proportion of new personal computers sold at Best Buy that will have Intel's latest coprocessor, you need to find the maximum value of p(t). This can be done by finding the derivative of p(t), setting it equal to 0, and solving for t.
Differentiate p(t) with respect to t:
p'(t) = (0.90'(1 + 3.5e^(-0.339*t)) - 0.90(1 + 3.5e^(-0.339*t))') / (1 + 3.5e^(-0.339*t))^2
Simplifying further:
p'(t) = (0 - (-0.339)(0.90)(3.5e^(-0.339*t))) / (1 + 3.5e^(-0.339*t))^2
p'(t) = (0.3051e^(-0.339*t)) / (1 + 3.5e^(-0.339*t))^2
To find the maximum, set p'(t) = 0 and solve for t:
0 = (0.3051e^(-0.339*t)) / (1 + 3.5e^(-0.339*t))^2
0 = 0.3051e^(-0.339*t)
Since e^(any non-zero value) is never 0, this equation has no real solutions. Therefore, there is no maximum value for p(t) in the given model.
c) To find when 75% of the new personal computers sold at Best Buy will have Intel's latest coprocessor, you need to solve the equation p(t) = 0.75.
0.75 = 0.90/(1 + 3.5e^(-0.339*t))
Rearrange the equation to isolate e^(-0.339*t):
0.75(1 + 3.5e^(-0.339*t)) = 0.90
1 + 3.5e^(-0.339*t) = 0.90/0.75
1 + 3.5e^(-0.339*t) = 1.2
3.5e^(-0.339*t) = 1.2 - 1
3.5e^(-0.339*t) = 0.2
e^(-0.339*t) = 0.2/3.5
e^(-0.339*t) ≈ 0.0571
Take the natural logarithm (ln) of both sides to solve for t:
ln(e^(-0.339*t)) = ln(0.0571)
-0.339*t = ln(0.0571)
Solve for t:
t ≈ ln(0.0571)/-0.339
Using a calculator, t ≈ 8.35
Therefore, approximately after 8.35 months, 75% of the new personal computers sold at Best Buy will have Intel's latest coprocessor.